sjamaar revised 2015

Manifolds and Differential Forms Reyer Sjamaar D/e.sc/p.sc/a.sc/r.sc/t.sc/m.sc/e.sc/n.sc/t.sc /o.sc/f.sc M/a.sc/t.sc/h.sc/e.sc/m.sc/a.sc/t.sc/i.sc/c.sc/s.sc, C/o.sc/r.sc/n.sc/e.sc/l.sc/l.sc U/n.sc/i.sc/v.sc/e.sc/r.sc/s.sc/i.sc/t.sc/y.sc, I/t.sc/h.sc/a.sc/c.sc/a.sc, N/e.sc /w.sc Y/o.sc/r.sc/k.sc /one.taboldstyle/four.taboldstyle/eight.taboldstyle/five.taboldstyle/three.taboldstyle- /four.taboldstyle/two.taboldstyle/zero.taboldstyle/one.taboldstyle E-mail address :[email protected]/l.Var/l.Var.edu URL:http://www.math.corne/l.Var/l.Var.edu/~sjamaar Revised edition, /two.taboldstyle/zero.taboldstyle/one.taboldstyle/five.taboldstyle Copyright ©Reyer Sjamaar, /two.taboldstyle/zero.taboldstyle/zero.taboldstyle/one.taboldstyle, /two.taboldstyle/zero.taboldstyle/one.taboldstyle/five.taboldstyle. Paper or electronic copies for pe rsonal use may be made without explicit permission from the author. All other rights reserved. Contents Preface v Chapter /one.taboldstyle. Introduction /one.taboldstyle /one.taboldstyle./one.taboldstyle. Manifolds /one.taboldstyle /one.taboldstyle./two.taboldstyle. Equations /seven.taboldstyle /one.taboldstyle./three.taboldstyle. Parametrizations /nine.taboldstyle /one.taboldstyle./four.taboldstyle. Configuration spaces /one.taboldstyle/zero.taboldstyle Exercises /one.taboldstyle/four.taboldstyle Chapter /two.taboldstyle. Differential forms on Euclidean space /one.taboldstyle/seven.taboldstyle /two.taboldstyle./one.taboldstyle. Elementary properties /one.taboldstyle/seven.taboldstyle /two.taboldstyle./two.taboldstyle. The exterior derivative /two.taboldstyle/zero.taboldstyle /two.taboldstyle./three.taboldstyle. Closed and exact forms /two.taboldstyle/two.taboldstyle /two.taboldstyle./four.taboldstyle. The Hodge star operator /two.taboldstyle/four.taboldstyle /two.taboldstyle./five.taboldstyle. div, grad and curl /two.taboldstyle/five.taboldstyle Exercises /two.taboldstyle/seven.taboldstyle Chapter /three.taboldstyle. Pulling back forms /three.taboldstyle/one.taboldstyle /three.taboldstyle./one.taboldstyle. Determinants /three.taboldstyle/one.taboldstyle /three.taboldstyle./two.taboldstyle. Pulling back forms /three.taboldstyle/eight.taboldstyle Exercises /four.taboldstyle/five.taboldstyle Chapter /four.taboldstyle. Integration of 1-forms /four.taboldstyle/nine.taboldstyle /four.taboldstyle./one.taboldstyle. Definition and elementary properties of the integral /four.taboldstyle/nine.taboldstyle /four.taboldstyle./two.taboldstyle. Integration of exact 1-forms /five.taboldstyle/one.taboldstyle /four.taboldstyle./three.taboldstyle. Angle functions and the winding number /five.taboldstyle/four.taboldstyle Exercises /five.taboldstyle/eight.taboldstyle Chapter /five.taboldstyle. Integration and Stokes’ theorem /six.taboldstyle/three.taboldstyle /five.taboldstyle./one.taboldstyle. Integration of forms over chains /six.taboldstyle/three.taboldstyle /five.taboldstyle./two.taboldstyle. The boundary of a chain /six.taboldstyle/six.taboldstyle /five.taboldstyle./three.taboldstyle. Cycles and boundaries /six.taboldstyle/eight.taboldstyle /five.taboldstyle./four.taboldstyle. Stokes’ theorem /seven.taboldstyle/zero.taboldstyle Exercises /seven.taboldstyle/one.taboldstyle Chapter /six.taboldstyle. Manifolds /seven.taboldstyle/three.taboldstyle /six.taboldstyle./one.taboldstyle. The definition /seven.taboldstyle/three.taboldstyle /six.taboldstyle./two.taboldstyle. The regular value theorem /eight.taboldstyle/zero.taboldstyle Exercises /eight.taboldstyle/six.taboldstyle Chapter /seven.taboldstyle. Differential forms on manifolds /eight.taboldstyle/nine.taboldstyle iii iv CONTENTS /seven.taboldstyle./one.taboldstyle. First definition /eight.taboldstyle/nine.taboldstyle /seven.taboldstyle./two.taboldstyle. Second definition /nine.taboldstyle/zero.taboldstyle Exercises /nine.taboldstyle/seven.taboldstyle Chapter /eight.taboldstyle. Volume forms /nine.taboldstyle/nine.taboldstyle /eight.taboldstyle./one.taboldstyle. n-Dimensional volume in RN/nine.taboldstyle/nine.taboldstyle /eight.taboldstyle./two.taboldstyle. Orientations /one.taboldstyle/zero.taboldstyle/two.taboldstyle /eight.taboldstyle./three.taboldstyle. Volume forms /one.taboldstyle/zero.taboldstyle/five.taboldstyle Exercises /one.taboldstyle/zero.taboldstyle/nine.taboldstyle Chapter /nine.taboldstyle. Integration and Stokes’ theorem for manifolds /one.taboldstyle/one.taboldstyle/one.taboldstyle /nine.taboldstyle./one.taboldstyle. Manifolds with boundary /one.taboldstyle/one.taboldstyle/one.taboldstyle /nine.taboldstyle./two.taboldstyle. Integration over orientable manifolds /one.taboldstyle/one.taboldstyle/five.taboldstyle /nine.taboldstyle./three.taboldstyle. Gauss and Stokes /one.taboldstyle/one.taboldstyle/eight.taboldstyle Exercises /one.taboldstyle/one.taboldstyle/nine.taboldstyle Chapter /one.taboldstyle/zero.taboldstyle. Applications to topology /one.taboldstyle/two.taboldstyle/three.taboldstyle /one.taboldstyle/zero.taboldstyle./one.taboldstyle. Brouwer’s fixed point theorem /one.taboldstyle/two.taboldstyle/three.taboldstyle /one.taboldstyle/zero.taboldstyle./two.taboldstyle. Homotopy /one.taboldstyle/two.taboldstyle/four.taboldstyle /one.taboldstyle/zero.taboldstyle./three.taboldstyle. Closed and exact forms re-examined /one.taboldstyle/two.taboldstyle/nine.taboldstyle Exercises /one.taboldstyle/three.taboldstyle/four.taboldstyle Appendix A. Sets and functions /one.taboldstyle/three.taboldstyle/seven.taboldstyle A./one.taboldstyle. Glossary /one.taboldstyle/three.taboldstyle/seven.taboldstyle A./two.taboldstyle. General topology of Euclidean space /one.taboldstyle/three.taboldstyle/nine.taboldstyle Exercises /one.taboldstyle/four.taboldstyle/zero.taboldstyle Appendix B. Calculus review /one.taboldstyle/four.taboldstyle/three.taboldstyle B./one.taboldstyle. The fundamental theorem of calculus /one.taboldstyle/four.taboldstyle/three.taboldstyle B./two.taboldstyle. Derivatives /one.taboldstyle/four.taboldstyle/three.taboldstyle B./three.taboldstyle. The chain rule /one.taboldstyle/four.taboldstyle/six.taboldstyle B./four.taboldstyle. The implicit function theorem /one.taboldstyle/four.taboldstyle/seven.taboldstyle B./five.taboldstyle. The substitution formula for integrals /one.taboldstyle/four.taboldstyle/nine.taboldstyle Exercises /one.taboldstyle/four.taboldstyle/nine.taboldstyle Appendix C. The Greek alphabet /one.taboldstyle/five.taboldstyle/three.taboldstyle Bibliography /one.taboldstyle/five.taboldstyle/five.taboldstyle Notation Index /one.taboldstyle/five.taboldstyle/seven.taboldstyle Index /one.taboldstyle/five.taboldstyle/nine.taboldstyle Preface These are the lecture notes for Math /three.taboldstyle/two.taboldstyle/one.taboldstyle/zero.taboldstyle (formerly named Mat h /three.taboldstyle/two.taboldstyle/one.taboldstyle), Mani- folds and Differential Forms, as taught at Cornell Universit y since the Fall of /two.taboldstyle/zero.taboldstyle/zero.taboldstyle/one.taboldstyle. The course covers manifolds and differential forms for an aud ience of undergrad- uates who have taken a typical calculus sequence at a North Am erican university, including basic linear algebra and multivariable calculus up to the integral theo- rems of Green, Gauss and Stokes. With a view to the fact that ve ctor spaces are nowadays a standard item on the undergraduate menu, the text is not restricted to curves and surfaces in three-dimensional space, but treats manifolds of arbitrary dimension. Some prerequisites are briefly reviewed within t he text and in appen- dices. The selection of material is similar to that in Spivak ’s book [ Spi/seven.taboldstyle/one.taboldstyle ] and in Flanders’ book [ Fla/eight.taboldstyle/nine.taboldstyle ], but the treatment is at a more elementary and informal level appropriate for sophomores and juniors. A large portion of the text consists of problem sets placed at the end of each chapter. The exercises range from easy substitution drills to fairly involved but, I hope, interesting computations, as well as more theoretica l or conceptual problems. More than once the text makes use of results obtained in the ex ercises. Because of its transitional nature between calculus and ana lysis, a text of this kind has to walk a thin line between mathematical informalit y and rigour. I have tended to err on the side of caution by providing fairly detai led definitions and proofs. In class, depending on the aptitudes and preference s of the audience and also on the available time, one can skip over many of the detai ls without too much loss of continuity. At any rate, most of the exercises do not r equire a great deal of formal logical skill and throughout I have tried to minimize the use of point-set topology. These notes, occasionally revised and updated, are availab le at http://www.math.corne/l.Var/l.Var.edu/~sjamaar/manifo/l.Vards/ . Corrections, suggestions and comments sent to [email protected]/l.Var/l.Var.edu will be received gratefully. Ithaca, New York, July /two.taboldstyle/zero.taboldstyle/one.taboldstyle/five.taboldstyle v CHAPTER /one.taboldstyle Introduction We start with an informal, intuitive introduction to manifo lds and how they arise in mathematical nature. Most of this material will be e xamined more thor- oughly in later chapters. /one.taboldstyle./one.taboldstyle. Manifolds Recall that Euclidean n-space Rnis the set of all column vectors with nreal entries x/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAx1 x2 ... xn/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA, which we shall call points orn-vectors and denote by lower case boldface letters. In R2orR3we often write x/equalx/parenleftBigg x y/parenrightBigg , resp. x/equalx/parenlefttpA/parenleftexA /parenleftbtAx y z/parenrighttpA/parenrightexA /parenrightbtA. For reasons having to do with matrix multiplication, column vectors are not to be confused with row vectors (x1x2···xn). Nevertheless, to save space we shall frequently write a column vector xas an n-tuple x/equalx(x1,x2,..., xn) with the entries separated by commas. Amanifold is a certain type of subset of Rn. A precise definition will follow in Chapter /six.taboldstyle, but one important consequence of the definition is that at ea ch of its points a manifold has a well-defined tangent space, which is a linear subspace of Rn. This fact enables us to apply the methods of calculus and lin ear algebra to the study of manifolds. The dimension of a manifold is by definition the dimension of any of its tangent spaces. The dimension of a manifold in Rncan be no higher than n. Dimension /one.taboldstyle. A one-dimensional manifold is, loosely speaking, a curve wi th- out kinks or self-intersections. Instead of the tangent “sp ace” at a point one usually speaks of the tangent line. A curve in R2is called a plane curve and a curve in R3 is aspace curve, but you can have curves in any Rn. Curves can be closed (as in the first picture below), unbounded (as indicated by the ar rows in the second picture), or have one or two endpoints (the third picture sho ws a curve with an endpoint, indicated by a black dot; the white dot at the other end indicates that /one.taboldstyle /two.taboldstyle /one.taboldstyle. INTRODUCTION that point does not belong to the curve; the curve “peters out ” without coming to an endpoint). Endpoints are also called boundary points . A circle with one point deleted is also an example of a manifol d. Think of a torn elastic band. By straightening out the elastic band we see that this manifo ld is really the same as an open interval. The four plane curves below are not manifolds. The teardrop h as a kink, where two distinct tangent lines occur instead of a single well-define d tangent line; the five- fold loop has five points of self-intersection, at each of whi ch there are two distinct tangent lines. The bow tie and the five-pointed star have well -defined tangent lines everywhere. Still they are not manifolds: the bow tie has a se lf-intersection and the cusps of the star have a jagged appearance which is proscribe d by the definition of a manifold (which we have not yet given). The points where the se curves fail to be manifolds are called singularities . The “good” points are called smooth . Singularities can sometimes be “resolved”. For instance, t he self-intersections of the Archimedean spiral, which is given in polar coordinates byris a constant times /one.taboldstyle./one.taboldstyle. MANIFOLDS /three.taboldstyle θ, where ris allowed to be negative, can be removed by uncoiling the spiral and wrapping it around a cone. You can convince yourself that the resulting space curve has no sing ularities by peeking at it along the direction of the x-axis or the y-axis. What you will see are the smooth curves shown in the (y,z)-plane and the (x,z)-plane. e1e2e3 Singularities are very interesting, but in this course we sh all focus on gaining a thorough understanding of the smooth points. Dimension /two.taboldstyle. A two-dimensional manifold is a smooth surface without self - intersections. It may have a boundary,which is always a one- dimensional manifold. You can have two-dimensional manifolds in the plane R2, but they are relatively boring. Examples are: an arbitrary open subset of R2, such as an open square, or /four.taboldstyle /one.taboldstyle. INTRODUCTION a closed subset with a smooth boundary. A closed square is not a manifold, because the corners are not smooth. /one.superior Two-dimensional manifolds in three-dimensional space inc lude a sphere (the sur- face of a ball), a paraboloid and a torus (the surface of a doug hnut). e1e2e3 The famous Möbius band is made by pasting together the two ends of a rectangular strip of paper giving one end a half twist. The boundary of the band consists of two boundary edges of the rectangle tied together and is ther efore a single closed /one.superiorTo be strictly accurate, the closed square is a topological manifold with boundary, but not a smooth manifold with boundary. In these notes we will consider only smooth manifolds. /one.taboldstyle./one.taboldstyle. MANIFOLDS /five.taboldstyle curve. Out of the Möbius band we can create a manifold without bounda ry by closing it up along the boundary edge. This can be done in two different wa ys. According to the direction in which we glue the edge to itself, we obtain theKlein bottle or the projective plane . A simple way to represent these three surfaces is by the foll owing gluing diagrams . The labels tell you which edges to glue together and the arro ws tell you in which direction. a a Möbius banda a bb Klein bottlea a bb projective plane One way to make a model of a Klein bottle is first to paste the top and bottom edges of the square together, which gives a tube, and then to j oin the resulting boundary circles, making sure the arrows match up. You will n otice this cannot be done without passing one end through the wall of the tube. The resulting surface intersects itself along a circle and therefore is not a manif old. A different model of the Klein bottle can be made by folding ove r the edge of a Möbius band until it touches the central circle. This create s a Möbius type band with a figure eight cross-section. Equivalently, take a leng th of tube with a figure eight cross-section and weld the ends together giving one en d a half twist. Again /six.taboldstyle /one.taboldstyle. INTRODUCTION the resulting surface has a self-intersection, namely the c entral circle of the original Möbius band. The self-intersection locus as well as a few of t he cross-sections are shown in black in the following wire mesh model. To represent the Klein bottle without self-intersections y ou need to embed it in four-dimensional space. The projective plane has the same p eculiarity, and it too has self-intersecting models in three-dimensional space. Perhaps the easiest model is constructed by merging the edges aand bshown in the gluing diagram for the projective plane, which gives the following gluing diagram . a a aa First fold the lower right corner over to the upper left corne r and seal the edges. This creates a pouch like a cherry turnover with two seams lab elled awhich meet at a corner. Now fuse the two seams to create a single seam labe lled a. Below is a wire mesh model of the resulting surface. It is obtained by we lding together two pieces along the dashed wires. The lower half shaped like a bo wl corresponds to the dashed circular disc in the middle of the square. The uppe r half corresponds to the complement of the disc and is known as a cross-cap . The wire shown in black corresponds to the edge a. The interior points of the black wire are ordinary self-intersection points. Its two endpoints are qualitati vely different singularities /one.taboldstyle./two.taboldstyle. EQUATIONS /seven.taboldstyle known as pinch points , where the surface is crinkled up. e1e2e3 Cartesian products. The Cartesian product M×Nof two manifolds Mand N may fail to be a manifold. (If you don’t remember what a Cartes ian product is, see Appendix A./one.taboldstylefor a review of set theory.) For instance, if M/equalxN/equalx[0,1], the unit interval, then M×Nis the unit square, which is not a manifold. However, if at lea st one of the two manifolds Mand Nhas no boundary, then M×Nis a manifold. The dimension of M×Nis the sum of the dimensions of Mand N. For instance, ifMis an interval and Na circle, then M×Nis a cylinder wall. If both Mand N are circles, then M×Nis a torus. We can also form Cartesian products of more than two factors: the product of ncopies of a circle with itself is an n-dimensional manifold known as an n-torus . Connected sums. LetMand Nbe2-manifolds. The connected sum is a2- manifold M#Nproduced by punching a circular hole in each of the manifolds Mand Nand then gluing the two boundary circles together. For insta nce, the connected sum of two tori is a pretzel-type surface with two h oles. /one.taboldstyle./two.taboldstyle. Equations Very commonly a manifold Mis given “implicitly”, namely as the solution set of a system φ1(x1,..., xn)/equalxc1, φ2(x1,..., xn)/equalxc2, ... φm(x1,..., xn)/equalxcm, ofmequations in nunknowns. Here φ1,φ2,...,φmare functions, c1,c2,...,cm are constants and x1,x2,...,xnare variables. By introducing the useful shorthand x/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAx1 x2 ... xn/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA, φ (x)/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAφ1(x) φ2(x) ... φm(x)/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA, c/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAc1 c2 ... cn/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA, /eight.taboldstyle /one.taboldstyle. INTRODUCTION we can represent this system as a single equation φ(x)/equalxc. The solution set Mis the set of all vectors xinRnwhich satisfy φ(x)/equalxcand is denoted by φ−1(c). (This notation is standard, but a bit unfortunate because i t suggests falsely that φis invertible, which it is usually not.) Thus M/equalxφ−1(c)/equalx{x∈Rn|φ(x)/equalxc}. It is in general difficult to find explicit solutions of a system of equations. (On the positive side, it is usually easy to decide whether any given point is a solution by plugging it into the equations.) Manifolds defined by linear equations (i.e. where φis a matrix) are called affine subspaces ofRnand are studied in linear algebra. More interesting manifolds arise from nonlinear equations . /one.taboldstyle./one.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. The simplest case is that of a single equation ( m/equalx1), such as x2+y2−z2/equalx0. Here we have a single scalar-valued function of three variab lesφ(x,y,z)/equalxx2+ y2−z2and c/equalx0. The solution set Mof the equation is a cone in R3, which is not a manifold because it has no well-defined tangent plane at the origin. We can determine the tangent plane at any other point of Mby recalling from calculus that the gradient of φis perpendicular to the surface. Hence for any nonzero x/equalx (x,y,z)∈Mthe tangent plane to Matxis the plane perpendicular to grad (φ)(x)/equalx (2x,2y,−2z). As we see from this example, the solution set of a system of equ ations may have singularities and is therefore not necessarily a manif old. In general, if Mis given by a single equation φ(x)/equalxcandxis a point of Mwith the property that grad (φ)(x)/nequal0, then xis a smooth point of Mand the tangent space at xis the orthogonal complement of grad (φ)(x). (Conversely, if xis a singular point of M, we must have grad (φ)(x)/equalx0!) The standard notation for the tangent space to M atxisTxM. Thus we can write TxM/equalxgrad (φ)(x)⊥. /one.taboldstyle./two.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Thesphere of radius rabout the origin in Rnis the set of all xin Rnsatisfying the single equation /bardblx/bardbl/equalxr. Here /bardblx/bardbl/equalx√ x·x/equalx/radicalBig x2 1+x2 2+···+x2 n is the norm orlength ofxand x·y/equalxxTy/equalxx1y1+x2y2+···+xnyn is the inner product ordot product ofxandy. The sphere of radius ris an n−1- dimensional manifold in Rn. The sphere of radius 1is called the unit sphere and is denoted by Sn−1. In particular, the one-dimensional unit “sphere” S1is the unit circle in the plane, and the zero-dimensional unit “sphere” S0is the subset{−1,1} of the real line. To determine the tangent spaces of the unit s phere it is easier to work with the equation /bardblx/bardbl2/equalx1instead of/bardblx/bardbl/equalx1. In other words, we let φ(x)/equalx/bardblx/bardbl2. Then grad (φ)(x)/equalx2x, which is nonzero for all xin the unit sphere. Therefore Sn−1is a manifold and for any xinSn−1we have TxSn−1/equalx(2x)⊥/equalxx⊥/equalx{y∈Rn|y·x/equalx0}, /one.taboldstyle./three.taboldstyle. PARAMETRIZATIONS /nine.taboldstyle a linear subspace of Rn. (In Exercise /one.taboldstyle./seven.taboldstyleyou will be asked to find a basis of the tangent space for a particular xand you will see that TxSn−1isn−1-dimensional.) /one.taboldstyle./three.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Consider the system of two equations in three unknowns, x2+y2/equalx1, y+z/equalx0. Here φ(x)/equalx/parenleftBigg x2+y2 y+z/parenrightBigg and c/equalx/parenleftBigg 1 0/parenrightBigg . The solution set of this system is the intersection of a cylin der of radius 1about thez-axis (given by the first equation) and a plane cutting the x-axis at a 45◦angle (given by the second equation). Hence the solution set is an e llipse. It is a manifold of dimension 1. We will discuss in Chapter /six.taboldstylehow to find the tangent spaces to manifolds given by more than one equation. Inequalities. Manifolds with boundary are often presented as solution set s of a system of equations together with one or more inequaliti es. For instance, the closed ball of radius rabout the origin in Rnis given by the single inequality /bardblx/bardbl≤r. Its boundary is the sphere of radius r. /one.taboldstyle./three.taboldstyle. Parametrizations A dual method for describing manifolds is the “explicit” way , namely by par- ametrizations. For instance, x/equalxcosθ, y/equalxsinθ parametrizes the unit circle in R2and x/equalxcosθcosφ, y/equalxsinθcosφ, z/equalxsinφ parametrizes the unit sphere in R3. (Hereφis the angle between a vector and the(x,y)-plane andθis the polar angle in the (x,y)-plane.) The explicit method has various merits and demerits, which are complementary to those of the im- plicit method. One obvious advantage is that it is easy to find points lying on a parametrized manifold simply by plugging in values for the p arameters. A disad- vantage is that it can be hard to decide if any given point is on the manifold or not, because this involves solving for the parameters. Parametr izations are often harder to come by than a system of equations, but are at times more use ful, for example when one wants to integrate over the manifold. Also, it is usu ally impossible to parametrize a manifold in such a way that every point is cover ed exactly once. Such is the case for the two-sphere. One commonly restricts t he polar coordinates (θ,φ)to the rectangle [0,2π]×[−π/2,π/2]to avoid counting points twice. Only the meridian θ/equalx0is then hit twice, but this does not matter for many purposes, such as computing the surface area or integrating a continuo us function. We will use parametrizations to give a formal definition of th e notion of a manifold in Chapter /six.taboldstyle. Note however that not every parametrization describes a manifold. /one.taboldstyle/zero.taboldstyle /one.taboldstyle. INTRODUCTION /one.taboldstyle./four.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Define c(t)/equalx(t2,t3)fort∈R. As truns through the real line, the point c(t)travels along a curve in the plane, which we call a pathorparametrized curve . The path chas no self-intersections: if t1/nequalt2then c(t1)/nequalc(t2). The cusp at the origin ( t/equalx0) is a singular point, but all other points ( t/nequal0) are smooth. The tangent line at a smooth point c(t)is the line spanned by the velocity vector c′(t)/equalx(2t,3t2). The slope of the tangent line is 3t2/2t/equalx3 2t. For t/equalx0the velocity vector is c′(0)/equalx0, which does not span a line. Nevertheless, the curve has a well -defined (horizontal) tangent line at the origin, which we can think of as the limit o f the tangent lines as ttends to 0. More examples of parametrizations are given in Exercises /one.taboldstyle./one.taboldstyle–/one.taboldstyle./three.taboldstyle. /one.taboldstyle./four.taboldstyle. Configuration spaces Frequently manifolds arise in more abstract ways that may be hard to capture in terms of equations or parametrizations. Examples are sol ution curves of differ- ential equations (see e.g. Exercise /one.taboldstyle./one.taboldstyle/one.taboldstyle) and configuration spaces. The configuration orstate of a mechanical system (such as a pendulum, a spinning top, th e solar system, a fluid, or a gas etc.) is a complete specification of th e position of each of its parts. (The configuration ignores any motions that the syste m may be undergoing. So a configuration is like a snapshot or a movie still. When the system moves, its configuration changes.) The configuration space orstate space of the system is an ab- stract space, the points of which are in one-to-one correspo ndence to all physically possible configurations of the system. Very often the configu ration space turns out to be a manifold. Its dimension is called the number of degrees of freedom of the system. /one.taboldstyle./five.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Aspherical pendulum is a weight or bob attached to a fixed centre by a rigid rod, free to swing in any direction in three-space. The state of the pendulum is entirely determined by the posit ion of the bob. The bob can move from any point at a fixed distance (equal to the len gth of the rod) from the centre to any other. The configuration space is therefore a two-dimensional sphere, and the spherical pendulum has two degrees of freedo m. The configuration space of even a fairly small system can be qu ite complicated. Even if the system is situated in three-dimensional space, i t may have many more /one.taboldstyle./four.taboldstyle. CONFIGURATION SPACES /one.taboldstyle/one.taboldstyle than three degrees of freedom. This is why higher-dimension al manifolds are common in physics and applied mathematics. /one.taboldstyle./six.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Take a spherical pendulum of length rand attach a second one of length sto the moving end of the first by a universal joint. The resulti ng system is adouble spherical pendulum . The state of this system can be specified by a pair of vectors (x,y),xbeing the vector pointing from the centre to the first weight a ndy the vector pointing from the first to the second weight. x y The vector xis constrained to a sphere of radius rabout the centre and yto a sphere of radius sabout the head of x. Aside from this limitation, every pair of vectors can occur (if we suppose the second rod is allowed to swing com pletely freely and move “through” the first rod) and describes a distinct configu ration. Thus there are four degrees of freedom. The configuration space is a four-di mensional manifold, namely the Cartesian product of two two-dimensional sphere s. /one.taboldstyle./seven.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. What is the number of degrees of freedom of a rigid body moving inR3? Select any triple of points A,B,Cin the solid that do not lie on one line. A AB AB C The point Acan move about freely and is determined by three coordinates , and so it has three degrees of freedom. But the position of Aalone does not determine the position of the whole solid. If Ais kept fixed, the point Bcan perform two independent swivelling motions. In other words, it moves on a sphere centred at A, which gives two more degrees of freedom. If Aand Bare both kept fixed, the point Ccan rotate about the axis AB, which gives one further degree of freedom. The positions of A,Band Cdetermine the position of the solid uniquely, so the total number of degrees of freedom is 3 + 2 + 1 /equalx6. Thus the configuration space of a rigid body is a six-dimensional manifold. Let us call thi s manifold Mand try /one.taboldstyle/two.taboldstyle /one.taboldstyle. INTRODUCTION to say something about its shape. Choose an arbitrary refere nce point in space. Inside the manifold Mwe have a subset M0consisting of all configurations which have the point Aplaced at the reference point. Configurations in M0have three fewer degrees of freedom, because only the points Band Ccan move, so M0is a three-dimensional manifold. Every configuration in Mcan be moved to a unique configuration in M0by a unique parallel translation of the solid which moves A to the reference point. In other words, the points of Mcan be viewed as pairs consisting of a point in M0and a vector in R3: the manifold Mis the Cartesian product M0×R3. See Exercise /eight.taboldstyle./six.taboldstylefor more information on M0. /one.taboldstyle./eight.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc (/t.sc/h.sc/e.sc /s.sc/p.sc/a.sc/c.sc/e.sc /o.sc/f.sc /q.sc/u.sc/a.sc/d.sc/r.sc/i.sc/l.sc/a.sc/t.sc/e.sc/r.sc/a.sc/l.sc/s.sc). Consider all quadrilaterals ABCD in the plane with fixed sidelengths a,b,c,d. A BCD abc d (Think of four rigid rods attached by hinges.) What are all th e possibilities? For simplicity let us disregard translations by keeping the firs t edge ABfixed in one place. Edges are allowed to cross each other, so the short edg eBCcan spin full circle about the point B. During this motion the point Dmoves back and forth on a circle of radius dcentred at A. A few possible positions are shown here. AsCmoves all the way around, the point Dreaches its greatest left- or rightward displacement when the edges BCand CDare collinear. Arrangements such as this are used in engines for converting a circular motion to a pumping motion, or vice versa. The position of the “crank” Cwholly determines that of the “rocker” D. This means that the configurations are in one-to-one corres pondence with the points on the circle of radius babout the point B, i.e. the configuration space is a circle. Actually, this is not completely accurate: for every choice ofC, there are two choices Dand D′for the fourth point! They are interchanged by reflection in t he /one.taboldstyle./four.taboldstyle. CONFIGURATION SPACES /one.taboldstyle/three.taboldstyle diagonal AC. A BCD D′ So there is in fact another circle’s worth of possible configu rations. It is not possible to move continuously from the first set of configurations to th e second; in fact they are each other’s mirror images. Thus the configuration space is a disjoint union of two circles. This is an example of a disconnected manifold consisting of two connected components . /one.taboldstyle./nine.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc (/q.sc/u.sc/a.sc/d.sc/r.sc/i.sc/l.sc/a.sc/t.sc/e.sc/r.sc/a.sc/l.sc/s.sc, /c.sc/o.sc/n.sc/t.sc/i.sc/n.sc/u.sc/e.sc/d.sc). Even this is not the full story: it is possible to move from one circle to the other when b+c/equalxa+d(and also when a+b/equalxc+d). A BCD abc d In this case, when BCpoints straight to the left, the quadrilateral collapses to a line segment: /one.taboldstyle/four.taboldstyle /one.taboldstyle. INTRODUCTION and when Cmoves further down, there are two possible directions for Dto go, back up: or further down: This means that when b+c/equalxa+dthe two components of the configuration space are merged at a point. The juncture represents the collapsed quadrilateral. This configuration space is not a manifold, but most configuration spaces occurring in natur e are (and an engineer designing an engine wouldn’t want to use this quadrilateral to make a piston drive a flywheel). More singularities appear in the case of a parall elogram ( a/equalxcand b/equalxd) and in the equilateral case ( a/equalxb/equalxc/equalxd). Exercises Computer software can be helpful with some of the exercises i n these notes. Useful free software packages include Microsoft Mathematics for W indows or Grapher for Mac OS. Packages such as Mathematica or MATHLAB are more powerfu l, but are not free. /one.taboldstyle./one.taboldstyle.The formulas x/equalxt−sint,y/equalx1−cost(t∈R) parametrize a plane curve. Graph this curve. You may use software and turn in computer output. Also include a few tangent lines at judiciously chosen points. (E.g. find all tangent li nes with slope 0,±1, and∞.) To compute tangent lines, recall that the tangent vector at a point (x,y)of the curve has components dx/dtand dy/dt. In your plot, identify all points where the curve is not a manifold. /one.taboldstyle./two.taboldstyle. Same questions as in Exercise /one.taboldstyle./one.taboldstylefor the parametrized curve x/equalx3t/(1 +t3), y/equalx3t2/(1 +t3), where 0<t<∞. /one.taboldstyle./three.taboldstyle. Parametrize in Cartesian coordinates the space curve wrapp ed around the cone shown in Section /one.taboldstyle./one.taboldstyle. /one.taboldstyle./four.taboldstyle.Draw gluing diagrams of the following surfaces. EXERCISES /one.taboldstyle/five.taboldstyle (i) A sphere. (ii) A torus with a hole punched in it. (iii) The connected sum of two tori. /one.taboldstyle./five.taboldstyle.Sketch the surfaces defined by the following gluing diagrams . One of these surfaces cannot be embedded in R3, so use a self-intersection where necessary. aa b b ababd c d a1b1a2b2 a1 b1 a2 b2a1b1a2b2 a1 b1 a2 b2 (There are at least two possible strategies. The first is to pr oceed in stages by gluing the a’s, then the b’s, etc., and trying to identify what you get at each step. The second is to decompose each diagram into a connected sum of simpler diagr ams.) /one.taboldstyle./six.taboldstyle.For the values of nindicated below graph the surface in R3defined by xn/equalxy2z. Determine all the points where the surface does not have a wel l-defined tangent plane. (As a preliminary step, determine the intersection of each surf ace with a general plane parallel to one of the coordinate planes. To investigate the tangent p lanes, write the equation of the surface asφ(x,y,z)/equalx0, whereφ(x,y,z)/equalxxn−y2z, and then find the gradient of φ. You may use output generated by computer.) (i)n/equalx0. (ii)n/equalx1. (iii) n/equalx2. (iv) n/equalx3. /one.taboldstyle./seven.taboldstyle. LetMbe the sphere of radius√nabout the origin in Rnand let xbe the point (1,1,..., 1)onM. Find a basis of the tangent space to Matx. (Use that TxMis the set of allysuch that y·x/equalx0. View this equation as a homogeneous linear equation in the e ntries y1,y2,...,ynofyand find the general solution by means of linear algebra.) /one.taboldstyle./eight.taboldstyle. What is the number of degrees of freedom of a bicycle? (Imagin e that it moves freely through empty space and is not constrained to the surf ace of the earth.) /one.taboldstyle./nine.taboldstyle.Choose two distinct positive real numbers aand b. What is the configuration space of all quadrilaterals ABCD such that ABand CDhave length aand BCand ADhave length b? (These quadrilaterals include all parallelograms with si desaand b.) What happens if a/equalxb? (As in Examples /one.taboldstyle./eight.taboldstyleand/one.taboldstyle./nine.taboldstyleassume that the edge ABis kept fixed in place so as to rule out translations.) /one.taboldstyle./one.taboldstyle/zero.taboldstyle. What is the configuration space of all pentagons ABCDE in the plane with fixed sidelengths a,b,c,d,e? (As in the case of quadrilaterals, for certain choices of si delengths /one.taboldstyle/six.taboldstyle /one.taboldstyle. INTRODUCTION singularities may occur. You may ignore these cases. To redu ce the number of degrees of freedom you may also assume the edge ABto be fixed in place.) A BCDE abcd e /one.taboldstyle./one.taboldstyle/one.taboldstyle. The Lotka-Volterra system is an early (ca. /one.taboldstyle/nine.taboldstyle/two.taboldstyle/five.taboldstyle) predator-p rey model. It is the pair of differential equations dx dt/equalx−rx+sxy, dy dt/equalxpy−qxy, where x(t)represents the number of prey and y(t)the number of predators at time t, while p,q,r,sare positive constants. In this problem we will consider the solution curves (also called trajectories) (x(t),y(t))of this system that are contained in the positive quadrant (x>0,y>0) and derive an implicit equation satisfied by these solution curves. (The Lotka-Volterra system is exceptional in this regard. Usual ly it is impossible to write down an equation for the solution curves of a differential equatio n.) (i) Show that the solutions of the system satisfy a single diff erential equation of the form dy/dx/equalxf(x)g(y), where f(x)is a function that depends only on xand g(y)a function that depends only on y. (ii) Solve the differential equation of part ( i) by separating the variables, i.e. by writing 1 g(y)dy/equalxf(x)dxand integrating both sides. (Don’t forget the integration constant.) (iii) Set p/equalxq/equalxr/equalxs/equalx1and plot a number of solution curves. Indicate the direction in which the solutions move. You may use computer software. A n online phase portrait generator can be found at http://www.on/l.Varinesciencetoo/l.Vars.com/too/l.Vars/phaseportra it. CHAPTER /two.taboldstyle Differential forms on Euclidean space The notion of a differential form encompasses such ideas as el ements of surface area, volume elements, the work exerted by a force, the flow of a fluid, and the curvature of a surface, space or hyperspace. An important op eration on differen- tial forms is exterior differentiation, which generalizes t he operators div, grad and curl of vector calculus. The study of differential forms, whi ch was initiated by É. Cartan in the years around /one.taboldstyle/nine.taboldstyle/zero.taboldstyle/zero.taboldstyle, is often termed the exterior differential calculus . A mathematically rigorous study of differential forms requi res the machinery of multilinear algebra, which is examined in Chapter /seven.taboldstyle. Fortunately, it is entirely pos- sible to acquire a solid working knowledge of differential fo rms without entering into this formalism. That is the objective of this chapter. /two.taboldstyle./one.taboldstyle. Elementary properties Adifferential form of degree kor a k-form onRnis an expression α/equalx/summationdisplay IfIdxI. (If you don’t know the symbol α, look up and memorize the Greek alphabet, Appendix C.) Here Istands for a multi-index (i1,i2,..., ik)ofdegree k, that is a “vector” consisting of kinteger entries ranging between 1and n. The fIare smooth functions on Rncalled the coefficients ofα, and dxIis an abbreviation for dxi1dxi2···dxik. (Instead of dxi1dxi2···dxikthe notation dxi1∧dxi2∧···∧ dxikis used by some authors to distinguish this kind of product from another kin d, called the tensor product.) For instance the expressions α/equalxsin(x1+ex4)dx1dx5+x2x2 5dx2dx3+ 6dx2dx4+ cos x2dx5dx3, β/equalxx1x3x5dx1dx6dx3dx2, represent a 2-form on R5, resp. a 4-form on R6. The formαconsists of four terms, corresponding to the multi-indices (1,5),(2,3),(2,4)and(5,3), whereasβconsists of one term, corresponding to the multi-index (1,6,3,2). Note, however, that αcould equally well be regarded as a 2-form on R6that does not involve the variable x6. To avoid such ambiguities it is good practice to state explicitly the domain of definition when writing a diffe rential form. Another reason for being precise about the domain of a form is that the coef- ficients fImay not be defined on all of Rn, but only on an open subset UofRn. In such a case we say αis ak-form on U. Thus the expression ln(x2+y2)z dzis /one.taboldstyle/seven.taboldstyle /one.taboldstyle/eight.taboldstyle /two.taboldstyle. DIFFERENTIAL FORMS ON EUCLIDEAN SPACE not a 1-form on R3, but on the open set U/equalxR3\{(x,y,z)|x2+y2/nequal0}, i.e. the complement of the z-axis. You can think of dxias an infinitesimal increment in the variable xiand of dxI as the volume of an infinitesimal k-dimensional rectangular block with sides dxi1, dxi2,...,dxik. (A precise definition will follow in Section /seven.taboldstyle./two.taboldstyle.) By volume we here mean oriented volume, which takes into account the order of the variables. Thus, if we interchange two variables, the sign changes: dxi1dxi2···dxiq···dxip···dxik/equalx−dxi1dxi2···dxip···dxiq···dxik, (/two.taboldstyle./one.taboldstyle) and so forth. This is called anticommutativity , orgraded commutativity , or the alter- nating property . In particular, this rule implies dxidxi/equalx−dxidxi, sodxidxi/equalx0for alli. Let us consider k-forms for some special values of k. A0-form on Rnis simply a smooth function (no dx’s). A general 1-form looks like f1dx1+f2dx2+···+fndxn. A general 2-form has the shape /summationdisplay i,jfi,jdxidxj/equalxf1,1dx1dx1+f1,2dx1dx2+···+f1,ndx1dxn +f2,1dx2dx1+f2,2dx2dx2+···+f2,ndx2dxn+··· +fn,1dxndx1+fn,2dxndx2+···+fn,ndxndxn. Because of the alternating property ( /two.taboldstyle./one.taboldstyle) the terms fi,idxidxivanish, and a pair of terms such as f1,2dx1dx2and f2,1dx2dx1can be grouped together: f1,2dx1dx2+ f2,1dx2dx1/equalx(f1,2−f2,1)dx1dx2. So we can write any 2-form as /summationdisplay 1≤i<j≤ngi,jdxidxj/equalxg1,2dx1dx2+···+g1,ndx1dxn +g2,3dx2dx3+···+g2,ndx2dxn+···+gn−1,ndxn−1dxn. Written like this, a 2-form has at most n−1 +n−2 +···+ 2 + 1/equalx1 2n(n−1) components. Likewise, a general n−1-form can be written as a sum of ncomponents, f1dx2dx3···dxn+f2dx1dx3···dxn+···+fndx1dx2···dxn−1 /equalxn/summationdisplay i/equalx1fidx1dx2···/hatwiderdxi···dxn, where /hatwiderdximeans “omit the factor dxi”. Every n-form on Rncan be written as f dx 1dx2···dxn. The special n-form dx1dx2···dxnis also known as the volume form . Forms of degree k>nonRnare always 0, because at least one variable has to repeat in any expression dxi1···dxik. By convention, forms of negative degree are 0. /two.taboldstyle./one.taboldstyle. ELEMENTARY PROPERTIES /one.taboldstyle/nine.taboldstyle In general a form of degree kcan be expressed as a sum α/equalx/summationdisplay IfIdxI, where the Iareincreasing multi-indices, 1≤i1<i2<···<ik≤n. We shall almost always represent forms in this manner. The maximum number of terms occurring inαis then the number of increasing multi-indices of degree k. An increasing multi-index of degree kamounts to a choice of knumbers among the numbers 1,2,...,n. The total number of increasing multi-indices of degree kis therefore equal to the binomial coefficient “ nchoose k”, /parenleftBigg n k/parenrightBigg /equalxn! k!(n−k)!. (Compare this to the number of allmulti-indices of degree k, which is nk.) Two k-formsα/equalx/summationtext IfIdxIandβ/equalx/summationtext IgIdxI(with Iranging over the increasing multi- indices of degree k) are considered equal if and only if fI/equalxgIfor all I. The collection of all k-forms on an open set Uis denoted byΩk(U). Since k-forms can be added together and multiplied by scalars, the collection Ωk(U)constitutes a vector space. A form is constant if the coefficients fIare constant functions. The set of constant k-forms is a linear subspace of Ωk(U)of dimension/parenleftbign k/parenrightbig. A basis of this subspace is given by the forms dxI, where Iranges over all increasing multi-indices of degree k. (The spaceΩk(U)itself is infinite-dimensional.) The(exterior) product of a k-formα/equalx/summationtext IfIdxIand an l-formβ/equalx/summationtext JgJdxJis defined to be the k+l-form αβ/equalx/summationdisplay I,JfIgJdxIdxJ. Usually many terms in a product cancel out or can be combined. For instance, (y dx+x dy)(x dx dz +y dy dz )/equalxy2dx dy dz +x2dy dx dz /equalx(y2−x2)dx dy dz. As an extreme example of such a cancellation, consider an arb itrary form αof degree k. Its p-th powerαpis of degree kp, which is greater than nifk>0and p>n. Therefore αn+1/equalx0 for any form αonRnof positive degree. The alternating property combines with the multiplication rule to give the following result. /two.taboldstyle./one.taboldstyle. P/r.sc/o.sc/p.sc/o.sc/s.sc/i.sc/t.sc/i.sc/o.sc/n.sc (/g.sc/r.sc/a.sc/d.sc/e.sc/d.sc /c.sc/o.sc/m.sc/m.sc/u.sc/t.sc/a.sc/t.sc/i.sc/v.sc/i.sc/t.sc/y.sc). βα/equalx(−1)klαβ for all k-formsαand all l-formsβ. /two.taboldstyle/zero.taboldstyle /two.taboldstyle. DIFFERENTIAL FORMS ON EUCLIDEAN SPACE P/r.sc/o.sc/o.sc/f.sc. LetI/equalx(i1,i2,..., ik)and J/equalx(j1,j2,..., jl). Successively applying the alternating property we get dxIdxJ/equalxdxi1dxi2···dxikdxj1dxj2dxj3···dxjl /equalx(−1)kdxj1dxi1dxi2···dxikdxj2dxj3···dxjl /equalx(−1)2kdxj1dxj2dxi1dxi2···dxikdxj3···dxjl ... /equalx(−1)kldxJdxI. For general forms α/equalx/summationtext IfIdxIandβ/equalx/summationtext JgJdxJwe get from this βα/equalx/summationdisplay I,JgJfIdxJdxI/equalx(−1)kl/summationdisplay I,JfIgJdxIdxJ/equalx(−1)klαβ, which establishes the result. QED A noteworthy special case is α/equalxβ. Then we get α2/equalx(−1)k2α2/equalx(−1)kα2. This equality is vacuous if kis even, but tells us that α2/equalx0ifkis odd. /two.taboldstyle./two.taboldstyle. C/o.sc/r.sc/o.sc/l.sc/l.sc/a.sc/r.sc/y.sc. α2/equalx0ifαis a form of odd degree. /two.taboldstyle./two.taboldstyle. The exterior derivative Iffis a0-form, that is a smooth function, we define d fto be the 1-form d f/equalxn/summationdisplay i/equalx1∂f ∂xidxi. Then we have the product orLeibniz rule : d(f g)/equalxf dg+g d f. Ifα/equalx/summationtext IfIdxIis ak-form, each of the coefficients fIis a smooth function and we define dαto be the k+ 1-form dα/equalx/summationdisplay Id fIdxI. The operation dis called exterior differentiation . An operator of this sort is called a first-order partial differential operator, because it invol ves the first partial deriva- tives of the coefficients of a form. /two.taboldstyle./two.taboldstyle. THE EXTERIOR DERIVATIVE /two.taboldstyle/one.taboldstyle /two.taboldstyle./three.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Ifα/equalx/summationtextn i/equalx1fidxiis a1-form on Rn, then dα/equalxn/summationdisplay i/equalx1d fidxi/equalxn/summationdisplay i,j/equalx1∂fi ∂xjdxjdxi /equalx/summationdisplay 1≤i<j≤n∂fi ∂xjdxjdxi+/summationdisplay 1≤j<i≤n∂fi ∂xjdxjdxi /equalx−/summationdisplay 1≤i<j≤n∂fi ∂xjdxidxj+/summationdisplay 1≤i<j≤n∂fj ∂xidxidxj (/two.taboldstyle./two.taboldstyle) /equalx/summationdisplay 1≤i<j≤n/parenleftbigg∂fj ∂xi−∂fi ∂xj/parenrightbigg dxidxj, where in line ( /two.taboldstyle./two.taboldstyle) in the first sum we used the alternating property and in the second sum we interchanged the roles of iand j. /two.taboldstyle./four.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Ifα/equalx/summationtext 1≤i<j≤nfi,jdxidxjis a2-form on Rn, then dα/equalx/summationdisplay 1≤i<j≤nd fi,jdxidxj/equalx/summationdisplay 1≤i<j≤nn/summationdisplay k/equalx1∂fi,j ∂xkdxkdxidxj /equalx/summationdisplay 1≤k<i<j≤n∂fi,j ∂xkdxkdxidxj+/summationdisplay 1≤i<k<j≤n∂fi,j ∂xkdxkdxidxj +/summationdisplay 1≤i<j<k≤n∂fi,j ∂xkdxkdxidxj /equalx/summationdisplay 1≤i<j<k≤n∂fj,k ∂xidxidxjdxk+/summationdisplay 1≤i<j<k≤n∂fi,k ∂xjdxjdxidxk +/summationdisplay 1≤i<j<k≤n∂fi,j ∂xkdxkdxidxj (/two.taboldstyle./three.taboldstyle) /equalx/summationdisplay 1≤i<j<k≤n/parenleftbigg∂fi,j ∂xk−∂fi,k ∂xj+∂fj,k ∂xi/parenrightbigg dxidxjdxk. (/two.taboldstyle./four.taboldstyle) Here in line ( /two.taboldstyle./three.taboldstyle) we rearranged the subscripts (for instance, in the first ter m we relabelled k−→i,i−→ jand j−→k) and in line ( /two.taboldstyle./four.taboldstyle) we applied the alternating property. An obvious but quite useful remark is that if αis an n-form on Rn, then dαis of degree n+ 1and so dα/equalx0. The operator dis linear and satisfies a generalized Leibniz rule. /two.taboldstyle./five.taboldstyle. P/r.sc/o.sc/p.sc/o.sc/s.sc/i.sc/t.sc/i.sc/o.sc/n.sc. (i)d(aα+bβ)/equalxa dα+b dβfor all k-formsαandβand all scalars aandb. (ii)d(αβ)/equalx(dα)β+(−1)kαdβfor all k-formsαandl-formsβ. P/r.sc/o.sc/o.sc/f.sc. The linearity property ( i) follows from the linearity of partial differen- tiation: ∂(a f+bg) ∂xi/equalxa∂f ∂xi+b∂g ∂xi /two.taboldstyle/two.taboldstyle /two.taboldstyle. DIFFERENTIAL FORMS ON EUCLIDEAN SPACE for al smooth functions f,gand constants a,b. Now letα/equalx/summationtext IfIdxIandβ/equalx/summationtext JgJdxJ. The Leibniz rule for functions and Proposition /two.taboldstyle./one.taboldstylegive d(αβ)/equalx/summationdisplay I,Jd(fIgJ)dxIdxJ/equalx/summationdisplay I,J(fIdgJ+gJd fI)dxIdxJ /equalx/summationdisplay I,J/parenleftbigd fIdxI(gJdxJ)+(−1)kfIdxI(dgJdxJ)/parenrightbig /equalx(dα)β+(−1)kαdβ, which proves part ( ii). QED Here is one of the most curious properties of the exterior der ivative. /two.taboldstyle./six.taboldstyle. P/r.sc/o.sc/p.sc/o.sc/s.sc/i.sc/t.sc/i.sc/o.sc/n.sc. d(dα)/equalx0for any form α. In short, d2/equalx0. P/r.sc/o.sc/o.sc/f.sc. Letα/equalx/summationtext IfIdxI. Then d(dα)/equalxd/parenleftbigg/summationdisplay In/summationdisplay i/equalx1∂fI ∂xidxidxI/parenrightbigg /equalx/summationdisplay In/summationdisplay i/equalx1d/parenleftbigg∂fI ∂xi/parenrightbigg dxidxI. Applying the formula of Example /two.taboldstyle./three.taboldstyle(replacing fiwith∂fI/∂xi) we find n/summationdisplay i/equalx1d/parenleftbigg∂fI ∂xi/parenrightbigg dxi/equalx/summationdisplay 1≤i<j≤n/parenleftbigg∂2fI ∂xi∂xj−∂2fI ∂xj∂xi/parenrightbigg dxidxj/equalx0, because for any smooth (indeed, C2) function fthe mixed partials ∂2f/∂xi∂xjand ∂2f/∂xj∂xiare equal. Hence d(dα)/equalx0. QED /two.taboldstyle./three.taboldstyle. Closed and exact forms A formαisclosed ifdα/equalx0. It is exact ifα/equalxdβfor some form β(of degree one less). /two.taboldstyle./seven.taboldstyle. P/r.sc/o.sc/p.sc/o.sc/s.sc/i.sc/t.sc/i.sc/o.sc/n.sc. Every exact form is closed. P/r.sc/o.sc/o.sc/f.sc. Ifα/equalxdβthen dα/equalxd(dβ)/equalx0by Proposition /two.taboldstyle./six.taboldstyle. QED /two.taboldstyle./eight.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc.−y dx+x dy is not closed and therefore cannot be exact. On the other hand y dx+x dyis closed. It is also exact, because d(x y)/equalxy dx+x dy. For a 0-form (function) fonRnto be closed all its partial derivatives must vanish, which means it is constant. A nonzero constant function is not exac t, because forms of degree−1are0. Is every closed form of positive degree exact? This question has interesting ramifications, which we shall explore in Chapters /four.taboldstyle,/five.taboldstyleand/one.taboldstyle/zero.taboldstyle. Amazingly, the answer depends strongly on the topology, that is the qualita tive “shape”, of the domain of definition of the form. Let us consider the simplest case of a 1-formα/equalx/summationtextn i/equalx1fidxi. Determining whetherαis exact means solving the equation dg/equalxαfor the function g. This amounts to ∂g ∂x1/equalxf1,∂g ∂x2/equalxf2, ...,∂g ∂xn/equalxfn, (/two.taboldstyle./five.taboldstyle) /two.taboldstyle./three.taboldstyle. CLOSED AND EXACT FORMS /two.taboldstyle/three.taboldstyle a system of first-order partial differential equations . Finding a solution is sometimes called integrating the system. By Proposition /two.taboldstyle./seven.taboldstylethis is not possible unless αis closed. By the formula in Example /two.taboldstyle./three.taboldstyleαis closed if and only if ∂fi ∂xj/equalx∂fj ∂xi for all 1≤i<j≤n. These identities must be satisfied for the system ( /two.taboldstyle./five.taboldstyle) to be solvable and are therefore called the integrability conditions for the system. /two.taboldstyle./nine.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Letα/equalxy dx+(zcosyz+x)dy+ycosyz dz . Then dα/equalxdy dx +/parenleftbigz(−ysinyz)+ cos yz/parenrightbigdz dy +dx dy +/parenleftbigy(−zsinyz)+ cos yz/parenrightbigdy dz/equalx0, soαis closed. Is αexact? Let us solve the equations ∂g ∂x/equalxy,∂g ∂y/equalxzcosyz+x,∂g ∂z/equalxycosyz by successive integration. The first equation gives g/equalxyx+c(y,z), where cis a function of yand zonly. Substituting into the second equation gives ∂c/∂y/equalx zcosyz, soc/equalxsinyz+k(z). Substituting into the third equation gives k′/equalx0, sok is a constant. So g/equalxx y+ sin yzis a solution and therefore αis exact. This method works always for a 1-form defined on all of Rn. (See Exercise /two.taboldstyle./eight.taboldstyle.) Hence every closed 1-form on Rnis exact. /two.taboldstyle./one.taboldstyle/zero.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. The1-form on the punctured plane R2\{0}defined by α0/equalx−y x2+y2dx+x x2+y2dy/equalx−y dx+x dy x2+y2. is called the angle form for reasons that will become clear in Section /four.taboldstyle./three.taboldstyle. From ∂ ∂xx x2+y2/equalxy2−x2 (x2+y2)2,∂ ∂yy x2+y2/equalxx2−y2 (x2+y2)2 it follows that the angle form is closed. This example is cont inued in Examples /three.taboldstyle./eight.taboldstyle, /four.taboldstyle./one.taboldstyleand/four.taboldstyle./six.taboldstyle, where we shall see that this form is notexact. For a 2-formα/equalx/summationtext 1≤i<j≤nfi,jdxidxjand a 1-formβ/equalx/summationtextn i/equalx1gidxithe equation dβ/equalxαamounts to the system ∂gj ∂xi−∂gi ∂xj/equalxfi,j. (/two.taboldstyle./six.taboldstyle) By the formula in Example /two.taboldstyle./four.taboldstylethe integrability condition dα/equalx0comes down to ∂fi,j ∂xk−∂fi,k ∂xj+∂fj,k ∂xi/equalx0 for all 1≤i<j<k≤n. We shall learn how to solve the system ( /two.taboldstyle./six.taboldstyle), and its higher-degree analogues, in Example /one.taboldstyle/zero.taboldstyle./one.taboldstyle/eight.taboldstyle . /two.taboldstyle/four.taboldstyle /two.taboldstyle. DIFFERENTIAL FORMS ON EUCLIDEAN SPACE /two.taboldstyle./four.taboldstyle. The Hodge star operator The binomial coefficient/parenleftbign k/parenrightbigis the number of ways of selecting k(unordered) objects from a collection of nobjects. Equivalently,/parenleftbign k/parenrightbigis the number of ways of partitioning a pile of nobjects into a pile of kobjects and a pile of n−kobjects. Thus we see that/parenleftBigg n k/parenrightBigg /equalx/parenleftBigg n n−k/parenrightBigg . This means that in a certain sense there are as many k-forms as n−k-forms. In fact, there is a natural way to turn k-forms into n−k-forms. This is the Hodge star operator . Hodge star of αis denoted by∗α(or sometimes α∗) and is defined as follows. Ifα/equalx/summationtext IfIdxI, then ∗α/equalx/summationdisplay IfI(∗dxI), with ∗dxI/equalxεIdxIc. Here, for any increasing multi-index I,Icdenotes the complementary increasing multi-index, which consists of all numbers between 1and nthat do not occur in I. The factorεIis a sign, εI/equalx1ifdxIdxIc/equalxdx1dx2···dxn, −1ifdxIdxIc/equalx−dx1dx2···dxn. In other words,∗dxIis the product of all the dxj’s that do not occur in dxI, times a factor±1which is chosen in such a way that dxI(∗dxI)is the volume form: dxI(∗dxI)/equalxdx1dx2···dxn. /two.taboldstyle./one.taboldstyle/one.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Letn/equalx6and I/equalx(2,6). Then Ic/equalx(1,3,4,5), sodxI/equalxdx2dx6 and dxIc/equalxdx1dx3dx4dx5. Therefore dxIdxIc/equalxdx2dx6dx1dx3dx4dx5 /equalxdx1dx2dx6dx3dx4dx5/equalx−dx1dx2dx3dx4dx5dx6, which shows that εI/equalx−1. Hence∗(dx2dx6)/equalx−dx1dx3dx4dx5. /two.taboldstyle./one.taboldstyle/two.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. OnR2we have∗dx/equalxdyand∗dy/equalx−dx. On R3we have ∗dx/equalxdy dz, ∗(dx dy )/equalxdz, ∗dy/equalx−dx dz/equalxdz dx,∗(dx dz )/equalx−dy, ∗dz/equalxdx dy, ∗(dy dz )/equalxdx. (This is the reason that 2-forms on R3are sometimes written as f dx dy +g dz dx + h dy dz , in contravention of our usual rule to write the variables in increasing order. In higher dimensions it is better to stick to the rule.) On R4we have ∗dx1/equalxdx2dx3dx4,∗dx3/equalxdx1dx2dx4, ∗dx2/equalx−dx1dx3dx4,∗dx4/equalx−dx1dx2dx3, /two.taboldstyle./five.taboldstyle. DIV, GRAD AND CURL /two.taboldstyle/five.taboldstyle and ∗(dx1dx2)/equalxdx3dx4,∗(dx2dx3)/equalxdx1dx4, ∗(dx1dx3)/equalx−dx2dx4,∗(dx2dx4)/equalx−dx1dx3, ∗(dx1dx4)/equalxdx2dx3,∗(dx3dx4)/equalxdx1dx2. OnRnwe have∗1/equalxdx1dx2···dxn,∗(dx1dx2···dxn)/equalx1, and ∗dxi/equalx(−1)i+1dx1dx2···/hatwiderdxi···dxn for1≤i≤n, ∗(dxidxj)/equalx(−1)i+j+1dx1dx2···/hatwiderdxi···/hatwiderdxj···dxn for1≤i<j≤n. /two.taboldstyle./five.taboldstyle. div, grad and curl Avector field on an open subset UofRnis a smooth map F:U→Rn. We can write Fin components as F(x)/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAF1(x) F2(x) ... Fn(x)/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA, or alternatively as F/equalx/summationtextn i/equalx1Fiei, where e1,e2,...,enare the standard basis vectors ofRn. Vector fields in the plane can be plotted by placing the vecto rF(x)with its tail at the point x. The diagrams below represent the vector fields −ye1+xe2and (−x+x y)e1+(y−x y)e2(which you may recognize from Exercise /one.taboldstyle./one.taboldstyle/one.taboldstyle). The arrows have been shortened so as not to clutter the pictures. The bla ck dots are the zeroes of the vector fields (i.e. points xwhere F(x)/equalx0). xy xy We can turn Finto a 1-formαby using the Fias coefficients: α/equalx/summationtextn i/equalx1Fidxi. For instance, the 1-formα/equalx−y dx+x dycorresponds to the vector field F/equalx−ye1+xe2. Let us introduce the symbolic notation dx/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAdx1 dx2 ... dxn/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA, /two.taboldstyle/six.taboldstyle /two.taboldstyle. DIFFERENTIAL FORMS ON EUCLIDEAN SPACE which we will think of as a vector-valued 1-form. Then we can write α/equalxF·dx. Clearly, Fis determined by αand vice versa. Thus vector fields and 1-forms are symbiotically associated to one another. vector field F←→ 1-formα:α/equalxF·dx. Intuitively, the vector-valued 1-form dxrepresents an infinitesimal displacement. IfFrepresents a force field, such as gravity or electricity acti ng on a particle, then α/equalxF·dxrepresents the work done by the force when the particle is displaced by an amount dx. (If the particle travels along a path, the total work done by the force is found by integratingαalong the path. We shall see how to do this in Section /four.taboldstyle./one.taboldstyle.) The correspondence between vector fields and 1-forms behaves in an interest- ing way with respect to exterior differentiation and the Hodg e star operator. For each function fthe1-form d f/equalx/summationtextn i/equalx1(∂f/∂xi)dxiis associated to the vector field grad (f)/equalxn/summationdisplay i/equalx1∂f ∂xiei/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA∂f ∂x1∂f ∂x2... ∂f ∂xn/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA. This vector field is called the gradient off. (Equivalently, we can view grad (f)as the transpose of the Jacobi matrix of f.) grad (f)←→ d f: d f/equalxgrad (f)·dx. Starting with a vector field Fand lettingα/equalxF·dx, we find ∗α/equalxn/summationdisplay i/equalx1Fi(∗dxi)/equalxn/summationdisplay i/equalx1Fi(−1)i+1dx1dx2···/hatwiderdxi···dxn, Using the vector-valued n−1-form ∗dx/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA∗dx1 ∗dx2 ... ∗dxn/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAdx2dx3···dxn −dx1dx3···dxn ... (−1)n+1dx1dx2···dxn−1/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA we can also write∗α/equalxF·∗dx. Intuitively, the vector-valued n−1-form∗dxrepresents an infinitesimal n−1-dimensional hypersurface perpendicular to dx. (This point of view will be justified in Section /eight.taboldstyle./three.taboldstyle, after the proof of Theorem /eight.taboldstyle./one.taboldstyle/six.taboldstyle.) In fluid mechanics, the flow of a fluid or gas in Rnis represented by a vector field F. The n−1-form∗αthen represents the flux, that is the amount of material passing through the hypersurface ∗dxper unit time. (The total amount of fluid passing through a hypersurface Sis found by integratingαover S. We shall see how to do this in Section /five.taboldstyle./one.taboldstyle.) We have d∗α/equalxd(F·∗dx)/equalxn/summationdisplay i/equalx1∂Fi ∂xi(−1)i+1dxidx1dx2···/hatwiderdxi···dxn /equalxn/summationdisplay i/equalx1∂Fi ∂xidx1dx2···dxi···dxn/equalx/parenleftbiggn/summationdisplay i/equalx1∂Fi ∂xi/parenrightbigg dx1dx2···dxn. EXERCISES /two.taboldstyle/seven.taboldstyle The function div(F)/equalx/summationtextn i/equalx1∂Fi/∂xiis the divergence ofF. Thus ifα/equalxF·dx, then d∗α/equalxd(F·∗dx)/equalxdiv(F)dx1dx2···dxn. An alternative way of writing this identity is obtained by ap plying∗to both sides, which gives div(F)/equalx∗d∗α. A very different identity is found by first applying dand then∗toα: dα/equalxn/summationdisplay i,j/equalx1∂Fi ∂xjdxjdxi/equalx/summationdisplay 1≤i<j≤n/parenleftbigg∂Fj ∂xi−∂Fi ∂xj/parenrightbigg dxidxj, and hence ∗dα/equalx/summationdisplay 1≤i<j≤n(−1)i+j+1/parenleftbigg∂Fj ∂xi−∂Fi ∂xj/parenrightbigg dx1dx2···/hatwiderdxi···/hatwiderdxj···dxn. In three dimensions ∗dαis a1-form and so is associated to a vector field, namely curl(F)/equalx/parenleftbigg∂F3 ∂x2−∂F2 ∂x3/parenrightbigg e1−/parenleftbigg∂F3 ∂x1−∂F1 ∂x3/parenrightbigg e2+/parenleftbigg∂F2 ∂x1−∂F1 ∂x2/parenrightbigg e3, thecurlofF. Thus, for n/equalx3, ifα/equalxF·dx, then curl(F)·dx/equalx∗dα. You need not memorize every detail of this discussion. The po int is rather to remember that exterior differentiation in combination with the Hodge star unifies and extends to arbitrary dimensions the classical different ial operators of vector calculus. Exercises /two.taboldstyle./one.taboldstyle. Consider the forms α/equalxx dx−y dy,β/equalxz dx dy +x dy dz andγ/equalxz dy onR3. Calculate (i)αβ,αβγ; (ii)dα,dβ,dγ. /two.taboldstyle./two.taboldstyle.Compute the exterior derivative of the following forms. Rec all that a hat indicates that a term has to be omitted. (i)ex y+z2dx. (ii)/summationtextn i/equalx1x2 idx1···/hatwiderdxi···dxn. /two.taboldstyle./three.taboldstyle.Calculate dsinf(x)2, where f:Rn→Ris an arbitrary smooth function. /two.taboldstyle./four.taboldstyle.Define functions ξandηby ξ(x,y)/equalxx/radicalBig x2+y2, η (x,y)/equalxy/radicalBig x2+y2. Show thatα0/equalx−ηdξ+ξdη, whereα0denotes the angle form defined in Example /two.taboldstyle./one.taboldstyle/zero.taboldstyle. /two.taboldstyle./five.taboldstyle.Write the coordinates on R2nas(x1,y1,x2,y2,..., xn,yn). Let ω/equalxdx1dy1+dx2dy2+···+dxndyn/equalxn/summationdisplay i/equalx1dxidyi. Computeωn/equalxωω···ω(n-fold product). (First work out the cases n/equalx1,2,3.) /two.taboldstyle/eight.taboldstyle /two.taboldstyle. DIFFERENTIAL FORMS ON EUCLIDEAN SPACE /two.taboldstyle./six.taboldstyle.Write the coordinates on R2n+1as(x1,y1,x2,y2,..., xn,yn,z). Let α/equalxdz+x1dy1+x2dy2+···+xndyn/equalxdz+n/summationdisplay i/equalx1xidyi. Computeα(dα)n/equalxα(dαdα···dα). (Use the result of Exercise ( /two.taboldstyle./five.taboldstyle).) /two.taboldstyle./seven.taboldstyle.Check that each of the following forms α∈Ω1(R3)is closed and find a function g such that dg/equalxα. (i)α/equalx(yex y−zsin(xz))dx+(xex y+z2)dy+(−xsin(xz)+ 2yz+ 3z2)dz. (ii)α/equalx2xy3z4dx+(3x2y2z4−zeysin(zey))dy+(4x2y3z3−eysin(zey)+ez)dz. /two.taboldstyle./eight.taboldstyle.Letα/equalx/summationtextn i/equalx1fidxibe aclosed 1-form on Rn. Define a function gby g(x)/equalx/integraldisplayx1 0f1(t,x2,x3,..., xn)dt+/integraldisplayx2 0f2(0,t,x3,x4,..., xn)dt +/integraldisplayx3 0f3(0,0,t,x4,x5,..., xn)dt+···+/integraldisplayxn 0fn(0,0,..., 0,t)dt. Show that dg/equalxα. (Apply the fundamental theorem of calculus, formula ( B./three.taboldstyle), differentiate under the integral sign and don’t forget to use dα/equalx0. If you get confused, first do the case n/equalx2, where g(x)/equalx/integraltextx1 0f1(t,x2)dt+/integraltextx2 0f2(0,t)dt.) /two.taboldstyle./nine.taboldstyle.Letα/equalx/summationtextn i/equalx1fidxibe a closed 1-form whose coefficients fiare smooth functions defined on Rn\{0}that are all homogeneous of the same degree p/nequal−1. Let g(x)/equalx1 p+ 1n/summationdisplay i/equalx1xifi(x). Show that dg/equalxα. (Use dα/equalx0and apply the identity proved in Exercise B./six.taboldstyleto each fi.) /two.taboldstyle./one.taboldstyle/zero.taboldstyle. Letαandβbe closed forms. Prove that αβis also closed. /two.taboldstyle./one.taboldstyle/one.taboldstyle. Letαbe closed and βexact. Prove that αβis exact. /two.taboldstyle./one.taboldstyle/two.taboldstyle. Calculate∗α,∗β,∗γ,∗(αβ), whereα,βandγare as in Exercise /two.taboldstyle./one.taboldstyle. /two.taboldstyle./one.taboldstyle/three.taboldstyle. Letα/equalxx1dx2+x3dx4,β/equalxx1x2dx3dx4+x3x4dx1dx2andγ/equalxx2dx1dx3dx4 be forms on R4. Calculate (i)αβ,αγ; (ii)dβ,dγ; (iii)∗α,∗γ. /two.taboldstyle./one.taboldstyle/four.taboldstyle. Consider the form α/equalx−x2 2dx1+x2 1dx2onR2. (i) Find∗αand∗d∗dα(where∗d∗dαis shorthand for∗(d(∗(dα))).) (ii) Repeat the calculation, regarding αas a form on R3. (iii) Again repeat the calculation, now regarding αas a form on R4. /two.taboldstyle./one.taboldstyle/five.taboldstyle. Prove that∗∗α/equalx(−1)kn+kαfor every k-formαonRn. /two.taboldstyle./one.taboldstyle/six.taboldstyle. Recall that for any increasing multi-index I/equalx(i1,i2,..., ik)the number εI/equalx±1 is determined by the requirement that dxIdxIc/equalxεIdx1dx2...dxn. Let us define|I|/equalxi1+i2+···+ik. Show that εI/equalx(−1)|I|+/parenleftbigk+1 2/parenrightbig . EXERCISES /two.taboldstyle/nine.taboldstyle /two.taboldstyle./one.taboldstyle/seven.taboldstyle. Letα/equalx/summationtext IaIdxIandβ/equalx/summationtext JbJdxJbeconstant k-forms on Rn, i.e. forms with constant coefficients aIand bJ. (We also assume, as usual, that the multi-indices Iand Jare increasing.) The inner product ofαandβis the number defined by (α,β)/equalx/summationdisplay IaIbI. For instance, if α/equalx7dx1dx2+√ 2dx1dx3+ 11 dx2dx3andβ/equalx5dx1dx3−3dx2dx3, then (α,β)/equalx√ 2·5 + 11·(−3)/equalx5√ 2−33. Prove the following assertions. (i) The dxIform an orthonormal basis of the space of constant k-forms. (ii)(α,α)≥0for allαand (α,α)/equalx0if and only if α/equalx0. (iii)α(∗β)/equalx(α,β)dx1dx2···dxn. (iv)α(∗β)/equalxβ(∗α). (v) The Hodge star operator is orthogonal, i.e. (α,β)/equalx(∗α,∗β). /two.taboldstyle./one.taboldstyle/eight.taboldstyle. TheLaplacian∆fof a smooth function fon an open subset of Rnis defined by ∆f/equalx∂2f ∂x2 1+∂2f ∂x2 2+···+∂2f ∂x2n. Prove the following formulas. (i)∆f/equalx∗d∗d f. (ii)∆(f g)/equalx(∆f)g+f∆g+ 2∗(d f(∗dg)). (Use Exercise /two.taboldstyle./one.taboldstyle/seven.taboldstyle(iv).) /two.taboldstyle./one.taboldstyle/nine.taboldstyle. Letα/equalx/summationtextn i/equalx1fidxibe a1-form on Rn. (i) Find formulas for ∗α,d∗α,∗d∗α, and d∗d∗α. (ii) Find formulas for dα,∗dα,d∗dα, and∗d∗dα. (iii) Finally compute d∗d∗α+(−1)n∗d∗dα. Try to write the answer in terms of the Laplace operator ∆defined in Exercise /two.taboldstyle./one.taboldstyle/eight.taboldstyle. /two.taboldstyle./two.taboldstyle/zero.taboldstyle. Letαbe the 1-form/bardblx/bardbl2px·∗dxonRn, where pis a real constant. Compute dα. Show thatαis closed if and only if p/equalx−1 2n. /two.taboldstyle./two.taboldstyle/one.taboldstyle. (i) Let Ube an open subset of Rnand let f:U→Rbe a function satisfying grad (f)(x)/nequal0for all xinU. On Udefine a vector field n, ann−1-formνand a 1-formαby n(x)/equalx/bardblgrad (f)(x)/bardbl−1grad (f)(x), ν/equalxn·∗dx, α/equalx/bardblgrad (f)(x)/bardbl−1d f. Prove that dx1dx2···dxn/equalxανonU. (ii) Let r:Rn→Rbe the function r(x)/equalx/bardblx/bardbl(distance to the origin). Deduce from part ( i) that dx1dx2···dxn/equalx(dr)νonRn\{0}, whereν/equalx/bardblx/bardbl−1x·∗dx. /two.taboldstyle./two.taboldstyle/two.taboldstyle. TheMinkowski orrelativistic inner product on Rn+1is given by (x,y)/equalxn/summationdisplay i/equalx1xiyi−xn+1yn+1. A vector x∈Rn+1isspacelike if(x,x)>0,lightlike if(x,x)/equalx0, and timelike if(x,x)<0. (i) Give examples of (nonzero) vectors of each type. (ii) Show that for every x/nequal0there is a ysuch that (x,y)/nequal0. /three.taboldstyle/zero.taboldstyle /two.taboldstyle. DIFFERENTIAL FORMS ON EUCLIDEAN SPACE A Hodge star operator corresponding to this inner product is defined as follows: if α/equalx/summationtext IfIdxI, then ∗α/equalx/summationdisplay IfI(∗dxI), with ∗dxI/equalxεIdxIcifIcontains n+ 1, −εIdxIcifIdoes not contain n+ 1. (HereεIand Icare as in the definition of the ordinary Hodge star.) (iii) Find∗1,∗dxifor1≤i≤n+ 1, and∗(dx1dx2···dxn). (iv) Compute the “relativistic Laplacian” (usually called the d’Alembertian or wave operator)∗d∗d ffor any smooth function fonRn+1. (v) For n/equalx3(ordinary space-time) find ∗(dxidxj)for1≤i<j≤4. /two.taboldstyle./two.taboldstyle/three.taboldstyle. One of the greatest advances in theoretical physics of the ni neteenth century was Maxwell’s formulation of the equations of electromagnetis m: curl(E)/equalx−1 c∂B ∂t(Faraday’s Law) , curl(H)/equalx4π cJ+1 c∂D ∂t(Ampère’s Law) , div(D)/equalx4πρ (Gauss’ Law) , div(B)/equalx0 (no magnetic monopoles) . Here cis the speed of light, Eis the electric field, His the magnetic field, Jis the density of electric current, ρis the density of electric charge, Bis the magnetic induction and Dis the dielectric displacement. E,H,J,BandDare vector fields and ρis a function on R3and all depend on time t. The Maxwell equations look particularly simple in differen tial form notation, as we shall now see. In space-time R4with coordinates (x1,x2,x3,x4), where x4/equalxct, introduce forms α/equalx(E1dx1+E2dx2+E3dx3)dx4+B1dx2dx3+B2dx3dx1+B3dx1dx2, β/equalx−(H1dx1+H2dx2+H3dx3)dx4+D1dx2dx3+D2dx3dx1+D3dx1dx2, γ/equalx1 c(J1dx2dx3+J2dx3dx1+J3dx1dx2)dx4−ρdx1dx2dx3. (i) Show that Maxwell’s equations are equivalent to dα/equalx0, dβ+ 4πγ/equalx0. (ii) Conclude that γis closed and that div(J)+∂ρ/∂ t/equalx0. (iii) In vacuum one has E/equalxDandH/equalxB. Show that in vacuum β/equalx∗α, the relativistic Hodge star of αdefined in Exercise /two.taboldstyle./two.taboldstyle/two.taboldstyle. (iv)Free space is a vacuum without charges or currents. Show that the Maxwel l equations in free space are equivalent to dα/equalxd∗α/equalx0. (v) Let f,g:R→Rbe any smooth functions and define E(x)/equalx/parenlefttpA/parenleftexA /parenleftbtA0 f(x1−x4) g(x1−x4)/parenrighttpA/parenrightexA /parenrightbtA, B(x)/equalx/parenlefttpA/parenleftexA /parenleftbtA0 −g(x1−x4) f(x1−x4)/parenrighttpA/parenrightexA /parenrightbtA. Show that the corresponding 2-formαsatisfies the free Maxwell equations dα/equalx d∗α/equalx0. Such solutions are called electromagnetic waves . Explain why. In what direction do these waves travel? CHAPTER /three.taboldstyle Pulling back forms /three.taboldstyle./one.taboldstyle. Determinants The determinant of a square matrix is the oriented volume of t he parallelepiped spanned by its column vectors. It is therefore not surprisin g that differential forms are closely related to determinants. This section is a revie w of some fundamental facts concerning determinants. Let A/equalx/parenlefttpA/parenleftexA/parenleftexA /parenleftbtAa1,1... a1,n ...... an,1... an,n/parenrighttpA/parenrightexA/parenrightexA /parenrightbtA be an n×n-matrix with column vectors a1,a2,...,an. The parallelepiped spanned by the columns is by definition the set of all linear combinati ons/summationtextn i/equalx1ciai, where the coefficients cirange over the unit interval [0,1]. A parallelepiped spanned by a single vector is called a line segment and a parallelepiped spanned by two vectors is called a parallelogram . The determinant of Ais variously denoted by det(A)/equalxdet(a1,a2,..., an)/equalxdet(ai,j)1≤i,j≤n/equalx/barex/barex/barex/barex/barex/barex/barex/barexa1,1... a1,n ...... an,1... an,n/barex/barex/barex/barex/barex/barex/barex/barex. Expansion on the j-th column. You may have seen the following definition of the determinant: det(A)/equalxn/summationdisplay i/equalx1(−1)i+1ai,jdet(Ai,j). Here Ai,jdenotes the (n−1)×(n−1)-matrix obtained from Aby crossing out thei-th row and the j-th column. This is a recursive definition, which reduces the calculation of any determinant to that of determinants o f smaller size. The recursion starts at n/equalx1; the determinant of a 1×1-matrix (a)is simply defined to be the number a. It is a useful rule, but it has two serious flaws: first, it is ex tremely inefficient computationally (except for matrices containin g lots of zeroes), and second, it obscures the relationship with volumes of parall elepipeds. Axioms. A far better definition is available. The determinant can be c om- pletely characterized by a few simple axioms, which make goo d sense in view of its geometrical significance and which comprise an efficient a lgorithm for calcu- lating any determinant. To motivate these axioms we first con sider the case of a 2×2-matrix A(n/equalx2). Then the columns a1anda2are vectors in the plane, and instead of the oriented “volume” we speak of the oriented are a of the parallelogram spanned by a1anda2. (This notion is familiar from calculus: the integral/integraltextb af(x)dx /three.taboldstyle/one.taboldstyle /three.taboldstyle/two.taboldstyle /three.taboldstyle. PULLING BACK FORMS of a function fis the oriented area between its graph and the x-axis.) How is the ori- ented area affected by various transformations of the vector s? Adding any multiple ofa1to the second column a2has the effect of performing a shear transformation on the parallelogram, which does not change its oriented are a: a1a2 0a1a2+ba1 0 Multiplying the first column by a scalar chas the effect of stretching (if c>1) or compressing (if 0<c<1) the parallelogram and changing its oriented area by a factor of c: a1a2 0ca1a2 0 What if cis negative? In the picture below the parallelogram on the le ft is positively oriented in the sense that the angle from edge a1to edge a2is counterclockwise (positive). Because cis negative, the parallelogram on the right is negatively oriented in the sense that the angle from edge ca1to edge a2is clockwise (negative). Therefore multiplying a1by a negative calso changes the oriented area by a factor ofc: a1a2 0 ca1a2 0 Similarly, interchanging the columns of Ahas the effect of reversing the orientation of the parallelogram, which changes the sign of its oriented area: a1a2 0a1a2 0 To generalize this to higher dimensions, recall the elementary column operations , which come in three types: adding a multiple of any column of Ato any other /three.taboldstyle./one.taboldstyle. DETERMINANTS /three.taboldstyle/three.taboldstyle column (type I); multiplying a column by a nonzero constant ( type II); and inter- changing any two columns (type III). As suggested by the pict ures above, type I does not affect the determinant, type II multiplies it by the c orresponding constant, and type III causes a sign change. We turn these observations into a definition as follows. /three.taboldstyle./one.taboldstyle. D/e.sc/f.sc/i.sc/n.sc/i.sc/t.sc/i.sc/o.sc/n.sc. Adeterminant is a function detwhich assigns to every n×n- matrix Aa number det(A)subject to the following axioms: (i) If Eis an elementary column operation, then det(E(A))/equalxkdet(A), where k/equalx1ifEis of type I, cifEis of type II (multiplication of a column by c), −1ifEis of type III . (ii)det(I)/equalx1. Axiom ( ii) is a normalization convention, which is justified by the rea sonable requirement that the unit cube inRn(i.e. the parallelepiped spanned by the columns of the identity matrix I) should have oriented volume 1. /three.taboldstyle./two.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. The following calculation is a sequence of column operation s, at the end of which we apply the normalization axiom. /barex/barex/barex/barex/barex/barex/barex1 1 1 4 10 9 1 5 4/barex/barex/barex/barex/barex/barex/barex/equalx/barex/barex/barex/barex/barex/barex/barex1 0 0 4 6 5 1 4 3/barex/barex/barex/barex/barex/barex/barex/equalx/barex/barex/barex/barex/barex/barex/barex1 0 0 4 1 5 1 1 3/barex/barex/barex/barex/barex/barex/barex/equalx/barex/barex/barex/barex/barex/barex/barex1 0 0 3 1 2 0 1 0/barex/barex/barex/barex/barex/barex/barex /equalx2/barex/barex/barex/barex/barex/barex/barex1 0 0 3 1 1 0 1 0/barex/barex/barex/barex/barex/barex/barex/equalx2/barex/barex/barex/barex/barex/barex/barex1 0 0 0 0 1 0 1 0/barex/barex/barex/barex/barex/barex/barex/equalx−2/barex/barex/barex/barex/barex/barex/barex1 0 0 0 1 0 0 0 1/barex/barex/barex/barex/barex/barex/barex/equalx−2. As this example suggests, the axioms of Definition /three.taboldstyle./one.taboldstylesuffice to calculate any n×n-determinant. In other words, there is at most one function detwhich obeys these axioms. More precisely, we have the following result. /three.taboldstyle./three.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc (/u.sc/n.sc/i.sc/q.sc/u.sc/e.sc/n.sc/e.sc/s.sc/s.sc /o.sc/f.sc /d.sc/e.sc/t.sc/e.sc/r.sc/m.sc/i.sc/n.sc/a.sc/n.sc/t.sc/s.sc). Letdetanddet′be two functions satisfying Axioms (i)–(ii). Then det(A)/equalxdet′(A)for all n×n-matrices A. P/r.sc/o.sc/o.sc/f.sc. Leta1,a2,...,anbe the column vectors of A. Suppose first that A isnotinvertible. Then the columns of Aare linearly dependent. For simplicity let us assume that the first column is a linear combination of t he others: a1/equalx c2a2+···+cnan. Repeatedly applying type I column operations gives det(A)/equalxdet/parenleftbiggn/summationdisplay i/equalx2ciai,a2,..., ai,..., an/parenrightbigg /equalxdet(0,a2,..., ai,..., an). Applying a type II operation gives det(0,a2,..., ai,..., an)/equalxdet(−0,a2,..., ai,..., an) /equalx−det(0,a2,..., ai,..., an), and therefore det(A)/equalx0. For the same reason det′(A)/equalx0, sodet(A)/equalxdet′(A). Now assume that Ais invertible. Then Ais column equivalent to the identity /three.taboldstyle/four.taboldstyle /three.taboldstyle. PULLING BACK FORMS matrix, i.e. it can be transformed to Iby successive elementary column operations. LetE1,E2,...,Embe these elementary operations, so that EmEm−1···E2E1(A)/equalxI. By Axiom ( i) each operation Eihas the effect of multiplying the determinant by a certain factor ki, so Axiom ( ii) yields 1/equalxdet(I)/equalxdet(EmEm−1···E2E1(A))/equalxkmkm−1···k2k1det(A). Applying the same reasoning to det′(A)we get 1/equalxkmkm−1···k2k1det′(A). Hence det(A)/equalx1/(k1k2···km)/equalxdet′(A). QED /three.taboldstyle./four.taboldstyle. R/e.sc/m.sc/a.sc/r.sc/k.sc (/c.sc/h.sc/a.sc/n.sc/g.sc/e.sc /o.sc/f.sc /n.sc/o.sc/r.sc/m.sc/a.sc/l.sc/i.sc/z.sc/a.sc/t.sc/i.sc/o.sc/n.sc). Suppose that det′is a function that satisfies Axiom ( i) but is normalized differently: det′(I)/equalxc. Then the proof of Theorem /three.taboldstyle./three.taboldstyleshows that det′(A)/equalxcdet(A)for all n×n-matrices A. The counterpart of this uniqueness theorem is an existence t heorem, which states that Axioms /three.taboldstyle./one.taboldstyle(i)–(ii) are consistent. We will establish consistence by dis- playing an explicit formula for the determinant of any n×n-matrix that does not involve any column reductions. Unlike Definition /three.taboldstyle./one.taboldstyle, this formula is not very practical for the purpose of calculating large determinant s, but it has other uses, notably in the theory of differential forms. /three.taboldstyle./five.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc (/e.sc/x.sc/i.sc/s.sc/t.sc/e.sc/n.sc/c.sc/e.sc /o.sc/f.sc /d.sc/e.sc/t.sc/e.sc/r.sc/m.sc/i.sc/n.sc/a.sc/n.sc/t.sc/s.sc). Every n×n-matrix Ahas a well-defined determinant. It is given by the formula det(A)/equalx/summationdisplay σ∈Snsign(σ)a1,σ(1)a2,σ(2)···an,σ(n). This requires a little explanation. Snstands for the collection of all permutations of the set{1,2,..., n}. A permutation is a way of ordering the numbers 1,2,..., n. Permutations are usually written as n-tuples containing each of these numbers exactly once. Thus for n/equalx2there are only two permutations: (1,2)and(2,1). For n/equalx3all possible permutations are (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1). For general nthere are n(n−1)(n−2)···3·2·1/equalxn! permutations. An alternative way of thinking of a permutati on is as a bijective (i.e. one-to-one and onto) map from the set {1,2,..., n}to itself. For example, for n/equalx5 a possible permutation is (5,3,1,2,4), and we think of this as a shorthand notation for the map σgiven byσ(1)/equalx5, σ(2)/equalx3,σ(3)/equalx1,σ(4)/equalx2andσ(5)/equalx4. The permutation (1,2,3,..., n−1,n) then corresponds to the identity map of the set {1,2,..., n}. Ifσis the identity permutation, then clearly σ(i)< σ(j)whenever i<j. However, if σis not the identity permutation, it cannot preserve the orde r in this way. An inversion ofσis any pair of numbers iand jsuch that 1≤i<j≤nand σ(i)> σ(j). The length ofσ, denoted by l(σ), is the number of inversions of σ. A permutation is called evenoroddaccording to whether its length is even, resp. odd. /three.taboldstyle./one.taboldstyle. DETERMINANTS /three.taboldstyle/five.taboldstyle For instance, the permutation (5,3,1,2,4)has length 6and so is even. The signof σis sign(σ)/equalx(−1)l(σ)/equalx1ifσis even, −1ifσis odd. Thus sign(5,3,1,2,4)/equalx1. The permutations of {1,2}are(1,2), which has sign 1, and (2,1), which has sign−1, while for n/equalx3we have the table below. σ l(σ)sign(σ) (1,2,3)0 1 (1,3,2)1−1 (2,1,3)1−1 (2,3,1)2 1 (3,1,2)2 1 (3,2,1)3−1 Thinking of permutations in Snas bijective maps from {1,2,..., n}to itself, we can form the composition σ◦τof any two permutations σandτinSn. For permutations we usually write as στinstead ofσ◦τand call it the product ofσand τ. This is the permutation produced by firstperforming τandthenσ! For instance, ifσ/equalx(5,3,1,2,4)andτ/equalx(5,4,3,2,1), then τσ/equalx(1,3,5,4,2), στ /equalx(4,2,1,3,5). A basic fact concerning signs, which we shall not prove here, is sign(στ)/equalxsign(σ)sign(τ). (/three.taboldstyle./one.taboldstyle) In particular, the product of two even permutations is even a nd the product of an even and an odd permutation is odd. The determinant formula in Theorem /three.taboldstyle./five.taboldstylecontains n!terms, one for each per- mutationσ. Each term is a product which contains exactly one entry from each row and each column of A. For instance, for n/equalx5the permutation (5,3,1,2,4) contributes the term a1,5a2,3a3,1a4,2a5,4. For 2×2- and 3×3-determinants Theorem /three.taboldstyle./five.taboldstylegives the well-known formulæ /barex/barex/barex/barex/barexa1,1a1,2 a2,1a2,2/barex/barex/barex/barex/barex/equalxa1,1a2,2−a1,2a2,1, /barex/barex/barex/barex/barex/barex/barexa1,1a1,2a1,3 a2,1a2,2a2,3 a3,1a3,2a3,3/barex/barex/barex/barex/barex/barex/barex/equalxa1,1a2,2a3,3−a1,1a2,3a3,2−a1,2a2,1a3,3+a1,2a2,3a3,1 +a1,3a2,1a3,2−a1,3a2,2a3,1. P/r.sc/o.sc/o.sc/f.sc /o.sc/f.sc T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc /three.taboldstyle./five.taboldstyle.We need to check that the right-hand side of the deter- minant formula in Theorem /three.taboldstyle./five.taboldstyleobeys Axioms ( i)–(ii) of Definition /three.taboldstyle./one.taboldstyle. Let us for the moment denote the right-hand side by f(A). Axiom ( ii) is the easiest to verify: ifA/equalxI, then a1,σ(1)a2,σ(2)···an,σ(n)/equalx1ifσ/equalxidentity, 0otherwise, /three.taboldstyle/six.taboldstyle /three.taboldstyle. PULLING BACK FORMS and therefore f(I)/equalx1. Next we consider how f(A)behaves when we interchange two columns of A. We assert that each term in f(A)changes sign. To see this, let τ be the permutation in Snthat interchanges the two numbers iand jand leaves all others fixed. Then f(a1,..., aj,..., ai,..., an) /equalx/summationdisplay σ∈Snsign(σ)a1,τσ(1)a2,τσ(2)···an,τσ(n) /equalx/summationdisplay ρ∈Snsign(τρ)a1,ρ(1)a2,ρ(2)···an,ρ(n) substituteρ/equalxτσ /equalx/summationdisplay ρ∈Snsign(τ)sign(ρ)a1,ρ(1)a2,ρ(2)···an,ρ(n)by formula ( /three.taboldstyle./one.taboldstyle) /equalx−/summationdisplay ρ∈Snsign(ρ)a1,ρ(1)a2,ρ(2)···an,ρ(n) by Exercise /three.taboldstyle./five.taboldstyle /equalx−f(a1,..., ai,..., aj,..., an). To see what happens when we multiply a column of Abyc, observe that for every permutation σthe product a1,σ(1)a2,σ(2)···an,σ(n) contains exactly one entry from each row and each column in A. So if we multiply thei-th column of Abyc, each term in f(A)is multiplied by c. Therefore f(a1,a2,..., cai,..., an)/equalxc f(a1,a2,..., ai,..., an). By a similar argument we have f(a1,a2,..., ai+a′ i,..., an)/equalxf(a1,a2,..., ai,..., an)+f(a1,a2,..., a′ i,..., an) for any vector a′ i. In particular we can take a′ i/equalxajfor some j/nequali, which gives f(a1,a2,..., ai+aj,..., aj,..., an)/equalxf(a1,a2,..., ai,..., aj,..., an) +f(a1,a2,..., aj,..., aj,..., an) /equalxf(a1,a2,..., ai,..., aj,..., an), because f(a1,a2,..., ai,..., aj,..., an)/equalx0. This shows that fsatisfies the condi- tions of Definition /three.taboldstyle./one.taboldstyle. QED We can calculate the determinant of any matrix by column redu cing it to the identity matrix, but there are many different ways of perform ing this reduction. Theorem /three.taboldstyle./five.taboldstyleimplies that different column reductions lead to the same ans wer for the determinant. The following corollary of Theorem /three.taboldstyle./five.taboldstyleamounts to a reformulation of Defini- tion/three.taboldstyle./one.taboldstyle. Recall that e1,e2,...,endenote the standard basis vectors of Rn, i.e. the columns of the identity n×n-matrix. /three.taboldstyle./six.taboldstyle. C/o.sc/r.sc/o.sc/l.sc/l.sc/a.sc/r.sc/y.sc. The determinant possesses the following properties. These properties characterize the determinant uniquely. (i)detis multilinear (i.e. linear in each column): det(a1,a2,..., cai+c′a′ i,..., an) /equalxcdet(a1,a2,..., ai,..., an)+c′det(a1,a2,..., a′ i,..., an) /three.taboldstyle./one.taboldstyle. DETERMINANTS /three.taboldstyle/seven.taboldstyle for all scalars c,c′and all vectors a1,a2,...,ai,a′ i,...,an; (ii)detis alternating (or antisymmetric): det(a1,..., ai,..., aj,..., an)/equalx−det(a1,..., aj,..., ai,..., an) for all vectors a1,a2,...,anand for all pairs of distinct indices i/nequalj; (iii)normalization: det(e1,e2,..., en)/equalx1. P/r.sc/o.sc/o.sc/f.sc. Property ( i) was established in the proof of Theorem /three.taboldstyle./five.taboldstyle, while prop- erties ( ii)–(iii) are simply a restatement of part of Definition /three.taboldstyle./one.taboldstyle. Therefore the determinant has properties ( i)–(iii). Conversely, properties ( i)–(iii) taken together imply Axioms ( i)–(ii) of Definition /three.taboldstyle./one.taboldstyle. Therefore, by Theorem /three.taboldstyle./three.taboldstyle, properties (i)–(iii) characterize the determinant uniquely. QED Here are some further rules obeyed by determinants. Each can be deduced from Definition /three.taboldstyle./one.taboldstyleor from Theorem /three.taboldstyle./five.taboldstyle. (Recall that the transpose of an n×n-matrix A/equalx(ai,j)is the matrix ATwhose i,j-th entry is aj,i.) /three.taboldstyle./seven.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. LetAandBben×n-matrices. (i)det(A)/equalxa1,1a2,2···an,nifAis upper triangular (i.e. ai,j/equalx0fori>j). (ii)det(AB)/equalxdet(A)det(B). (iii)det(AT)/equalxdet(A). (iv)(Expansion on the j-th column) det(A)/equalx/summationtextn i/equalx1(−1)i+jai,jdet(Ai,j)for all j/equalx1,2,...,n. Here Ai,jdenotes the (n−1)×(n−1)-matrix obtained from Aby deleting the i-th row and the j-th column. (v)Letσ∈Snbe a permutation. Then det/parenleftbigaσ(1),aσ(2),..., aσ(n)/parenrightbig/equalxsign(σ)det(a1,a2,..., an). When calculating determinants in practice one combines col umn reductions with these rules. For instance, rule ( i) tells us we need not bother reducing A all the way to the identity matrix, like we did in Example /three.taboldstyle./two.taboldstyle, but that an upper triangular form suffices. Rule ( iii) tells us we may use row operations as well as column operations. Volume change. We conclude this discussion with a slightly different geo- metric view of determinants. A square matrix Acan be regarded as a linear map A:Rn→Rn. The unit cube in Rn, [0,1]n/equalx{x∈Rn|0≤xi≤1fori/equalx1,2,...,n}, hasn-dimensional volume 1. (For n/equalx1it is usually called the unit interval and for n/equalx2theunit square .) Its image A/parenleftbig[0,1]n/parenrightbigunder the map Ais the parallelepiped spanned by the vectors Ae1,Ae2,...,Aen, which are the columns of A. Hence A/parenleftbig[0,1]n/parenrightbighasn-dimensional volume vol/parenleftbigA([0,1]n)/parenrightbig/equalx|det(A|/equalx|det(A)|vol([0,1]n). /three.taboldstyle/eight.taboldstyle /three.taboldstyle. PULLING BACK FORMS This rule generalizes as follows: if Xis a measurable subset of Rn, then vol(A(X))/equalx|det(A)|vol(X). X e1e2A A(X) Ae1Ae2 (A set is measurable if it has a well-defined, finite or infinite, n-dimensional volume. Explaining exactly what this means is rather hard, but it suffi ces for our purposes to know that all open and all closed subsets of Rnare measurable.) So |det(A)| can be interpreted as a volume change factor . The sign of the determinant tells you whether Apreserves ( +) or reverses (−) the orientation of Rn. (See Section /eight.taboldstyle./two.taboldstylefor more on orientations.) /three.taboldstyle./two.taboldstyle. Pulling back forms By substituting new variables into a differential form we obt ain a new form of the same degree but possibly in a different number of variable s. /three.taboldstyle./eight.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. In Example /two.taboldstyle./one.taboldstyle/zero.taboldstylewe defined the angle form on R2\{0}to be α0/equalx−y dx+x dy x2+y2. By substituting x/equalxcostand y/equalxsintinto the angle form we obtain the following 1-form on R: −sint dcost+ cos t dsint cos2t+ sin2t/equalx/parenleftbig(−sint)(−sint)+ cos2t/parenrightbigdt/equalxdt. We can take any k-form and substitute any number of variables into it to obtai n a new k-form. This works as follows. Suppose αis ak-form defined on an open subset VofRm. Let us denote the coordinates on Rmbyy1,y2,...,ymand let us write, as usual, α/equalx/summationdisplay IfIdyI, where the functions fIare defined on V. Suppose we want to substitute “new” variables x1,x2,...,xnand that the old variables are given in terms of the new by functions y1/equalxφ1(x1,..., xn), y2/equalxφ2(x1,..., xn), ... ym/equalxφm(x1,..., xn). /three.taboldstyle./two.taboldstyle. PULLING BACK FORMS /three.taboldstyle/nine.taboldstyle As usual we write y/equalxφ(x), where φ(x)/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAφ1(x) φ2(x) ... φm(x)/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA. We assume that the functions φiare smooth and defined on a common domain U, which is an open subset of Rn. We regard φas a map from UtoV. (In Example /three.taboldstyle./eight.taboldstylewe have U/equalxR,V/equalxR2\{0}andφ(t)/equalx(cost,sint).) The pullback ofαalongφ is then the k-formφ∗(α)onUobtained by substituting yi/equalxφi(x1,..., xn)for all i in the formula for α. That is to say, φ∗(α)is defined by φ∗(α)/equalx/summationdisplay Iφ∗(fI)φ∗(dyI). Hereφ∗(fI)is defined by φ∗(fI)/equalxfI◦φ, the composition of φand fI. This means φ∗(fI)(x)/equalxfI(φ(x)); in other words, φ∗(fI)is the function resulting from fIby substituting y/equalxφ(x). The pullback φ∗(dyI)is defined by replacing each yiwithφi. That is to say, if I/equalx(i1,i2,..., ik) we put φ∗(dyI)/equalxφ∗(dyi1dyi2···dyik)/equalxdφi1dφi2···dφik. The picture below is a schematic representation of the subst itution process. The formα/equalx/summationtext IfIdyIis ak-form in the variables y1,y2,...,ym; its pullback φ∗(α)/equalx/summationtext JgJdxJis ak-form in the variables x1,x2,...,xn. In Theorem /three.taboldstyle./one.taboldstyle/three.taboldstylebelow we will give an explicit formula for the coefficients gJin terms of fIandφ. U V Rn Rmφφ∗(α)α xy=φ(x) /three.taboldstyle./nine.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. The formula φ/parenleftBigg x1 x2/parenrightBigg /equalx/parenleftBigg x3 1x2 ln(x1+x2)/parenrightBigg /four.taboldstyle/zero.taboldstyle /three.taboldstyle. PULLING BACK FORMS defines a map φ:U→R2, where U/equalx{x∈R2|x1+x2>0}. The components of φare given by φ1(x1,x2)/equalxx3 1x2andφ2(x1,x2)/equalxln(x1+x2). Accordingly, φ∗(dy1)/equalxdφ1/equalxd(x3 1x2)/equalx3x2 1x2dx1+x3 1dx2, φ∗(dy2)/equalxdφ2/equalxdln(x1+x2)/equalx(x1+x2)−1(dx1+dx2), φ∗(dy1dy2)/equalxdφ1dφ2/equalx(3x2 1x2dx1+x3 1dx2)(x1+x2)−1(dx1+dx2) /equalx3x2 2x2−x3 1 x1+x2dx1dx2. Observe that the pullback operation turns k-forms on the target space Vinto k-forms on the source space U. Thus, while φ:U→Vis a map from UtoV,φ∗is a map φ∗:Ωk(V)→Ωk(U), the opposite way from what you might naively expect. (Recall thatΩk(U)stands for the collection of all k-forms on U.) The property that φ∗“turns the arrow around” is called contravariance . Pulling back forms is nicely compatible with the other operations that we learned about (except the Hodge sta r). /three.taboldstyle./one.taboldstyle/zero.taboldstyle. P/r.sc/o.sc/p.sc/o.sc/s.sc/i.sc/t.sc/i.sc/o.sc/n.sc. Letφ:U→Vbe a smooth map, where Uis open in RnandV is open in Rm. The pullback operation is (i)linear:φ∗(aα+bβ)/equalxaφ∗(α)+bφ∗(β)for all scalars aandband all k-forms αandβonV; (ii)multiplicative: φ∗(αβ)/equalxφ∗(α)φ∗(β)for all k-formsαandl-formsβonV; (iii)natural:φ∗(ψ∗(α))/equalx(ψ◦φ)∗(α), whereψ:V→Wis a second smooth map with Wopen in Rl, andαis ak-form on W. The term “natural” in property ( iii) is a mathematical catchword meaning that a certain operation (in this case the pullback) is well- behaved with respect to composition of maps. P/r.sc/o.sc/o.sc/f.sc. Ifα/equalx/summationtext IfIdyIandβ/equalx/summationtext IgIdyIare two forms of the same degree, then aα+bβ/equalx/summationtext I(a fI+bgI)dyI, so φ∗(aα+bβ)/equalx/summationdisplay Iφ∗(a fI+bgI)φ∗(dyI). Now φ∗(a fI+bgI)(x)/equalx(a fI+bgI)(φ(x))/equalxa fI(φ(x))+bgI(φ(x)) /equalxaφ∗(fI)(x)+bφ∗(gI)(x), soφ∗(aα+bβ)/equalx/summationtext I(aφ∗(fI)+bφ∗(gI))φ∗(dyI)/equalxaφ∗(α)+bφ∗(β). This proves part (i). For the proof of part ( ii) consider two forms α/equalx/summationtext IfIdyIandβ/equalx/summationtext JgJdyJ (not necessarily of the same degree). Then αβ/equalx/summationtext I,JfIgJdyIdyJ, so φ∗(αβ)/equalx/summationdisplay I,Jφ∗(fIgJ)φ∗(dyIdyJ). Now φ∗(fIgJ)(x)/equalx(fIgJ)(φ(x))/equalxfI(φ(x))gJ(φ(x))/equalx(φ∗(fI)φ∗(gJ))(x), /three.taboldstyle./two.taboldstyle. PULLING BACK FORMS /four.taboldstyle/one.taboldstyle soφ∗(fIgJ)/equalxφ∗(fI)φ∗(gJ). Furthermore, φ∗(dyIdyJ)/equalxφ∗(dyi1dyi2···dyikdyj1dyj2···dyjl) /equalxdφi1dφi2···dφikdφj1···dφjl/equalxφ∗(dyI)φ∗(dyJ), so φ∗(αβ)/equalx/summationdisplay I,Jφ∗(fI)φ∗(gJ)φ∗(dyI)φ∗(dyJ) /equalx/parenleftbigg/summationdisplay Iφ∗(fI)φ∗(dyI)/parenrightbigg/parenleftbigg/summationdisplay Iφ∗(gJ)φ∗(dyJ)/parenrightbigg /equalxφ∗(α)φ∗(β), which establishes part ( ii). For the proof of property ( iii) first consider a function fonW. Then φ∗(ψ∗(f))(x)/equalxψ∗(f)(φ(x))/equalxf(ψ(φ(x)))/equalx(f◦ψ◦φ)(x) /equalx(f◦(ψ◦φ))(x)/equalx(ψ◦φ)∗(f)(x), soφ∗(ψ∗(f))/equalx(ψ◦φ)∗(f). Next consider a 1-formα/equalxdzionW, where z1,z2,..., zlare the variables on Rl. Thenψ∗(α)/equalxdψi/equalx/summationtextm j/equalx1∂ψi ∂yjdyj, so φ∗(ψ∗(α))/equalxm/summationdisplay j/equalx1φ∗/parenleftbigg∂ψi ∂yj/parenrightbigg φ∗(dyj)/equalxm/summationdisplay j/equalx1φ∗/parenleftbigg∂ψi ∂yj/parenrightbigg dφj /equalxm/summationdisplay j/equalx1φ∗/parenleftbigg∂ψi ∂yj/parenrightbiggn/summationdisplay k/equalx1∂φj ∂xkdxk/equalxn/summationdisplay k/equalx1/parenleftBiggm/summationdisplay j/equalx1φ∗/parenleftbigg∂ψi ∂yj/parenrightbigg∂φj ∂xk/parenrightBigg dxk. By the chain rule, formula ( B./six.taboldstyle), the sum/summationtextm j/equalx1φ∗(∂ψi/∂yj)∂φj/∂xkis equal to ∂φ∗(ψi)/∂xk. Therefore φ∗(ψ∗(α))/equalxn/summationdisplay k/equalx1∂φ∗(ψi) ∂xkdxk/equalxdφ∗(ψi) /equalxd((ψ◦φ)i)/equalx(ψ◦φ)∗(dzi)/equalx(ψ◦φ)∗(α). Because every form on Wis a sum of products of forms of type fand dzi, property (iii) in general follows from the two special cases α/equalxfandα/equalxdzi, together with properties ( i) and ( iii). QED Another application of the chain rule yields the following i mportant result. /three.taboldstyle./one.taboldstyle/one.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. Letφ:U→Vbe a smooth map, where Uis open in RnandVis open in Rm. Thenφ∗(dα)/equalxdφ∗(α)forα∈Ωk(V). In short, φ∗d/equalxdφ∗. /four.taboldstyle/two.taboldstyle /three.taboldstyle. PULLING BACK FORMS P/r.sc/o.sc/o.sc/f.sc. First let fbe a function. Then φ∗(d f)/equalxφ∗/parenleftBiggm/summationdisplay i/equalx1∂f ∂yidyi/parenrightBigg /equalxm/summationdisplay i/equalx1φ∗/parenleftbigg∂f ∂yi/parenrightbigg dφi/equalxm/summationdisplay i/equalx1φ∗/parenleftbigg∂f ∂yi/parenrightbiggn/summationdisplay j/equalx1∂φi ∂xjdxj /equalxn/summationdisplay j/equalx1m/summationdisplay i/equalx1φ∗/parenleftbigg∂f ∂yi/parenrightbigg∂φi ∂xjdxj. By the chain rule, formula ( B./six.taboldstyle), the quantity/summationtextm i/equalx1φ∗(∂f/∂yi)∂φi/∂xjis equal to ∂φ∗(f)/∂xj. Hence φ∗(d f)/equalxn/summationdisplay j/equalx1∂φ∗(f) ∂xjdxj/equalxdφ∗(f), so the theorem is true for functions. Next let α/equalx/summationtext IfIdyI. Then dα/equalx/summationtext Id fIdyI, so φ∗(dα)/equalx/summationdisplay Iφ∗(d fIdyI)/equalxφ∗(d fI)φ∗(dyI)/equalx/summationdisplay Idφ∗(fI)dφi1dφi2···dφik,(/three.taboldstyle./two.taboldstyle) becauseφ∗(d fI)/equalxdφ∗(fI). On the other hand, dφ∗(α)/equalx/summationdisplay Id/parenleftbigφ∗(fI)φ∗(dyI)/parenrightbig/equalx/summationdisplay Id/parenleftbigφ∗(fI)dφi1dφi2···dφik/parenrightbig /equalx/summationdisplay Idφ∗(fI)dφi1dφi2···dφik+/summationdisplay Iφ∗(fI)d(dφi1dφi2···dφik) /equalx/summationdisplay Idφ∗(fI)dφi1dφi2···dφik. (/three.taboldstyle./three.taboldstyle) Here we have used the Leibniz rule for forms, Proposition /two.taboldstyle./five.taboldstyle(ii), plus the fact that the form dφi1dφi2···dφikis always closed. (See Exercise /two.taboldstyle./one.taboldstyle/zero.taboldstyle.) Comparing ( /three.taboldstyle./two.taboldstyle) with ( /three.taboldstyle./three.taboldstyle) we see that φ∗(dα)/equalxdφ∗(α). QED Here is an application of Theorem /three.taboldstyle./one.taboldstyle/one.taboldstyle. An open subset UofRnis called connected if for every pair of points xandyinUthere exists a path c: [a,b]→U satisfying c(a)/equalxxand c(b)/equalxy. /three.taboldstyle./one.taboldstyle/two.taboldstyle. L/e.sc/m.sc/m.sc/a.sc. LetUbe a connected open subset of Rnand let f:U→Rbe a smooth function. Then d f/equalx0if and only if fis constant. P/r.sc/o.sc/o.sc/f.sc. Iffis constant, all its partial derivatives vanish, so d f/equalx0. Conversely, suppose d f/equalx0. To prove that fis constant it is enough to show that f(x)/equalxf(y) for any pair of points x,yinU. Choose a path c: [a,b]→Uwith the property c(a)/equalxxand c(b)/equalxy. Let h: [a,b]→Rbe the smooth function h/equalxc∗(f). Then h′(t)dt/equalxdh/equalxd(c∗(f))/equalxc∗(d f)/equalx0by Theorem /three.taboldstyle./one.taboldstyle/one.taboldstyle, soh′(t)/equalx0fora≤t≤b. Therefore his constant (by one-variable calculus), so f(x)/equalxf(c(a))/equalxh(a)/equalx h(b)/equalxf(c(b))/equalxf(y). QED We finish this section by giving an explicit formula for the pu llbackφ∗(α), which establishes a connection between forms and determina nts. Let us do this first in degrees 1and2. The pullback of a 1-formα/equalx/summationtextm i/equalx1fidyiis φ∗(α)/equalxm/summationdisplay i/equalx1φ∗(fi)φ∗(dyi)/equalxm/summationdisplay i/equalx1φ∗(fi)dφi. /three.taboldstyle./two.taboldstyle. PULLING BACK FORMS /four.taboldstyle/three.taboldstyle Now dφi/equalx/summationtextn j/equalx1∂φi ∂xjdxjand so φ∗(α)/equalxm/summationdisplay i/equalx1/parenleftBigg φ∗(fi)n/summationdisplay j/equalx1∂φi ∂xjdxj/parenrightBigg /equalxn/summationdisplay j/equalx1m/summationdisplay i/equalx1φ∗(fi)∂φi ∂xjdxj/equalxn/summationdisplay j/equalx1gjdxj, with gj/equalx/summationtextm i/equalx1φ∗(fi)∂φi ∂xj. For a 2-formα/equalx/summationtext 1≤i<j≤mfi,jdyidyjwe get φ∗(α)/equalx/summationdisplay 1≤i<j≤mφ∗(fi,j)φ∗(dyidyj)/equalx/summationdisplay 1≤i<j≤mφ∗(fi,j)dφidφj. To expressφ∗(α)in terms of the x-variables we use dφidφj/equalxn/summationdisplay k,l/equalx1∂φi ∂xk∂φj ∂xldxkdxl/equalx/summationdisplay 1≤k<l≤n/parenleftbigg∂φi ∂xk∂φj ∂xl−∂φi ∂xl∂φj ∂xk/parenrightbigg dxkdxl, where ∂φi ∂xk∂φj ∂xl−∂φi ∂xl∂φj ∂xk/equalx/barex/barex/barex/barex/barex/barex∂φi ∂xk∂φi ∂xl∂φj ∂xk∂φj ∂xl/barex/barex/barex/barex/barex/barex is the determinant of the 2×2-submatrix obtained from the Jacobi matrix Dφby extracting rows iand jand columns kand l. So we get φ∗(α)/equalx/summationdisplay 1≤i<j≤m/parenleftBigg φ∗(fi,j)/summationdisplay 1≤k<l≤n/barex/barex/barex/barex/barex/barex∂φi ∂xk∂φi ∂xl∂φj ∂xk∂φj ∂xl/barex/barex/barex/barex/barex/barexdxkdxl/parenrightBigg /equalx/summationdisplay 1≤k<l≤n/summationdisplay 1≤i<j≤mφ∗(fi,j)/barex/barex/barex/barex/barex/barex∂φi ∂xk∂φi ∂xl∂φj ∂xk∂φj ∂xl/barex/barex/barex/barex/barex/barexdxkdxl/equalx/summationdisplay 1≤k<l≤ngk,ldxkdxl with gk,l/equalx/summationdisplay 1≤i<j≤mφ∗(fi,j)/barex/barex/barex/barex/barex/barex∂φi ∂xk∂φi ∂xl∂φj ∂xk∂φj ∂xl/barex/barex/barex/barex/barex/barex. For an arbitrary k-formα/equalx/summationtext IfIdyIwe obtain φ∗(α)/equalx/summationdisplay Iφ∗(fI)φ∗(dyi1dyi2···dyik)/equalx/summationdisplay Iφ∗(fI)dφi1dφi2···dφik. To write the product dφi1dφi2···dφikin terms of the x-variables we use dφil/equalxn/summationdisplay pl/equalx1∂φil ∂xpldxpl forl/equalx1,2,...,k. This gives dφi1dφi2···dφik/equalxn/summationdisplay p1,p2,...,pk/equalx1∂φi1 ∂xp1∂φi2 ∂xp2···∂φik ∂xpkdxp1dxp2···dxpk /equalx/summationdisplay P∂φi1 ∂xp1∂φi2 ∂xp2···∂φik ∂xpkdxP, in which the summation is over allnkmulti-indices P/equalx(p1,p2,..., pk). If a multi- index Phas repeating entries, then dxP/equalx0. If the entries of Pare all distinct, we can /four.taboldstyle/four.taboldstyle /three.taboldstyle. PULLING BACK FORMS rearrange them in increasing order by means of a permutation σ. In other words, we have P/equalx(p1,p2,..., pk)/equalx(jσ(1),jσ(2),..., jσ(k)), where J/equalx(j1,j2,..., jk)is an increasing multi-index and σ∈Skis a permutation. Thus we can rewrite the sum over all multi-indices Pas a double sum over all increasing multi-indices Jand all permutations σ: dφi1dφi1···dφik/equalx/summationdisplay J/summationdisplay σ∈Sk∂φi1 ∂xjσ(1)∂φi2 ∂xjσ(2)···∂φik ∂xjσ(k)dxjσ(1)dxjσ(2)···dxjσ(k) /equalx/summationdisplay J/summationdisplay σ∈Sksign(σ)∂φi1 ∂xjσ(1)∂φi2 ∂xjσ(2)···∂φik ∂xjσ(k)dxJ (/three.taboldstyle./four.taboldstyle) /equalx/summationdisplay Jdet(DφI,J)dxJ. (/three.taboldstyle./five.taboldstyle) In (/three.taboldstyle./four.taboldstyle) we used the result of Exercise /three.taboldstyle./eight.taboldstyleand in ( /three.taboldstyle./five.taboldstyle) we applied Theorem /three.taboldstyle./five.taboldstyle. The notation DφI,Jstands for the (I,J)-submatrix ofDφ, that is the k×k-matrix obtained from the Jacobi matrix by extracting rows i1,i2,...,ikand columns j1, j2,...,jk. To sum up, we find φ∗(α)/equalx/summationdisplay Iφ∗(fI)/summationdisplay Jdet(DφI,J)dxJ/equalx/summationdisplay J/parenleftBigg/summationdisplay Iφ∗(fI)det(DφI,J)/parenrightBigg dxJ. This proves the following result. /three.taboldstyle./one.taboldstyle/three.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. Letφ:U→Vbe a smooth map, where Uis open in RnandVis open in Rm. Letα/equalx/summationtext IfIdyIbe ak-form on V. Thenφ∗(α)is the k-form on Ugiven by φ∗(α)/equalx/summationtext JgJdxJwith gJ/equalx/summationdisplay Iφ∗(fI)det(DφI,J). This formula is seldom used to calculate pullbacks in practi ce and you don’t need to memorize the details of the proof. It is almost always easier to apply the definition of pullback directly. However, the formula ha s some important theoretical uses, one of which we record here. Assume that k/equalxm/equalxn, that is to say, the number of new variables is equal to the number of old variables, and we are pulling back a form of t op degree. Then α/equalxf dy 1dy2···dyn, φ∗(α)/equalxφ∗(f)det(Dφ)dx1dx2···dxn. Iff/equalx1(constant function) then φ∗(f)/equalx1, so we see that det(Dφ(x))can be interpreted as the ratio between the oriented volumes of two infinitesimal paral- lelepipeds positioned at x: one with edges dx1,dx2,...,dxnand another with edges dφ1,dφ2,...,dφn. Thus the Jacobi determinant is a measurement of how much the map φchanges oriented volume from point to point. /three.taboldstyle./one.taboldstyle/four.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. Letφ:U→Vbe a smooth map, where UandVare open in Rn. Then the pullback of the volume form on Vis equal to the Jacobi determinant times the volume form on U, φ∗(dy1dy2···dyn)/equalxdet(Dφ)dx1dx2···dxn. EXERCISES /four.taboldstyle/five.taboldstyle Exercises /three.taboldstyle./one.taboldstyle.Deduce Theorem /three.taboldstyle./seven.taboldstyle(i) from Theorem /three.taboldstyle./five.taboldstyle. /three.taboldstyle./two.taboldstyle. Calculate the following determinants using column and/or r ow operations and Theorem /three.taboldstyle./seven.taboldstyle(i). /barex/barex/barex/barex/barex/barex/barex/barex/barex1 3 1 1 2 1 5 2 1−1 2 3 4 1−3 7/barex/barex/barex/barex/barex/barex/barex/barex/barex,/barex/barex/barex/barex/barex/barex/barex/barex/barex1 1−2 4 0 1 1 3 2−1 1 0 3 1 2 5/barex/barex/barex/barex/barex/barex/barex/barex/barex. /three.taboldstyle./three.taboldstyle.In this exercise we write Cnfor the unit cube [0,1]n. (i) Draw a picture of (a two-dimensional projection of) Cnforn/equalx0,1,2,3,4,5. (ii) Let fk(n)be the number of k-dimensional faces of Cn. Show that fk(n)/equalx2n−k/parenleftbign k/parenrightbig. (Let p(s)/equalx2 +s. Note that the coefficients of paref0(1)and f1(1). Show that fk(n)is the coefficient of skin the polynomial p(s)kby noting that each face of Cnis a Cartesian product of faces of the unit interval C1.) /three.taboldstyle./four.taboldstyle.List all permutations in S4with their lengths and signs. /three.taboldstyle./five.taboldstyle.Determine the length and the sign of the following permutati ons. (i) A permutation of the form (1,2,..., i−1,j,..., j−1,i,..., n)where 1≤i<j≤n. (Such a permutation is called a transposition . It interchanges iand jand leaves all other numbers fixed.) (ii)(n,n−1,n−2,..., 3,2,1). /three.taboldstyle./six.taboldstyle.Find all permutations in Snof length 1. /three.taboldstyle./seven.taboldstyle.Calculateσ−1,τ−1,στandτσ, where (i)σ/equalx(3,6,1,2,5,4)andτ/equalx(5,2,4,6,3,1); (ii)σ/equalx(2,1,3,4,5,..., n−1,n)andτ/equalx(n,2,3,..., n−2,n−1,1)(i.e. the transpo- sitions interchanging 1and2, resp. 1and n). /three.taboldstyle./eight.taboldstyle.Show that dxiσ(1)dxiσ(2)···dxiσ(k)/equalxsign(σ)dxi1dxi2···dxik for any multi-index (i1,i2,..., ik)and any permutation σinSk. (First show that the identity is true ifσis a transposition. Then show it is true for an arbitrary perm utationσby writing σas a product σ1σ2···σlof transpositions and using formula ( /three.taboldstyle./one.taboldstyle) and Exercise /three.taboldstyle./five.taboldstyle(i).) /three.taboldstyle./nine.taboldstyle. Show that for n≥2the permutation group Snhasn!/2even permutations and n!/2odd permutations. /three.taboldstyle./one.taboldstyle/zero.taboldstyle. (i) Show that every permutation has the same length and sign a s its inverse. (ii) Deduce Theorem /three.taboldstyle./seven.taboldstyle(iii) from Theorem /three.taboldstyle./five.taboldstyle. /three.taboldstyle./one.taboldstyle/one.taboldstyle. The i-th simple permutation is defined by σi/equalx(1,2,..., i−1,i+ 1,i,i+ 2,..., n). Soσiinterchanges iand i+ 1and leaves all other numbers fixed. Snhas n−1simple permutations, namely σ1,σ2,...,σn−1. Prove the Coxeter relations (i)σ2 i/equalx1for1≤i<n, (ii)(σiσi+1)3/equalx1for1≤i<n−1, (iii) (σiσj)2/equalx1for1≤i,j<nand i+ 1<j. /three.taboldstyle./one.taboldstyle/two.taboldstyle. Letσbe a permutation of {1,2,..., n}. The permutation matrix corresponding to σis the n×n-matrix Aσwhose i-th column is the vector eσ(i). In other words, Aσei/equalxeσ(i). (i) Write down the permutation matrices for all permutation s inS3. (ii) Show that Aστ/equalxAσAτ. /four.taboldstyle/six.taboldstyle /three.taboldstyle. PULLING BACK FORMS (iii) Show that det(Aσ)/equalxsign(σ). /three.taboldstyle./one.taboldstyle/three.taboldstyle. (i) Suppose that Ahas the shape A/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAa1,1a1,2... a1,n 0 a2,2... a2,n ......... 0 an,2... an,n/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA, i.e. all entries below a11are0. Deduce from Theorem /three.taboldstyle./five.taboldstylethat det(A)/equalxa1,1/barex/barex/barex/barex/barex/barex/barex/barexa2,2... a2,n ...... an,2... an,n/barex/barex/barex/barex/barex/barex/barex/barex. (ii) Deduce from this the expansion rule, Theorem /three.taboldstyle./seven.taboldstyle(iv). /three.taboldstyle./one.taboldstyle/four.taboldstyle. Show that /barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex1 1 ... 1 x1 x2... xn x2 1x2 2... x2n ......... xn−1 1xn−1 2... xn−1n/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/equalx/productdisplay i<j(xj−xi) for any numbers x1,x2,...,xn. (Starting at the bottom, from each row subtract x1times the row above it. This creates a new determinant whose first colum n is the standard basis vector e1. Expand on the first column and note that each column of the rem aining determinant has a common factor.) /three.taboldstyle./one.taboldstyle/five.taboldstyle. Letφ(x1,x2,x3)/equalx(x1x2,x1x3,x2x3). Find (i)φ∗(dy1),φ∗(dy2),φ∗(dy3); (ii)φ∗(y1y2y3),φ∗(dy1dy2); (iii)φ∗(dy1dy2dy3). /three.taboldstyle./one.taboldstyle/six.taboldstyle. Letφ(x1,x2)/equalx(x3 1,x2 1x2,x1x2 2,x3 2). Find (i)φ∗(y1+ 3y2+ 3y3+y4); (ii)φ∗(dy1),φ∗(dy2),φ∗(dy3),φ∗(dy4); (iii)φ∗(dy2dy3). /three.taboldstyle./one.taboldstyle/seven.taboldstyle. Computeψ∗(x dy dz +y dz dx +z dx dy ), whereψis the map R2→R3defined in Exercise B./eight.taboldstyle. /three.taboldstyle./one.taboldstyle/eight.taboldstyle. LetP3(r,θ,φ )/equalx(rcosθcosφ,rsinθcosφ,rsinφ)be spherical coordinates in R3. Calculate P∗ 3(α)for the following forms α: dx,dy,dz,dx dy,dx dy dz. /three.taboldstyle./one.taboldstyle/nine.taboldstyle (/s.sc/p.sc/h.sc/e.sc/r.sc/i.sc/c.sc/a.sc/l.sc /c.sc/o.sc/o.sc/r.sc/d.sc/i.sc/n.sc/a.sc/t.sc/e.sc/s.sc /i.sc/n.sc n/d.sc/i.sc/m.sc/e.sc/n.sc/s.sc/i.sc/o.sc/n.sc/s.sc). In this problem let us write a point in Rnas(r,θ1,...,θ n−1). Let P1be the function P1(r)/equalxr. For each n≥1define a map Pn+1:Rn+1→Rn+1by Pn+1(r,θ1,...,θ n)/equalx(cosθn)Pn(r,θ1,...,θ n−1),rsinθn). (This is an example of a recursive definition. If you know P1, you can compute P2, and then P3, etc.) (i) Show that P2and P3are the usual polar, resp. spherical coordinates on R2, resp. R3. (ii) Give an explicit formula for Pn. EXERCISES /four.taboldstyle/seven.taboldstyle (iii) Let pnbe the first column vector of the Jacobi matrix of Pn. Show that Pn/equalxrpn. (iv) Show that the Jacobi matrix of Pn+1is a(n+ 1)×(n+ 1)-matrix of the form DPn+1/equalx/parenleftBigg Au vw/parenrightBigg , where Ais an n×n-matrix, uis a column vector, vis a row vector and wis a function given respectively by A/equalx(cosθn)DPn, u/equalx−(sinθn)Pn, v/equalx(sinθn0 0···0), w/equalxrcosθn. (v) Show that det(DPn+1)/equalxrcosn−1θndet(DPn)forn≥1. (Expand det(DPn+1) with respect to the last row, using the formula in part ( iv), and apply the result of part ( iii).) (vi) Using the formula in part ( v) calculate det(DPn)forn/equalx1,2,3,4. (vii) Find an explicit formula for det(DPn)for general n. (viii) Show that det(DPn)/nequal0ifr/nequal0and−1 2π<θ i<1 2πfori/equalx2,3,...,n−1. /three.taboldstyle./two.taboldstyle/zero.taboldstyle. Letφ:Rn→Rnbe an orthogonal linear map. Prove that φ∗(∗α)/equalx∗φ∗(α)for all k-formsαonRn. CHAPTER /four.taboldstyle Integration of 1-forms Like functions, forms can be integrated as well as differenti ated. Differentiation and integration are related via a multivariable version of t he fundamental theorem of calculus, known as Stokes’ theorem. In this chapter we inv estigate the case of 1-forms. /four.taboldstyle./one.taboldstyle. Definition and elementary properties of the integral LetUbe an open subset of Rn. Apathorparametrized curve inUis a smooth mapping c:I→Ufrom an interval Iinto U. Our goal is to integrate forms over paths, so to avoid problems with improper integrals we will a ssume the interval I to be closed and bounded, I/equalx[a,b]. Letαbe a1-form on Uand let c: [a,b]→U be a path in U. The pullback c∗(α)is a1-form on [a,b], and can therefore be written asc∗(α)/equalxh dt, where tis the coordinate on Rand his a smooth function on [a,b]. Theintegral ofαover cis now defined by /integraldisplay cα/equalx/integraldisplay [a,b]c∗(α)/equalx/integraldisplayb ah(t)dt. More explicitly, writing αin components, α/equalx/summationtextn i/equalx1fidxi, we have c∗(α)/equalxn/summationdisplay i/equalx1c∗(fi)dci/equalxn/summationdisplay i/equalx1c∗(fi)dci dtdt, (/four.taboldstyle./one.taboldstyle) so/integraldisplay cα/equalxn/summationdisplay i/equalx1/integraldisplayb afi(c(t))c′ i(t)dt. /four.taboldstyle./one.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetUbe the punctured plane R2\{0}. Let c: [0,2π]→Ube the usual parametrization of the circle, c(t)/equalx(cost,sint), and letα0be the angle form, α0/equalx−y dx+x dy x2+y2. Then c∗(α0)/equalxdt(see Example /three.taboldstyle./eight.taboldstyle), so/integraltext cα0/equalx/integraltext2π 0dt/equalx2π. A path c: [a,b]→Ucan be reparametrized by substituting a new variable, t/equalxp(s), where sranges over another interval [¯a,¯b]. We shall assume pto be a one-to-one mapping from [¯a,¯b]onto [a,b]satisfying p′(s)/nequal0for¯a≤s≤¯b. Such apis called a reparametrization . The path c◦p: [¯a,¯b]→U has the same image as the original path c, but it is traversed at a different rate. Since p′(s)/nequal0for all s∈[¯a,¯b]we have either p′(s)>0for all s(in which case pis /four.taboldstyle/nine.taboldstyle /five.taboldstyle/zero.taboldstyle /four.taboldstyle. INTEGRATION OF 1-FORMS increasing) or p′(s)<0for all s(in which case pis decreasing). If pis increasing, we say that it preserves the orientation of the path (or that the paths cand c◦phave thesame orientation ); if pis decreasing, we say that it reverses the orientation (or that cand c◦phave opposite orientations ). In the orientation-reversing case, c◦p traverses the path in the opposite direction to c. /four.taboldstyle./two.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. The path c: [0,2π]→R2defined by c(t)/equalx(cost,sint)repre- sents the unit circle in the plane, traversed at a constant ra te (angular velocity) of 1radian per second. Let p(s)/equalx2s. Then pmaps [0,π]to[0,2π]and c◦p, re- garded as a map [0,π]→R2, represents the same circle, but traversed at 2radians per second. (It is important to restrict the domain of pto the interval [0,π]. If we allowed sto range over [0,2π], then (cos 2 s,sin 2 s)would traverse the circle twice. This is not considered a reparametrization of the ori ginal path c.) Now letp(s)/equalx−s. Then c◦p: [0,2π]→R2traverses the unit circle in the clockwise direction. This reparametrization reverses the orientati on; the angular velocity is now−1radian per second. Finally let p(s)/equalx2πs2. Then pmaps [0,1]to[0,2π] and c◦p: [0,1]→R2runs once counterclockwise through the unit circle, but at a variable angular velocity. It turns out that the integral of a form along a path is almost c ompletely independent of the parametrization. /four.taboldstyle./three.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. Letαbe a1-form on Uandc: [a,b]→Ua path in U. Let p: [¯a,¯b]→ [a,b]be a reparametrization. Then /integraldisplay c◦pα/equalx/integraltext cαifppreserves the orientation , −/integraltext cαifpreverses the orientation . P/r.sc/o.sc/o.sc/f.sc. It follows from the definition of the integral and from the nat urality of pullbacks (Proposition /three.taboldstyle./one.taboldstyle/zero.taboldstyle(iii)) that /integraldisplay c◦pα/equalx/integraldisplay [¯a,¯b](c◦p)∗(α)/equalx/integraldisplay [¯a,¯b]p∗(c∗(α)). Now let us write c∗(α)/equalxh dtand t/equalxp(s). Then p∗(c∗(α))/equalxp∗(g dt)/equalx(p∗(g)dp/equalx p∗(g)(dp/ds)ds, so /integraldisplay c◦pα/equalx/integraldisplay [¯a,¯b]p∗(g)dp dsds/equalx/integraldisplay¯b ¯ag(p(s))p′(s)ds. On the other hand,/integraltext cα/equalx/integraltextb ag(t)dt, so by the substitution formula, Theorem B./nine.taboldstyle, we have/integraltext c◦pα/equalx±/integraltext cα, where the +occurs if p′>0and the−ifp′<0. QED Interpretation of the integral. Integrals of 1-forms play an important role in physics and engineering. A path c: [a,b]→Umodels a particle travelling through the region U. Recall from Section /two.taboldstyle./five.taboldstylethat to a 1-formα/equalx/summationtextn i/equalx1Fidxicorresponds a vector field F/equalx/summationtextn i/equalx1Fiei, which can be thought of as a force field acting on the particle. Symbolically we write α/equalxF·dx, where we think of dxas an infinitesimal vector tangent to the path. Thus αrepresents the work done by the force field along an infinitesimal vector dx. From ( /four.taboldstyle./one.taboldstyle) we see that c∗(α)/equalxF(c(t))·c′(t)dt. We /four.taboldstyle./two.taboldstyle. INTEGRATION OF EXACT 1-FORMS /five.taboldstyle/one.taboldstyle define the work along the path cdone by the force Fto be the integral /integraldisplay cα/equalx/integraldisplay cF·dx/equalx/integraldisplayb aF(c(t))·c′(t)dt. In particular, the work done by the force is zero if the force i s perpendicular to the path, as in the picture on the left. The work done by the force i n the picture on the right is negative. c c Theorem /four.taboldstyle./three.taboldstylecan be translated into this language as follows: the work don e by the force does not depend on the rate at which the particle travel s along its path, but only on the path itself and on the direction of travel. The field Fisconservative if it can be written as the gradient of a function, F/equalxgrad (g). The function−gis called a potential for the field and is interpreted as the potential energy of the particle. In terms of forms this m eans thatα/equalxdg, i.e. αis exact. /four.taboldstyle./two.taboldstyle. Integration of exact 1-forms Integrating an exact 1-formα/equalxdgis easy once the function gis known. /four.taboldstyle./four.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc (/f.sc/u.sc/n.sc/d.sc/a.sc/m.sc/e.sc/n.sc/t.sc/a.sc/l.sc /t.sc/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc /o.sc/f.sc /c.sc/a.sc/l.sc/c.sc/u.sc/l.sc/u.sc/s.sc /i.sc/n.sc Rn).Letα/equalxdgbe an exact 1-form on an open subset UofRn. Let c: [a,b]→Ube a path. Then /integraldisplay cα/equalxg(c(b))−g(c(a)). P/r.sc/o.sc/o.sc/f.sc. By Theorem /three.taboldstyle./one.taboldstyle/one.taboldstyle we have c∗(α)/equalxc∗(dg)/equalxdc∗(g). Writing h(t)/equalx c∗(g)(t)/equalxg(c(t))we have c∗(α)/equalxdh, so /integraldisplay cα/equalx/integraldisplay [a,b]c∗(α)/equalx/integraldisplayb adh/equalxh(b)−h(a), where we used the (ordinary) fundamental theorem of calculu s, formula ( B./one.taboldstyle). Hence/integraltext cα/equalxg(c(b))−g(c(a)). QED The physical interpretation of this result is that when a par ticle moves in a conservative force field, its potential energy decreases by the amount of work done /five.taboldstyle/two.taboldstyle /four.taboldstyle. INTEGRATION OF 1-FORMS by the field. This clarifies what it means for a field to be conser vative: it means that the work done is entirely converted into mechanical ene rgy and that none is dissipated by friction into heat, radiation, etc. Thus the f undamental theorem of calculus “explains” the law of conservation of energy. Theorem /four.taboldstyle./four.taboldstylealso gives us a necessary criterion for a 1-form on Uto be exact. A path c: [a,b]→Uis called closed ifc(a)/equalxc(b). /four.taboldstyle./five.taboldstyle. C/o.sc/r.sc/o.sc/l.sc/l.sc/a.sc/r.sc/y.sc. Letαbe an exact 1-form defined on an open subset UofRn. Then/integraltext cα/equalx0for every closed path cinU. P/r.sc/o.sc/o.sc/f.sc. Letc: [a,b]→Ube a closed path and let gbe a smooth function on U satisfying dg/equalxα. Then/integraltext cα/equalxg(c(b))−g(c(a))/equalx0by Theorem /four.taboldstyle./four.taboldstyle. QED This corollary can be used to detect closed 1-forms that are not exact. /four.taboldstyle./six.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. The angle form α0∈Ω1(R2\{0})of Example /two.taboldstyle./one.taboldstyle/zero.taboldstyleis closed, but not exact. Indeed, its integral around the circle is 2π/nequal0, so by Corollary /four.taboldstyle./five.taboldstyleα0 is not exact. Note the contrast with closed 1-forms on Rn, which are always exact! (See Exercise /two.taboldstyle./eight.taboldstyle.) The main content of the next theorem is that the necessary cri terion of Corollary /four.taboldstyle./five.taboldstyleis in fact sufficient for a 1-form to be exact. /four.taboldstyle./seven.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. Letαbe a1-form on a connected open subset UofRn. Then the following statements are equivalent. (i)αis exact. (ii)/integraltext cα/equalx0for all closed paths c. (iii)/integraltext cαdepends only on the endpoints of cfor every path cinU. P/r.sc/o.sc/o.sc/f.sc. (i)/equalx⇒(ii): this is Corollary /four.taboldstyle./five.taboldstyle. (ii)/equalx⇒(iii): assume/integraltext cα/equalx0for all closed paths c. Let c1: [a1,b1]→U and c2: [a2,b2]→U be two paths with the same endpoints, i.e. c1(a1)/equalxc2(a2)and c1(b1)/equalxc2(b2). We need to show that/integraltext c1α/equalx/integraltext c2α. After reparametrizing c1and c2we may assume that a1/equalxa2/equalx0and b1/equalxb2/equalx1. Define a new path cby c(t)/equalxc1(t) for0≤t≤1, c2(2−t)for1≤t≤2. (First traverse c1, then traverse c2backwards.) Then cis closed, so/integraltext cα/equalx0. But Theorem /four.taboldstyle./three.taboldstyleimplies/integraltext cα/equalx/integraltext c1α−/integraltext c2α, so/integraltext c1α/equalx/integraltext c2α. (iii)/equalx⇒(i): assume that, for all c,/integraltext cαdepends only on the endpoints of c. We must define a function gsuch thatα/equalxdg. Fix a point x0inU. For each point xinUchoose a path cx: [0,1]→Uwhich joins x0tox. Define g(x)/equalx/integraldisplay cxα. We assert that dgis well-defined and equal to α. Writeα/equalx/summationtextn i/equalx1fidxi. We must show that∂g/∂xi/equalxfi. From the definition of partial differentiation, ∂g ∂xi(x)/equalxlim h→0g(x+hei)−g(x) h/equalxlim h→01 h/parenleftbigg/integraldisplay cx+heiα−/integraldisplay cxα/parenrightbigg . /four.taboldstyle./two.taboldstyle. INTEGRATION OF EXACT 1-FORMS /five.taboldstyle/three.taboldstyle Now consider a path ˜ccomposed of two pieces: for −1≤t≤0travel from x0tox along the path cxand then for 0≤t≤1travel from xtox+heialong the straight line given by l(t)/equalxx+thei. Then ˜chas the same endpoints as cx+hei. Therefore/integraltext cx+heiα/equalx/integraltext ˜cα, and hence ∂g ∂xi(x)/equalxlim h→01 h/parenleftbigg/integraldisplay ˜cα−/integraldisplay cxα/parenrightbigg /equalxlim h→01 h/parenleftbigg/integraldisplay cxα+/integraldisplay lα−/integraldisplay cxα/parenrightbigg /equalxlim h→01 h/integraldisplay lα/equalxlim h→01 h/integraldisplay [0,1]l∗(α).(/four.taboldstyle./two.taboldstyle) Letδi,jbe the Kronecker delta , which is defined by δi,i/equalx1andδi,j/equalx0ifi/nequalj. Then we can write lj(t)/equalxxj+δi,jth, and hence l′ j(t)/equalxδi,jh. This shows that l∗(α)/equalxn/summationdisplay j/equalx1fj(x+thei)dlj/equalxn/summationdisplay j/equalx1fj(x+thei)l′ j(t)dt /equalxn/summationdisplay j/equalx1fj(x+thei)δi,jh dt/equalxh fi(x+thei)dt.(/four.taboldstyle./three.taboldstyle) Taking equations ( /four.taboldstyle./two.taboldstyle) and ( /four.taboldstyle./three.taboldstyle) together we find ∂g ∂xi(x)/equalxlim h→01 h/integraldisplay1 0h fi(x+thei)dt/equalxlim h→0/integraldisplay1 0fi(x+thei)dt /equalx/integraldisplay1 0lim h→0fi(x+thei)dt/equalx/integraldisplay1 0fi(x)dt/equalxfi(x). This formula shows that gis smooth and that dg/equalxα. This proves that ( iii)/equalx⇒ (i), and at the same time it proves that the function g(x)/equalx/integraltext cxαis an antiderivative ofα. QED Notice that the proof of the theorem tells us how to find an anti derivative of an exact 1-formα, namely by integrating αalong an arbitrary path running from a base point x0to a variable point x. This useful fact deserves to be recorded separately. /four.taboldstyle./eight.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. Letαbe an exact 1-form defined on a connected open subset UofRn. Choose a base point x0inUand for each xinUchoose a path cxjoining x0tox. Then the function g(x)/equalx/integraltext cxαis smooth and satisfies dg/equalxα. /four.taboldstyle./nine.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Letα/equalx/summationtextn i/equalx1fidxibe a closed 1-form defined on all of Rn. Then we know that αis exact. One method for finding an antiderivative is explain ed in Exercise /two.taboldstyle./eight.taboldstyle. Theorem /four.taboldstyle./eight.taboldstylesuggests a quicker way: let us choose the origin 0 to be the base point of Rnand for each xletcx: [0,1]→Rnbe the straight path cx(t)/equalxtx. Then g(x)/equalx/integraltext cxα/equalx/integraltext1 0c∗ x(α)is an antiderivative of α. Since c∗ x(α)/equalxn/summationdisplay i/equalx1fi(tx)d(txi)/equalxn/summationdisplay i/equalx1xifi(tx)dt we arrive at g(x)/equalxn/summationdisplay i/equalx1xi/integraldisplay1 0fi(tx)dt. /five.taboldstyle/four.taboldstyle /four.taboldstyle. INTEGRATION OF 1-FORMS (In Exercise /four.taboldstyle./three.taboldstyleyou will be asked to verify directly that dg/equalxα.) For instance, let α/equalxy dx+(zcosyz+x)dy+ycosyz dz be the closed 1-form of Example /two.taboldstyle./nine.taboldstyle. Then g(x)/equalxx/integraldisplay1 0t y dt +y/integraldisplay1 0(tzcost2yz+tx)dt+z/integraldisplay1 0t ycost2yz dt /equalx2x y/integraldisplay1 0t dt+ 2yz/integraldisplay1 0tcost2yz dt /equalxx y+ sin yz. See Exercises /four.taboldstyle./four.taboldstyle–/four.taboldstyle./six.taboldstylefor further applications of this theorem. /four.taboldstyle./three.taboldstyle. Angle functions and the winding number In this section we will have a closer look at the angle form and see that it carries interesting information of a topological nature. Letx/equalx(x,y)be a nonzero vector in the plane and let θ(x)be the angle between the positive x-axis and x. Elementary trigonometry tells us that cosθ(x)/equalxx/radicalbig x2+y2, sinθ(x)/equalxy/radicalbig x2+y2. (/four.taboldstyle./four.taboldstyle) These equations determine θ(x)up to an integer multiple of 2π, and we will call any particular solution a choice of angle for x. Now let Ube an open subset of the punctured plane R2\{0}. Is it possible to make a choice of angle θ(x)for each xinUwhich varies smoothly with x? For U/equalxR2\{0}this would appear to be impossible: the usual choice of angle in the punctured plane ( 0≤θ(x)<2π) has a discontinuity along the positive x-axis, and there seems to be no way to get rid of this discontinuity by making a cleverer choice of angle. But it may be possible if Uis a smaller open subset, such as the complement of the positive x-axis in R2\{0}. Let us define an angle function on U to be a smooth function θ:U→Rwith the property that θ(x)has property ( /four.taboldstyle./four.taboldstyle) for all x∈U. Our next result states that an angle function on Uexists if and only ifα0is exact on U, whereα0is the angle form α0/equalx−y dx+x dy x2+y2 introduced in Example /two.taboldstyle./one.taboldstyle/zero.taboldstyle. /four.taboldstyle./one.taboldstyle/zero.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. LetUbe a connected open subset of R2\{0}. (i)Letθ:U→Rbe an angle function. Then dθ/equalxα0. (ii)Assume thatα0is exact on U. Then there exists an angle function θonU, which can be found as follows: choose a base point x0inU, choose an angle θ0forx0, and for each x∈Uchoose a path cxinUfromx0tox. Thenθ(x)/equalxθ0+/integraltext cxα0. P/r.sc/o.sc/o.sc/f.sc. (i) Define functions ξandηon the punctured plane by ξ(x,y)/equalxx/radicalbig x2+y2, η (x,y)/equalxy/radicalbig x2+y2. (/four.taboldstyle./five.taboldstyle) Observe that /parenleftBigg ξ(x) η(x)/parenrightBigg /equalxx /bardblx/bardbl, (/four.taboldstyle./six.taboldstyle) /four.taboldstyle./three.taboldstyle. ANGLE FUNCTIONS AND THE WINDING NUMBER /five.taboldstyle/five.taboldstyle the unit vector pointing in the direction of x/equalx(x,y). In particular ξ(x)2+η(x)2/equalx1. (/four.taboldstyle./seven.taboldstyle) We will use the result α0/equalx−ηdξ+ξdη (/four.taboldstyle./eight.taboldstyle) of Exercise /two.taboldstyle./four.taboldstyle. Ifθis an angle function on U, thenξ(x)/equalxcosθ(x)andη(x)/equalx sinθ(x)for all xinU. Substituting this into ( /four.taboldstyle./eight.taboldstyle) gives α0/equalx−sinθdcosθ+ cosθdsinθ/equalxsin2θdθ+ cos2θdθ/equalxdθ. (ii) Now assume that α0is exact on U. It follows from Theorem /four.taboldstyle./eight.taboldstylethat the function defined by θ(x)/equalxθ0+/integraltext cxα0is smooth and satisfies dθ/equalxα0. (/four.taboldstyle./nine.taboldstyle) To prove that θis an angle function on Uit is enough to show that the difference vector/parenleftBigg cosθ(x) sinθ(x)/parenrightBigg −/parenleftBigg ξ(x) η(x)/parenrightBigg has length 0for all x. The length squared of the difference vector is equal to (cosθ−ξ)2+(sinθ−η)2/equalxcos2θ+ sin2θ−2(ξcosθ+ηsinθ)+ξ2+η2 /equalx2−2(ξcosθ+ηsinθ), where we used ( /four.taboldstyle./seven.taboldstyle). Hence we need to show that the function u(x)/equalxξ(x)cosθ(x)+η(x)sinθ(x) is a constant equal to 1. For x/equalxx0we have u(x0)/equalxξ(x0)cosθ0+η(x0)sinθ0/equalxcos2θ0+ sin2θ0/equalx1, because by assumption θ0is a choice of angle for x0. Furthermore, the exterior derivative of uis du/equalxcosθdξ−ξsinθdθ+ sinθdη+ηcosθdθ /equalxcosθdξ+ sinθdη+(−ξsinθ+ηcosθ)α0 by (/four.taboldstyle./nine.taboldstyle) /equalxcosθdξ+ sinθdη+(−ξsinθ+ηcosθ)(−ηdξ+ξdη) by (/four.taboldstyle./eight.taboldstyle) /equalx/parenleftbig(1−η2)cosθ+ξηsinθ/parenrightbigdξ+/parenleftbigξηcosθ+(1−ξ2)sinθ/parenrightbigdη) /equalx(ξcosθ+ηsinθ)ξdξ+(ξcosθ+ηsinθ))ηdη by (/four.taboldstyle./seven.taboldstyle) /equalx1 2(ξcosθ+ηsinθ)d(ξ2+η2) /equalx0 by (/four.taboldstyle./seven.taboldstyle). Hence, by Lemma /three.taboldstyle./one.taboldstyle/two.taboldstyle,uis a constant function, so u(x)/equalx1for all xinU. QED /four.taboldstyle./one.taboldstyle/one.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. There exists no angle function on the punctured plane, becau se α0is not exact on R2\{0}(as was shown in Example /four.taboldstyle./six.taboldstyle). /five.taboldstyle/six.taboldstyle /four.taboldstyle. INTEGRATION OF 1-FORMS Angle functions along a path. Now let us modify the angle problem by con- sidering a path c: [a,b]→R2\{0}in the punctured plane and wondering whether we can make a choice of angle for c(t)which depends smoothly on t∈[a,b]. We define an angle function along cto be a smooth function ϑ: [a,b]→Rwith the property that cosϑ(t)/equalxξ(c(t)), sinϑ(t)/equalxη(c(t)) for all t∈[a,b]. The following example shows the difference between this not ion and that of an angle function on an open subset. /four.taboldstyle./one.taboldstyle/two.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Define c: [a,b]→R2\{0}byc(t)/equalx(cost,sint). This is a path travelling along the unit circle at constant angular veloci ty1from time ato time b. The function ϑ(t)/equalxtis an angle function along c. The difference ϑ(b)−ϑ(a)/equalxb−a is the total angle swept out by the path. In fact an angle function exists along every path in the punct ured plane. /four.taboldstyle./one.taboldstyle/three.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. Letc: [a,b]→R2\{0}be a path. Choose an angle ϑaforc(a)and fora≤t≤bdefine ϑ(t)/equalxϑa+/integraldisplayt ac∗(α0). Theϑis an angle function along c. P/r.sc/o.sc/o.sc/f.sc. This proof is nearly the same as that of Theorem /four.taboldstyle./one.taboldstyle/zero.taboldstyle, so we will skip some details. We will show that the difference vector /parenleftBigg cosϑ(t) sinϑ(t)/parenrightBigg −/parenleftBigg ξ(c(t)) η(c(t))/parenrightBigg has length 0for all t∈[a,b]. The square of the length is equal to (cosϑ−f)2+(sinϑ−g)2/equalxcos2ϑ+ sin2ϑ−2(fcosϑ+gsinϑ)+f2+g2 /equalx2−2u, where we introduced the abbreviations f(t)/equalxξ(c(t)), g(t)/equalxη(c(t)), u(t)/equalxf(t)cosϑ(t)+g(t)sinϑ(t), and where we used that f2+g2/equalx1. (/four.taboldstyle./one.taboldstyle/zero.taboldstyle) Therefore it is enough to show that the function uis a constant equal to 1. Now ϑ(a)/equalxϑa, so f(a)/equalxcosϑ(a)and g(a)/equalxsinϑ(a)(becauseϑais a choice of angle forc(a)), and hence u(a)/equalxcos2ϑa+ sin2ϑa/equalx1. So we can finish the proof by showing that u′(t)/equalx0for all t. Using ( /four.taboldstyle./eight.taboldstyle) we find c∗(α0)/equalxc∗(−ηdξ+ξdη)/equalx−c∗(η)dc∗(ξ)+c∗(ξ)dc∗(η) /equalx−g d f+f dg/equalx(−g f′+f g′)dt, and therefore, by the fundamental theorem of calculus, ϑ′/equalx−g f′+f g′. (/four.taboldstyle./one.taboldstyle/one.taboldstyle) /four.taboldstyle./three.taboldstyle. ANGLE FUNCTIONS AND THE WINDING NUMBER /five.taboldstyle/seven.taboldstyle This yields u′/equalxf′cosϑ+g′sinϑ+(−fsinϑ+gcosϑ)ϑ′ /equalxf′cosϑ+g′sinϑ+(−fsinϑ+gcosϑ)(−g f′+f g′) by (/four.taboldstyle./one.taboldstyle/one.taboldstyle) /equalx/parenleftbig(1−g2)cosϑ+f gsinϑ/parenrightbigf′+/parenleftbigf gcosϑ+(1−f2)sinϑ/parenrightbigg′ /equalx(fcosϑ+gsinϑ)f f′+(fcosϑ+gsinϑ))g g′by (/four.taboldstyle./one.taboldstyle/zero.taboldstyle) /equalx1 2(fcosϑ+gsinϑ)(f2+g2)′ /equalx0 by (/four.taboldstyle./one.taboldstyle/zero.taboldstyle). QED It is helpful to think of the vector (ξ(c(t)),η(c(t)))/equalxc(t)//bardblc(t)/bardblas a compass held by a traveller wandering through a magnetized puncture d plane. The punc- ture represents the magnetic north pole, so the needle indic ates the direction of the traveller’s position vector c(t). 0 0 Astincreases from atob, the needle starts at the angle ϑ(a)/equalxϑa, it moves around the compass, and ends up at the final angle ϑ(b). The difference ϑ(b)−ϑ(a) measures the net angle swept out by the needle, where a counte rclockwise motion counts as positive and a clockwise motion counts as negative . The formula for ϑgiven by Theorem /four.taboldstyle./one.taboldstyle/three.taboldstyleshows that this difference can also be expressed as an integral, ϑ(b)−ϑ(a)/equalx/integraldisplayb ac∗(α0)/equalx/integraldisplay cα0. (/four.taboldstyle./one.taboldstyle/two.taboldstyle) /four.taboldstyle./one.taboldstyle/four.taboldstyle. C/o.sc/r.sc/o.sc/l.sc/l.sc/a.sc/r.sc/y.sc. Ifc: [a,b]→Uis a closed path, then/integraltext cα0/equalx2πk, where kis an integer. P/r.sc/o.sc/o.sc/f.sc. Because of ( /four.taboldstyle./one.taboldstyle/two.taboldstyle) it suffices to show that ϑ(b)−ϑ(a)/equalx2πk. By Theorem /four.taboldstyle./one.taboldstyle/three.taboldstyle,ϑis an angle function along c, so the assumption that cis closed ( c(a)/equalxc(b)) implies/parenleftBigg cosϑ(a) sinϑ(a)/parenrightBigg /equalx/parenleftBigg ξ(c(a)) η(c(a))/parenrightBigg /equalx/parenleftBigg ξ(c(b)) η(c(b)/parenrightBigg /equalx/parenleftBigg cosϑ(b) sinϑ(b)/parenrightBigg . In other words cosϑ(a)/equalxcosϑ(b)andsinϑ(a)/equalxsinϑ(b), soϑ(a)andϑ(b)differ by an integer multiple of 2π. QED /five.taboldstyle/eight.taboldstyle /four.taboldstyle. INTEGRATION OF 1-FORMS The integer k/equalx(2π)−1/integraltext cα0measures how many times the path loops around the origin. It is called the winding number of the closed path cabout the origin, and we will denote it by w(c,0). w(c,0)/equalxwinding number of cabout origin /equalx1 2π/integraldisplay cα0. (/four.taboldstyle./one.taboldstyle/three.taboldstyle) /four.taboldstyle./one.taboldstyle/five.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. It follows from the calculation in Example /four.taboldstyle./one.taboldstylethat the winding number of the circle c(t)/equalx(cost,sint)(0≤t≤2π) is equal to 1. Exercises /four.taboldstyle./one.taboldstyle.Consider the path c: [0,1 2π]→R2defined by c(t)/equalx(acost,bsint), where aand bare positive constants. Let α/equalxxy dx +x2y dy. (i) Sketch the path cfora/equalx2and b/equalx1. (ii) Find/integraltext cα(for arbitrary aand b). /four.taboldstyle./two.taboldstyle.Restate Theorem /four.taboldstyle./seven.taboldstylein terms of force fields, potentials and energy. Explain why the result is plausible on physical grounds. /four.taboldstyle./three.taboldstyle.Letα/equalx/summationtextn i/equalx1fidxibe aclosed 1-form defined on all of Rn. (i) Verify that the function g(x)/equalx/summationtextn i/equalx1xi/integraltext1 0fi(tx)dtgiven in Example /four.taboldstyle./nine.taboldstylesatisfies dg/equalxα. (ii) In Exercise /two.taboldstyle./eight.taboldstylea different formula for the antiderivative gofαwas given. Show that that formula can be interpreted as an integral g(x)/equalx/integraltext cxαofαalong a suitable path cxfrom the origin to x. /four.taboldstyle./four.taboldstyle.Consider the 1-formα/equalx/bardblx/bardbla/summationtextn i/equalx1xidxionRn\{0}, where ais a real constant. For every x/nequal0letcxbe the line segment starting at the origin and ending at x. (i) Show that αis closed for any value of a. (ii) Determine for which values of athe function g(x)/equalx/integraltext cxαis well-defined and compute it. (iii) For the values of ayou found in part ( ii) check that dg/equalxα. /four.taboldstyle./five.taboldstyle. Letα∈Ω1(Rn\{0})be the 1-form of Exercise /four.taboldstyle./four.taboldstyle. Now let cxbe the half-line pointing from xradially outward to infinity. Parametrize cxby travelling from infinity inward to x. (You can do this by using an infinite time interval (−∞,0]in such a way that cx(0)/equalxx.) (i) Determine for which values of athe function g(x)/equalx/integraltext cxαis well-defined and compute it. (ii) For the values of ayou found in part ( i) check that dg/equalxα. (iii) Show how to recover from this computation the potentia l energy for Newton’s gravitational force. (See Exercise B./five.taboldstyle.) /four.taboldstyle./six.taboldstyle. Letα∈Ω1(Rn\{0})be as in Exercise /four.taboldstyle./four.taboldstyle. There is one value of awhich is not covered by Exercises /four.taboldstyle./four.taboldstyleand/four.taboldstyle./five.taboldstyle. For this value of afind a smooth function gonRn−{0} such that dg/equalxα. /four.taboldstyle./seven.taboldstyle.Calculate directly from the definition the winding number ab out the origin of the path c: [0,2π]→R2given by c(t)/equalx(coskt,sinkt). /four.taboldstyle./eight.taboldstyle.Letx0be a point in R2and ca closed path which does not pass through x0. How would you define the winding number w(c,x0)ofcaround x0? Try to formulate two different definitions: a “geometric” definition and a formula in terms of an integral over c of a certain 1-form analogous to formula ( /four.taboldstyle./one.taboldstyle/three.taboldstyle). EXERCISES /five.taboldstyle/nine.taboldstyle /four.taboldstyle./nine.taboldstyle.Letc: [0,1]→R2\{0}be a closed path with winding number k. Determine the winding numbers of the following paths ˜c: [0,1]→R2\{0}by using the formula, and then explain the answer by appealing to geometric intuition. (i)˜c(t)/equalxc(1−t); (ii)˜c(t)/equalxρ(t)c(t), whereρ: [0,1]→(0,∞)is a function satisfying ρ(0)/equalxρ(1); (iii) ˜c(t)/equalx/bardblc(t)/bardbl−1c(t); (iv) ˜c(t)/equalxφ(c(t)), whereφ(x,y)/equalx(y,x); (v)˜c(t)/equalxφ(c(t)), whereφ(x,y)/equalx1 x2+y2(x,−y). /four.taboldstyle./one.taboldstyle/zero.taboldstyle. For each of the following closed paths c: [0,2π]→R2\{0}set up the integral defining the winding number about the origin. Evaluate the in tegral if you can (but don’t give up too soon). If not, sketch the path (the use of software is allowed) and obtain the answer geometrically. (i)c(t)/equalx(acost,bsint), where a>0and b>0; (ii)c(t)/equalx(cost−2,sint); (iii) c(t)/equalx(cos3t,sin3t); (iv) c(t)/equalx/parenleftbig(acost+b)cost+(b−a)/2,(acost+b)sint/parenrightbig, where 0<b<a. /four.taboldstyle./one.taboldstyle/one.taboldstyle. Letb>0and a/nequal0be constants with |a|/nequalb. Define a path c: [0,2π]→R2\{0} by c(t)/equalx/parenleftbigg (a+b)cost+acosa+b at,(a+b)sint+asina+b at/parenrightbigg . (i) Sketch the path cfora/equalx±b/3. (ii) For what values of aand bis the path closed? (iii) Assume cis closed. Set up the integral defining the winding number of caround the origin and evaluate it. If you get stuck, find the answer ge ometrically. /four.taboldstyle./one.taboldstyle/two.taboldstyle. LetUbe an open subset of R2and let F/equalxF1e1+F2e2:U→R2be a smooth vector field. The differential form β/equalxF1dF2−F2dF1 F2 1+F2 2 is well-defined at all points xofUwhere F(x)/nequal0. Let cbe a parametrized circle contained inU, traversed once in the counterclockwise direction. Assume thatF(x)/nequal0for all x∈c. Theindex ofFrelative to cis index (F,c)/equalx1 2π/integraldisplay cβ. Prove the following assertions. (i)β/equalxF∗(α0), whereα0is the angle form (−y dx+x dy)/slashbig/parenleftbigx2+y2/parenrightbig; (ii)βis closed; (iii)index (F,c)/equalxw(F◦c,0), the winding number of the path F◦cabout the origin; (iv)index (F,c)is an integer. /six.taboldstyle/zero.taboldstyle /four.taboldstyle. INTEGRATION OF 1-FORMS /four.taboldstyle./one.taboldstyle/three.taboldstyle. (i) Find the indices of the following vector fields around the indicated circles. (ii) Draw diagrams of three vector fields in the plane with res pective indices 0,2and 4around suitable circles. /four.taboldstyle./one.taboldstyle/four.taboldstyle. Letc: [a,b]→R2be a path. For each t∈[a,b]the velocity vector c′(t)is tangent tocat the point c(t). The map c′: [a,b]→R2is called the derived path ofc. Let us assume that both cand c′are closed, i.e. c(a)/equalxc(b)and c′(a)/equalxc′(b). Let us also assume that the path cisregular in the sense that c′(t)/nequal0for all t. The quantity τ(c)/equalx1 2π/integraldisplay c′α0 is then well-defined and is called the turning number ofc. Hereα0denotes the angle form onR2\{0}. (i) Show that the turning number is an integer. Discuss its ge ometric meaning and how it differs from the winding number. (ii) Show that the turning number is independent of the direc tion of travel in the sense thatτ(˜c)/equalxτ(c), where ˜c(t)/equalxc(a+b−t)fora≤t≤b. EXERCISES /six.taboldstyle/one.taboldstyle (iii) For each of the following four paths cfind the turning number τ(c), as well as the winding number w(c,•)about the black point •. CHAPTER /five.taboldstyle Integration and Stokes’ theorem /five.taboldstyle./one.taboldstyle. Integration of forms over chains In this chapter we generalize the theory of Chapter /four.taboldstyleto higher dimensions. In the same way that 1-forms are integrated over parametrized curves, k-forms can be integrated over k-dimensional parametrized regions. Let Ube an open subset of Rnand letαbe a k-form on U. The simplest k-dimensional analogue of an interval is arectangular block inRkwhose edges are parallel to the coordinate axes. This is a set of the form R/equalx[a1,b1]×[a2,b2]×···× [ak,bk]/equalx{t∈Rk|ai≤ti≤bifor1≤i≤k}, where ai<bi. The k-dimensional analogue of a parametrized path is a smooth map c:R→U. Although the image c(R)may look very different from the block R, we think of the map cas a parametrization of the subset c(R)ofU: each choice of a point tinRgives rise to a point c(t)inc(R). The pullback c∗(α)is ak-form onRand therefore looks like h(t)dt1dt2···dtkfor some function h:R→R. The integral ofαover cis defined by /integraldisplay cα/equalx/integraldisplay Rc∗(α)/equalx/integraldisplaybk ak···/integraldisplayb2 a2/integraldisplayb1 a1h(t)dt1dt2···dtk. (The definition of the integral makes sense if we replace the r ectangular block R by more general shapes in Rk, such as skew blocks, k-dimensional balls, cylinders, etc. In fact any compact subset of Rkwill do.) Fork/equalx1this reproduces the definition given in Chapter /four.taboldstyle. The case k/equalx0is also worth examining. A zero-dimensional “block” Rin R0/equalx{0}is just the point 0. We can therefore think of a map c:R→Uas a collection{x}consisting of a single point x/equalxc(0)inU. The integral of a 0-form (function) fover cis by definition the value of fatx, /integraldisplay cf/equalxf(x). As in the one-dimensional case, integrals of k-forms are almost completely unaffected by a change of variables. Let ¯R/equalx[¯a1,¯b1]×[¯a2,¯b2]×···× [¯ak,¯bk] be a second rectangular block. A reparametrization is a map p:¯R→Rsatisfying the following conditions: pis bijective (i.e. one-to-one and onto) and the k×k-matrix Dp(s)is invertible for all s∈¯R. Then det(Dp(s))/nequal0for all s∈¯R, so either det(Dp(s))>0for all sordet(Dp(s))<0for all s. In these cases we say that the reparametrizion preserves , respectively reverses the orientation of c. /six.taboldstyle/three.taboldstyle /six.taboldstyle/four.taboldstyle /five.taboldstyle. INTEGRATION AND STOKES’ THEOREM /five.taboldstyle./one.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. Letαbe ak-form on Uandc:R→Ua smooth map. Let p:¯R→R be a reparametrization. Then /integraldisplay c◦pα/equalx/integraltext cαifppreserves the orientation , −/integraltext cαifpreverses the orientation . P/r.sc/o.sc/o.sc/f.sc. Almost verbatim the same proof as for k/equalx1(Theorem /four.taboldstyle./three.taboldstyle). It follows from the definition of the integral and from the naturality of pullbacks, Proposition /three.taboldstyle./one.taboldstyle/zero.taboldstyle(iii), that/integraldisplay c◦pα/equalx/integraldisplay ¯R(c◦p)∗(α)/equalx/integraldisplay ¯Rp∗(c∗(α)). Now let us write c∗(α)/equalxh dt 1dt2···dtkandt/equalxp(s). Then p∗(c∗(α))/equalxp∗(h dt 1dt2···dtk)/equalxp∗(h)det(Dp)ds1ds2···dsk by Theorem /three.taboldstyle./one.taboldstyle/four.taboldstyle, so /integraldisplay c◦pα/equalx/integraldisplay ¯Rh(p(s))det(Dp(s))ds1ds2···dsk. On the other hand,/integraltext cα/equalx/integraltext Rh(t)dt1dt2···dtk, so by the substitution formula, Theorem B./nine.taboldstyle, we have/integraltext c◦pα/equalx±/integraltext cα, where the +occurs if det(Dp)>0and the− ifdet(Dp)<0. QED /five.taboldstyle./two.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Theunit interval is the interval [0,1]in the real line. Any path c: [a,b]→Ucan be reparametrized to a path c◦p: [0,1]→Uby means of the reparametrization p(s)/equalx(b−a)s+a. Similarly, the unit cube inRkis the rectangular block [0,1]k/equalx{t∈Rk|ti∈[0,1]for1≤i≤k}. LetRbe any other block, given by ai≤ti≤bi. Define p: [0,1]k→Rbyp(s)/equalx As+a, where A/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAb1−a1 0... 0 0 b2−a2... 0 ............ 0 0 ... bk−ak/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtAand a/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAa1 a2 ... ak/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA. (“Squeeze the unit cube until it has the same edgelengths as Rand then translate it to the same position as R.”) Then pis one-to-one and onto and Dp(s)/equalxA, sodet(Dp(s))/equalxdet(A)/equalxvol(R)>0for all s, so pis an orientation-preserving reparametrization. Hence/integraltext c◦pα/equalx/integraltext cαfor any k-formαonU. /five.taboldstyle./three.taboldstyle. R/e.sc/m.sc/a.sc/r.sc/k.sc. A useful fact you learned in calculus is that one may intercha nge the order of integration in a multiple integral, as in the for mula /integraldisplayb2 a2/integraldisplayb1 a1f(t1,t2)dt1dt2/equalx/integraldisplayb1 a1/integraldisplayb2 a2f(t1,t2)dt2dt1. (/five.taboldstyle./one.taboldstyle) (This follows for instance from the substitution formula, T heorem B./nine.taboldstyle.) On the other hand, we have also learned that f(t1,t2)dt2dt1/equalx−f(t1,t2)dt1dt2. How can this be squared with formula ( /five.taboldstyle./one.taboldstyle)? The explanation is as follows. Let α/equalxf(t1,t2)dt1dt2. Then the left-hand side of formula ( /five.taboldstyle./one.taboldstyle) is the integral of αover c: [a1,b1]×[a2,b2]→ /five.taboldstyle./one.taboldstyle. INTEGRATION OF FORMS OVER CHAINS /six.taboldstyle/five.taboldstyle R2, the parametrization of the rectangle given by c(t1,t2)/equalx(t1,t2). The right-hand side is the integral of −αnot over c, but over c◦p, where p: [a2,b2]×[a1,b1]→[a1,b1]×[a2,b2] is the reparametrization p(s1,s2)/equalx(s2,s1). Since preverses the orientation, Theo- rem/five.taboldstyle./one.taboldstylesays that/integraltext c◦pα/equalx−/integraltext cα; in other words/integraltext cα/equalx/integraltext c◦p(−α), which is exactly formula ( /five.taboldstyle./one.taboldstyle). Analogously we have /integraldisplay [0,1]kf(t1,t2,..., tk)dt1dt2···dtk/equalx/integraldisplay [0,1]kf(t1,t2,..., tk)dtidt1dt2···/hatwidedti···dtk for any i. We see from Example /five.taboldstyle./two.taboldstylethat an integral over any rectangular block can be written as an integral over the unit cube. For this reason, fr om now on we shall usually take Rto be the unit cube. A smooth map c: [0,1]k→Uis called a k-cube inU(or sometimes a singular k-cube, the word singular meaning that the map cis not assumed to be one-to-one, so that the image can have self- intersections.) It is often necessary to integrate over regions that are made up of several pieces. Ak-chain inUis a formal linear combination of k-cubes, c/equalxa1c1+a2c2+···+apcp, where a1,a2,...,apare real coefficients and c1,c2,...,cparek-cubes. For any k-formαwe then define /integraldisplay cα/equalxp/summationdisplay q/equalx1aq/integraldisplay cqα. (In the language of linear algebra, the k-chains form a vector space with a basis consisting of the k-cubes. Integration, which is a priori only defined on cubes, is extended to chains in such a way as to be linear.) Recall that a 0-cube is nothing but a singleton {x}consisting of a single point xinU. Thus a 0-chain is a formal linear combination of points, c/equalx/summationtextp q/equalx1aq{xq}. A good way to think of a 0-chain cis as a collection of ppoint charges, with an electric charge aqplaced at the point xq. (You must carefully distinguish between the formal linear combination/summationtextp q/equalx1aq{xq}, which represents a distribution of point charges, and the linear combination of vectors/summationtextp q/equalx1aqxq, which represents a vector inRn.) The integral of a function fover the 0-chain is by definition /integraldisplay cf/equalxp/summationdisplay q/equalx1aq/integraldisplay {xq}f/equalxp/summationdisplay q/equalx1aqf(xq). Likewise, a k-chain/summationtextp q/equalx1aqcqcan be pictured as a charge distribution, with an electric charge aqspread along the k-dimensional “patch” cq. /six.taboldstyle/six.taboldstyle /five.taboldstyle. INTEGRATION AND STOKES’ THEOREM /five.taboldstyle./two.taboldstyle. The boundary of a chain Consider a path (“ 1-cube”) c: [0,1]→U. Its boundary is by definition the 0-chain∂cdefined by∂c/equalx{c(1)}−{ c(0)}. −{c(0)}+{c(1)} c We will define the boundary ∂cof a k-cube c: [0,1]k→Ufork≥1similarly, namely as an alternating sum over the k−1-dimensional faces of c. There are 2k such faces, which can be conveniently labelled as follows. F ort/equalx(t1,t2,..., tk−1) in[0,1]k−1and for i/equalx1,2,...,kput ci,0(t)/equalxc(t1,t2,..., ti−1,0,ti,..., tk−1), ci,1(t)/equalxc(t1,t2,..., ti−1,1,ti,..., tk−1). (“Insert 0, resp. 1in the i-th slot”.) For instance, a 2-cube c: [0,1]2→Uhas four edges, namely the “left” edge c1,0, the “right” edge c1,1, the “bottom” edge c2,0, and the “top” edge c2,1. c c2,0c1,1c2,1 c1,0 The picture suggests that we should define ∂c/equalxc2,0+c1,1−c2,1−c1,0, because the orientation of the 2-cube goes “with” the edges c2,0and c1,1and “against” the edges c2,1and c1,0. (Alternatively we could reverse the orientations of the to p and left edges by defining ¯c2,1(t)/equalxc(1−t,1)and ¯c1,0(t)/equalxc(0,1−t), and then put ∂c/equalxc2,0+c1,1+¯c2,1+¯c1,0. This corresponds to the following picture: c c2,0c1,1¯c2,1 ¯c1,0 This would work equally well, but is technically less conven ient.) For any k≥1we now define the boundary of a k-cube c: [0,1]k→Uby ∂c/equalxk/summationdisplay i/equalx1(−1)i(ci,0−ci,1)/equalxk/summationdisplay i/equalx1/summationdisplay ρ/equalx0,1(−1)i+ρci,ρ. This definition is consistent with the one- and two-dimensio nal cases considered above. The boundary of a 0-cube is by convention equal to 0. For an arbitrary /five.taboldstyle./two.taboldstyle. THE BOUNDARY OF A CHAIN /six.taboldstyle/seven.taboldstyle k-chain cwe define the boundary ∂cby writing cas a formal linear combination c/equalx/summationtext qaqcqofk-cubes cqwith real coefficients aq, and by putting ∂c/equalx/summationtext qaq∂cq. This definition makes ∂a linear map from k-chains to k−1-chains. There are a number of curious similarities between the bound ary operator ∂ and the exterior derivative d, the most important of which is the following. (There are also many differences, such as the fact that draises the degree of a form by 1, whereas∂lowers the dimension of a chain by 1.) /five.taboldstyle./four.taboldstyle. P/r.sc/o.sc/p.sc/o.sc/s.sc/i.sc/t.sc/i.sc/o.sc/n.sc. ∂(∂c)/equalx0for every k-chain cinU. In short, ∂2/equalx0. P/r.sc/o.sc/o.sc/f.sc. By linearity of ∂we may assume without loss of generality that cis a single k-cube c: [0,1]k→U. Then the k−2-chain∂(∂c)is given by ∂(∂c)/equalx∂k/summationdisplay i/equalx1/summationdisplay ρ/equalx0,1(−1)i+ρci,ρ/equalxk/summationdisplay i/equalx1/summationdisplay ρ/equalx0,1(−1)i+ρ∂ci,ρ /equalxk/summationdisplay i/equalx1k−1/summationdisplay j/equalx1/summationdisplay ρ,σ/equalx0,1(−1)i+j+ρ+σ(ci,ρ)j,σ. The double sum over iand jcan be rearranged as a sum over i≤jand a sum over i>jto give ∂(∂c)/equalx/summationdisplay 1≤i≤j≤k−1/summationdisplay ρ,σ/equalx0,1(−1)i+j+ρ+σ(ci,ρ)j,σ +/summationdisplay 1≤j<i≤k/summationdisplay ρ,σ/equalx0,1(−1)i+j+ρ+σ(ci,ρ)j,σ.(/five.taboldstyle./two.taboldstyle) Lett/equalx(t1,t2,..., tk−2)∈[0,1]k−2and letρandσbe0or1. Then for 1≤i≤j≤k−1 we have (ci,ρ)j,σ(t1,t2,..., tk−2)/equalxci,ρ(t1,t2,..., tj−1,σ,tj,..., tk−2) /equalxc(t1,t2,..., ti−1,ρ,ti,..., tj−1,σ,tj,..., tk−2). On the other hand, (cj+1,σ)i,ρ(t1,t2,..., tk−2)/equalxcj+1,σ(t1,t2,..., ti−1,ρ,ti,..., tk−2) /equalxc(t1,t2,..., ti−1,ρ,ti,..., tj−1,σ,tj,..., tk−2), because in the vector (t1,t2,..., ti−1,ρ,ti,..., tk−2)the entry tjoccupies the j+1-st slot! We conclude that (ci,ρ)j,σ/equalx(cj+1,σ)i,ρfor1≤i≤j≤k−1. It follows that /summationdisplay 1≤i≤j≤k−1/summationdisplay ρ,σ/equalx0,1(−1)i+j+ρ+σ(ci,ρ)j,σ/equalx/summationdisplay 1≤i≤j≤k−1/summationdisplay ρ,σ/equalx0,1(−1)i+j+ρ+σ(cj+1,σ)i,ρ /equalx/summationdisplay 1≤s<r≤k/summationdisplay µ,ν/equalx0,1(−1)s+r−1+ν+µ(cr,µ)s,ν /equalx−/summationdisplay 1≤s<r≤k/summationdisplay µ,ν/equalx0,1(−1)s+r+µ+ν(cr,µ)s,ν, /six.taboldstyle/eight.taboldstyle /five.taboldstyle. INTEGRATION AND STOKES’ THEOREM where in the first line we substituted (ci,ρ)j,σ/equalx(cj+1,σ)i,ρand in the second line we substituted r/equalxj+ 1,s/equalxi,µ/equalxσ, andν/equalxρ. Thus the two terms on the right-hand side of ( /five.taboldstyle./two.taboldstyle) cancel out. QED /five.taboldstyle./three.taboldstyle. Cycles and boundaries Letk≥1. Ak-cube cisdegenerate ifc(t1,..., tk)is independent of tifor some i. A k-chain cisdegenerate if it is a linear combination of degenerate cubes. In particular, a degenerate 1-cube is a constant path. The work done by a force field on a motionless particle is 0. More generally we have the following. /five.taboldstyle./five.taboldstyle. L/e.sc/m.sc/m.sc/a.sc. Letαbe ak-form and ca degenerate k-chain. Then/integraltext cα/equalx0. P/r.sc/o.sc/o.sc/f.sc. By linearity we may assume that cis a degenerate cube. Suppose cis constant as a function of ti. Then c(t1,..., ti,..., tk)/equalxc(t1,..., 0,..., tk)/equalxg/parenleftbigf(t1,..., ti,..., tk)/parenrightbig, where f: [0,1]k→[0,1]k−1and g: [0,1]k−1→Uare given respectively by f(t1,..., ti,..., tk)/equalx(t1,..., ˆti,..., tk), g(s1,..., sk−1)/equalxc(s1,..., si−1,0,si+1,..., sk−1). Now g∗(α)is ak-form on [0,1]k−1and hence equal to 0, and so c∗(α)/equalxf∗(g∗(α))/equalx0. We conclude that/integraltext cα/equalx/integraltext [0,1]kc∗(α)/equalx0. QED So degenerate chains are irrelevant in so far as integration is concerned. This motivates the following definition. A k-chain cisclosed , or a cycle, if∂cis a degenerate k−1-chain. A k-chain cis aboundary ifc/equalx∂b+afor some k+ 1-chain band some degenerate k-chain a. /five.taboldstyle./six.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Ifc1and c2are paths arranged head to tail as in the picture below, then∂(c1+c2)/equalx0, soc1+c2is a1-cycle. The closed path csatisfies∂c/equalx0, so it is a1-cycle as well. c1c2 c /five.taboldstyle./seven.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. The2-cube c: [0,1]2→R3defined by c(t1,t2)/equalx(cos 2πt1sinπt2,cos 2πt1sinπt2,cosπt2). parametrizes the unit sphere. The left and right edges of the unit square are mapped onto a single meridian of the sphere, the bottom edge i s mapped to the /five.taboldstyle./three.taboldstyle. CYCLES AND BOUNDARIES /six.taboldstyle/nine.taboldstyle north pole, and the top edge to the south pole. c The sphere has no boundary, so one might expect that cis a cycle. Indeed, we have c1,0(t)/equalxc1,1(t)/equalx(sinπt,sinπt,cosπt),c2,0(t)/equalx(0,0,1),c2,1(t)/equalx(0,0,−1), and therefore ∂c/equalx−c1,0+c1,1+c2,0−c2,1/equalxc2,0−c2,1 is a degenerate 1-chain and cis a cycle. /five.taboldstyle./eight.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Letc: [0,1]→R2be the path c(t)/equalx(cos 2πt,sin 2πt). This is a closed 1-cube, which parametrizes the unit circle. The circle is the boundary of the disc of radius 1and therefore it is reasonable to expect that the 1-cube cis a boundary. To show that this is the case we will display a 2-cube band a constant 1-chain asatisfying c/equalx∂b+a. The 2-cube bis defined by “shrinking cto a point”, b(t1,t2)/equalx(1−t2)c(t1)for(t1,t2)in the unit square. Then b(t1,0)/equalxc(t1), b(0,t2)/equalxb(1,t2)/equalx(1−t2,0), b(t1,1)/equalx(0,0), so that∂b/equalxc−a, where ais the constant path located at the origin. Therefore c/equalx∂b+a, which proves that cis a boundary. You may wonder whether we really need the degenerate path a. Isn’t it possible to find a 2-chain bwith the property that c/equalx∂b? Exercise /five.taboldstyle./two.taboldstyleshows that this is not the case. /five.taboldstyle./nine.taboldstyle. L/e.sc/m.sc/m.sc/a.sc. The boundary of a degenerate k-chain is a degenerate k−1-chain. P/r.sc/o.sc/o.sc/f.sc. By linearity it suffices to consider the case of a degenerate k-cube c. Suppose cis constant as a function of ti. Then ci,0/equalxci,1, so ∂c/equalx/summationdisplay j/nequali(−1)j(cj,0−cj,1). Lett/equalx(t1,t2,..., tk−1). For j>ithe cubes cj,0(t)and cj,1(t)are independent of tiand for j<ithey are independent of ti−1. So∂cis a combination of degenerate k−1-cubes and hence is degenerate. QED /five.taboldstyle./one.taboldstyle/zero.taboldstyle. C/o.sc/r.sc/o.sc/l.sc/l.sc/a.sc/r.sc/y.sc. Every boundary is a cycle. P/r.sc/o.sc/o.sc/f.sc. Suppose c/equalx∂b+awith adegenerate. Then by Lemma /five.taboldstyle./five.taboldstyle∂c/equalx ∂(∂b)+∂a/equalx∂a, where we used Proposition /five.taboldstyle./four.taboldstyle. Lemma /five.taboldstyle./nine.taboldstylesays that∂ais degenerate, and therefore so is ∂c. QED In the same way that a closed form is not necessarily exact, it may happen that a1-cycle is not a boundary. See Example /five.taboldstyle./one.taboldstyle/three.taboldstyle. /seven.taboldstyle/zero.taboldstyle /five.taboldstyle. INTEGRATION AND STOKES’ THEOREM /five.taboldstyle./four.taboldstyle. Stokes’ theorem In the language of chains and boundaries we can rewrite the fu ndamental theorem of calculus, Theorem /four.taboldstyle./four.taboldstyle, as follows: /integraldisplay cdg/equalxg(c(1))−g(c(0))/equalx/integraldisplay {c(1)}g−/integraldisplay {c(0)}g/equalx/integraldisplay {c(1)}−{c(0)}g/equalx/integraldisplay ∂cg, i.e./integraltext cdg/equalx/integraltext ∂cg. This is the form in which the fundamental theorem of calculu s generalizes to higher dimensions. This generalization is p erhaps the neatest rela- tionship between the exterior derivative and the boundary o perator. It contains as special cases the classical integration formulas of vect or calculus (Green, Gauss and Stokes) and for that reason has Stokes’ name attached to i t, although it would perhaps be better to call it the “fundamental theorem of mult ivariable calculus”. /five.taboldstyle./one.taboldstyle/one.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc (S/t.sc/o.sc/k.sc/e.sc/s.sc’ /t.sc/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc). Letαbe ak−1-form on an open subset UofRn and let cbe ak-chain in U. Then /integraldisplay cdα/equalx/integraldisplay ∂cα. P/r.sc/o.sc/o.sc/f.sc. Let us assume, as we may, that c: [0,1]k→Uis a single k-cube. By the definition of the integral and by Theorem /three.taboldstyle./one.taboldstyle/one.taboldstylewe have /integraldisplay cdα/equalx/integraldisplay [0,1]kc∗(dα)/equalx/integraldisplay [0,1]kdc∗(α). Since c∗(α)is ak−1-form on [0,1]k, it can be written as c∗(α)/equalxk/summationdisplay i/equalx1gidt1dt2···/hatwidedti···dtk for certain functions g1,g2,...,gkdefined on [0,1]k. Therefore /integraldisplay cdα/equalxk/summationdisplay i/equalx1/integraldisplay [0,1]kd/parenleftbiggidt1dt2···/hatwidedti···dtk/parenrightbig/equalxk/summationdisplay i/equalx1(−1)i+1/integraldisplay [0,1]k∂gi ∂tidt1dt2···dtk. Changing the order of integration (see Remark /five.taboldstyle./three.taboldstyle) and subsequently applying the fundamental theorem of calculus in one variable, formula ( B./one.taboldstyle), gives /integraldisplay [0,1]k∂gi ∂tidt1dt2···dtk/equalx/integraldisplay [0,1]k∂gi ∂tidtidt1dt2···/hatwidedti···dtk /equalx/integraldisplay [0,1]k−1/parenleftbiggi(t1,..., ti−1,1,ti+1,..., tk) −gi(t1,..., ti−1,0,ti+1,..., tk)/parenrightbigdt1dt2···/hatwidedti···dtk. The forms gi(t1,..., ti−1,1,ti+1,..., tk)dt1dt2···/hatwidedti···dtkand gi(t1,..., ti−1,0,ti+1,..., tk)dt1dt2···/hatwidedti···dtk EXERCISES /seven.taboldstyle/one.taboldstyle are nothing but c∗ i,1(α), resp. c∗ i,0(α). Accordingly, /integraldisplay cdα/equalxk/summationdisplay i/equalx1(−1)i+1/integraldisplay [0,1]k∂gi ∂tidtidt1dt2···/hatwidedti···dtk /equalxk/summationdisplay i/equalx1(−1)i+1/integraldisplay [0,1]k−1/parenleftbigc∗ i,1(α)−c∗ i,0(α)/parenrightbig /equalxk/summationdisplay i/equalx1/summationdisplay ρ/equalx0,1(−1)i+ρ/integraldisplay [0,1]k−1c∗ i,ρ(α) /equalxk/summationdisplay i/equalx1/summationdisplay ρ/equalx0,1(−1)i+ρ/integraldisplay ci,ρα/equalx/integraldisplay ∂cα, which proves the result. QED /five.taboldstyle./one.taboldstyle/two.taboldstyle. C/o.sc/r.sc/o.sc/l.sc/l.sc/a.sc/r.sc/y.sc. LetUbe an open subset of Rn, letcbe ak-chain in U, and letαbe ak-form on U. Then/integraltext cα/equalx0if either of the following two conditions holds: (i)cis a cycle and αis exact; or (ii)cis a boundary and αis closed. P/r.sc/o.sc/o.sc/f.sc. (i) Ifcis a cycle, then ∂cis a degenerate k−1-chain. Ifαis exact, then α/equalxdβfor some k−1-formβ. Therefore/integraltext cα/equalx/integraltext cdβ/equalx/integraltext ∂cβ/equalx0by Stokes’ theorem and by Lemma /five.taboldstyle./five.taboldstyle. (ii) Ifcis a boundary, then c/equalx∂b+afor some k+1-chain band some degenerate k-chain a. Ifαis closed, then dα/equalx0. Therefore/integraltext cα/equalx/integraltext ∂b+aα/equalx/integraltext bdα+/integraltext aα/equalx0by Stokes’ theorem and by Lemma /five.taboldstyle./five.taboldstyle. QED /five.taboldstyle./one.taboldstyle/three.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. The unit circle c(t)/equalx(cos 2πt,sin 2πt)(0≤t≤1) is a 1-cycle in the punctured plane U/equalxR2\{0}. Considered as a chain in R2it is also a boundary, as we saw in Example /five.taboldstyle./eight.taboldstyle. However, we claim that it is not a boundary inU: it is impossible to find a 2-chain band a degenerate 1-chain aboth contained inUsuch that c/equalx∂b+a. Indeed, suppose this was possible. Then/integraltext cα0/equalx0by Corollary /five.taboldstyle./one.taboldstyle/two.taboldstyle, whereα0is the angle form, because α0is closed. On the other hand, by Example /four.taboldstyle./one.taboldstylewe have/integraltext cα0/equalx2π. This is a contradiction, so we conclude that cis not a boundary in U. The presence of the puncture in Uis responsible both for the existence of the non-exact closed 1-formα0(see Example /four.taboldstyle./six.taboldstyle) and for the non-bounding closed 1-chain c. Exercises /five.taboldstyle./one.taboldstyle.LetUbe an open subset of Rn,Van open subset of Rmandφ:U→Va smooth map. Let cbe a k-cube in Uandαak-form on V. Prove that/integraltext cφ∗(α)/equalx/integraltext φ◦cα. /five.taboldstyle./two.taboldstyle.LetUbe an open subset of Rn. (i) Let bbe a k+ 1-chain in U. In Section /five.taboldstyle./two.taboldstylewe defined the boundary of bas a certain linear combination of k-cubes,∂b/equalx/summationtext iaici. Prove that/summationtext iai/equalx0. (ii) Let cbe a k-cube in U. Prove that there exists no k+ 1-chain binUsatisfying ∂b/equalxc. /five.taboldstyle./three.taboldstyle. Define a 2-cube c: [0,1]2→R3byc(t1,t2)/equalx/parenleftbigt2 1,t1t2,t2 2/parenrightbig, and letα/equalxx1dx2+ x1dx3+x2dx3. /seven.taboldstyle/two.taboldstyle /five.taboldstyle. INTEGRATION AND STOKES’ THEOREM (i) Sketch the image of c. (ii) Calculate both/integraltext cdαand/integraltext ∂cαand check that they are equal. /five.taboldstyle./four.taboldstyle. Define a 3-cube c: [0,1]3→R3byc(t1,t2,t3)/equalx(t2t3,t1t3,t1t2), and letα/equalx x1dx2dx3. Calculate both/integraltext cdαand/integraltext ∂cαand check that they are equal. /five.taboldstyle./five.taboldstyle.Using polar coordinates in ndimensions (see Exercise /three.taboldstyle./one.taboldstyle/nine.taboldstyle) write the n−1-dimen- sional unit sphere Sn−1inRnas the image of an n−1-cube c. (The domain of cwill not be the unit cube in Rn−1, but a rectangular block R. Choose Rin such a way as to cover the sphere as economically as possible.) For n/equalx2,3,4, calculate the boundary ∂cand show that cis a cycle. (See also Example /five.taboldstyle./seven.taboldstyle.) /five.taboldstyle./six.taboldstyle.Deduce the following classical integration formulas from t he generalized version of Stokes’ theorem. All functions, vector fields, chains etc . are smooth and are defined in an open subset UofRn. (Some formulas hold only for special values of n, as indicated.) (i)/integraldisplay cgrad (g)·dx/equalxg(c(1))−g(c(0))for any function gand any path c. (ii) Green’s formula:/integraldisplay c/parenleftbigg∂g ∂x−∂f ∂y/parenrightbigg dx dy/equalx/integraldisplay ∂c(f dx+g dy)for any functions f,g and any 2-chain c. (Here n/equalx2.) (iii) Gauss’ formula:/integraldisplay cdiv(F)dx1dx2···dxn/equalx/integraldisplay ∂cF·∗dxfor any vector field Fand any n-chain c. (iv) Stokes’ formula:/integraldisplay ccurl(F)·∗dx/equalx/integraldisplay ∂cF·dxfor any vector field Fand any 2-chain c. (Here n/equalx3.) In parts ( iii) and ( iv) we use the notations dxand∗dxexplained in Section /two.taboldstyle./five.taboldstyle. We shall give a geometric interpretation of the entity ∗dxin terms of volume forms later on. (See Corollary /eight.taboldstyle./one.taboldstyle/seven.taboldstyle.) CHAPTER /six.taboldstyle Manifolds /six.taboldstyle./one.taboldstyle. The definition Intuitively, an n-dimensional manifold in the Euclidean space RNis a subset that in the neighbourhood of every point “looks like” Rnup to “smooth distor- tions”. The formal definition is given below and is a bit long. It will help to consider first the basic example of the surface of the earth, which is a t wo-dimensional sphere placed in three-dimensional space. Geographers describe t he earth by means of a world atlas, which is a collection of maps. Each map depicts a portion of the world, such as a country or an ocean. The correspondence betw een points on a map and points on the earth’s surface is not entirely faithfu l, because charting a curved surface on a flat piece of paper inevitably distorts th e distances between points. But the distortions are continuous, indeed differen tiable (in most tradi- tional cartographic projections). Maps of neighbouring ar eas overlap near their edges and the totality of all maps in a world atlas covers the w hole world. An arbitrary manifold is defined similarly, as an n-dimensional “world” rep- resented by an “atlas” consisting of “maps”. These maps are a special kind of parametrizations known as embeddings. /six.taboldstyle./one.taboldstyle. D/e.sc/f.sc/i.sc/n.sc/i.sc/t.sc/i.sc/o.sc/n.sc. LetUbe an open subset of Rn. An embedding ofUintoRNis aC∞mapψ:U→RNsatisfying the following conditions: (i)ψis one-to-one (i.e. if ψ(t1)/equalxψ(t2), then t1/equalxt2); (ii)Dψ(t)is one-to-one for all t∈U; (iii) the inverse of ψ, which is a map ψ−1:ψ(U)→U, is continuous. Theimage of the embedding is the set ψ(U)/equalx{ψ(t)|t∈U}consisting of all points of the form ψ(t)with t∈U. You should think of ψ(U)as an n-dimensional “patch” in RNparametrized by the map ψ. The inverse map ψ−1is called a chart orcoordinate map . It maps each point in the patch ψ(U)to an n-tuple of numbers, which we think of as the “coordinates” of the point. Conditio n (i) means that to distinct values of the parameter tmust correspond distinct points ψ(t)in the patch ψ(U). Thus the patch ψ(U)has no self-intersections. Condition ( ii) means that for each tinUallncolumns of the Jacobi matrix Dψ(t)must be independent. This condition is imposed to prevent the occurrence of cusps and o ther singularities in the imageψ(U). Since Dψ(t)hasNrows, condition ( ii) also implies that N≥n: the target space RNmust have dimension greater than or equal to that of the source space U, or elseψcannot be an embedding. Condition ( iii) can be restated as follows: if tiis any sequence of points in Usuch that lim i→∞ψ(ti)exists and is equal toψ(t)for some t∈U, then lim i→∞ti/equalxt. This is intended to avoid situations where the image ψ(U)doubles back on itself “at infinity”. (See Exercise /six.taboldstyle./four.taboldstylefor an /seven.taboldstyle/three.taboldstyle /seven.taboldstyle/four.taboldstyle /six.taboldstyle. MANIFOLDS example.) /six.taboldstyle./two.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetUbe an open subset of Rnand let f:U→Rmbe a smooth map. The graph offis the collection graph (f)/equalx/braceleftbigg/parenleftBigg t f(t)/parenrightBigg/barex/barex/barex/barext∈U/bracerightbigg . Since tis an n-vector and f(t)anm-vector, the graph is a subset of RNwith N/equalxn+m. We claim that the graph is the image of an embedding ψ:U→RN. Define ψ(t)/equalx/parenleftBigg t f(t)/parenrightBigg . Then by definition graph (f)/equalxψ(U). Furthermore ψis an embedding. Indeed, ψ(t1)/equalxψ(t2)implies t1/equalxt2, soψis one-to-one. Also, Dψ(t)/equalx/parenleftBigg In D f(t)/parenrightBigg , soDψ(t)hasnindependent columns. Finally the inverse of ψis given by ψ−1/parenleftBigg t f(t)/parenrightBigg /equalxt, which is continuous. Hence ψis an embedding. A manifold is an object patched together out of the images of s everal embed- dings. More precisely, /six.taboldstyle./three.taboldstyle. D/e.sc/f.sc/i.sc/n.sc/i.sc/t.sc/i.sc/o.sc/n.sc. Ann-dimensional manifold /one.superior(orn-manifold for short) in RNis a subset MofRNwith the property that for each x∈Mthere exist •an open subset VofRNcontaining x, •an open subset UofRn, •and an embedding ψ:U→RNsatisfyingψ(U)/equalxM∩V. Such an embedding ψis called a local parametrization of Matx. Its inverse ψ−1:ψ(U)→Uis achart of Matx. An atlasofMis a collection of local parametriza- tionsψi:Ui→RNofMwith the property that Mis the union of all the sets ψi(Ui). Thetangent space toMat a point x∈Mis the column space of Dψ(t), TxM/equalxDψ(t)(Rn), whereψ:U→RNis a local parametrization of Matxandtis the unique vector inUsatisfyingψ(t)/equalxx. The elements of TxMaretangent vectors toMatx. The codimension ofMinRNis the number N−n. /one.superiorIn the literature this is usually called a submanifold of Euclidean space. It is possible to define manifolds more abstractly, without reference to a surround ing vector space. However, it turns out that practically all abstract manifolds can be embedded int o a vector space of sufficiently high dimen- sion. Hence the abstract notion of a manifold is not substant ially more general than the notion of a submanifold of a vector space. /six.taboldstyle./one.taboldstyle. THE DEFINITION /seven.taboldstyle/five.taboldstyle Note that the tangent space TxMat each point xof an n-manifold Mis an n- dimensional linear subspace of RN. The reason is that for every local parametriza- tionψofMthe Jacobi matrix Dψ(t)hasnindependent columns. One-dimensional manifolds are called smooth curves , two-dimensional mani- folds smooth surfaces , and n-manifolds in Rn+1smooth hypersurfaces . In these cases the tangent spaces are usually called tangent lines ,tangent planes , and tangent hyper- planes , respectively. The following picture illustrates the definition. Here Mis a curve in the plane, so we have N/equalx2and n/equalx1. The open set Uis an interval in Rand Vis an open disc in R2. The mapψsends ttoxand parametrizes the portion of the curve inside V. Since n/equalx1, the Jacobi matrix Dψ(t)consists of a single column vector, which is tangent to the curve at x/equalxψ(t). The tangent line TxMis the line spanned by this vector. tU VM xTxM ψ /six.taboldstyle./four.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. An open subset UofRncan be regarded as a manifold of dimen- sion n(hence of codimension 0). Indeed, Uis the image of the map ψ:U→Rn given byψ(x)/equalxx, the identity map. The tangent space to Uat any point is Rn itself. /six.taboldstyle./five.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Let N≥nand define ψ:Rn→RNbyψ(x1,x2,..., xn)/equalx (x1,x2,..., xn,0,0,..., 0). It is easy to check that ψis an embedding. Hence the imageψ(Rn)is an n-manifold in RN. (Note that ψ(Rn)is just a linear subspace isomorphic to Rn; e.g. if N/equalx3and n/equalx2it is just the (x,y)-plane.) Combining this example with the previous one, we see that if Uis any open subset of Rn, then ψ(U)is a manifold in RNof codimension N−n. Its tangent space at any point is the linear subspace ψ(Rn)ofRN. /six.taboldstyle./six.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetM/equalxgraph (f), where f:U→Rmis a smooth map. As shown in Example /six.taboldstyle./two.taboldstyle,Mis the image of a single embedding ψ:U→Rn+m, so Mis an n-dimensional manifold in Rn+m, covered by a single chart. At a point (x,f(x))in the graph the tangent space is spanned by the columns of Dψ(x). For instance, if n/equalxm/equalx1,Mis one-dimensional and the tangent line to Mat(x,f(x)) is spanned by the vector (1,f′(x)). This is equivalent to the familiar fact that the /seven.taboldstyle/six.taboldstyle /six.taboldstyle. MANIFOLDS slope of the tangent line to the graph at xisf′(x). xf(x)/parenleftbigg1 f′(x)/parenrightbigggraph(f) Forn/equalx2and m/equalx1,Mis a surface in R3. The tangent plane to Mat a point (x,y,f(x,y))is spanned by the columns of Dψ(x,y), namely /parenlefttpA/parenleftexA /parenleftbtA1 0 ∂f ∂x(x,y)/parenrighttpA/parenrightexA /parenrightbtAand/parenlefttpA/parenleftexA/parenleftexA /parenleftbtA0 1 ∂f ∂y(x,y)/parenrighttpA/parenrightexA/parenrightexA /parenrightbtA. The figure below shows the graph of the cubic function f(x,y)/equalx1 2(x3+y3−3x y) from two different angles, together with a few points and tang ent vectors. e1e2e3 e1e2e3 /six.taboldstyle./seven.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Consider the path ψ:R→R2given byψ(t)/equalxeat(cosbt,sinbt), where aand bare nonzero constants. Let us check that ψis an embedding. Observe first that/bardblψ(t)/bardbl/equalxeat. Thereforeψ(t1)/equalxψ(t2)implies eat1/equalxeat2. The exponential function is one-to-one, so t1/equalxt2(since a/nequal0). This shows that ψis one-to-one. /six.taboldstyle./one.taboldstyle. THE DEFINITION /seven.taboldstyle/seven.taboldstyle The velocity vector is ψ′(t)/equalxeat/parenleftBigg acosbt−bsinbt asinbt+bcosbt/parenrightBigg /equalxeat/parenleftBigg cosbt−sinbt sinbt cosbt/parenrightBigg /parenleftBigg a b/parenrightBigg . The2×2-matrix in this formula is a rotation matrix and hence invert ible. The vector/parenleftbiga b/parenrightbigis nonzero, and therefore ψ′(t)/nequal0for all t. Moreover we have t/equalxa−1lneat/equalx a−1ln/bardblψ(t)/bardbl. Hence the inverse of ψis given byψ−1(x)/equalxa−1ln/bardblx/bardblforx∈ψ(R) and so is continuous. Therefore ψis an embedding and ψ(R)is a1-manifold. The imageψ(R)is a spiral, which winds infinitely many times around the orig in and which for t→−∞ converges to the origin. xy Even though ψ(R)is a manifold, the set ψ(R)∪{0}is not: it has a very nasty singularity at the origin! The manifolds of Examples /six.taboldstyle./four.taboldstyle–/six.taboldstyle./seven.taboldstyleeach have an atlas consisting of one single chart. Here are two examples where one needs more than one cha rt to cover a manifold. /six.taboldstyle./eight.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. The picture below shows the map ψ(t1,t2)/equalx/parenleftbig(R+rcost2)cost1,(R+rcost2)sint1,rsint2/parenrightbig. The domain Uis an open rectangle in the plane and the image is a portion of a torus in three-space. One can check that ψis an embedding, but we will not provide the details here. (If we chose Utoo big, the image would self-intersect and the map would not be an embedding.) For one particular value of tthe column vectors of the Jacobi matrix are also shown. As you can see, they span the tangent plane at /seven.taboldstyle/eight.taboldstyle /six.taboldstyle. MANIFOLDS the image point. x=ψ(t)Dψ(t)e1 Dψ(t)e2 ψ(U)tU e1e2e1e2e3ψ In this way we can cover the entire torus with images of rectan gles, thus showing that the torus is a 2-dimensional manifold. /six.taboldstyle./nine.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetMbe the unit sphere Sn−1inRn. Let U/equalxRn−1and let ψ:U→Rnbe the map ψ(t)/equalx1 /bardblt/bardbl2+ 1/parenleftbig2t+(/bardblt/bardbl2−1)en/parenrightbig given in Exercise B./eight.taboldstyle. As we saw in that exercise, the image of ψis the punctured sphere M\{en}, so if we let Vbe the open set Rn\{en}, thenψ(U)/equalxM∩V. Also we saw that ψhas an inverse φ:ψ(U)→U, the stereographic projection from the north pole. Therefore ψis one-to-one and its inverse is continuous (indeed, differentiable). Moreover, φ◦ψ(t)/equalxtimplies Dφ(ψ(t))Dψ(t)v/equalxvfor all vin Rn−1by the chain rule. Therefore, if vis in the nullspace of Dψ(t), v/equalxDφ(ψ(t))Dψ(t)v/equalxDφ(ψ(t))0/equalx0. Thus we see that ψis an embedding. To cover all of Mwe need a second map, for example the inverse of the stereographic projection fro m the south pole. This /six.taboldstyle./one.taboldstyle. THE DEFINITION /seven.taboldstyle/nine.taboldstyle is also an embedding and its image is M\{−en}/equalxM∩V, where V/equalxRn\{−en}. This finishes the proof that Mis an n−1-manifold in Rn. As these examples show, the definition of a manifold can be a li ttle awkward to work with in practice, even for a simple manifold. In practic e it can be rather hard to decide whether a given subset is a manifold using the defini tion alone. In the next section we will give a more manageable criterion for a se t to be a manifold. We conclude this section by taking a second look at tangent ve ctors. There are many different ways to parametrize a manifold Min the neighbourhood of a point x. (For instance, for a sphere we have a choice among a large num ber of different cartographic projections.) let ψ1:U1→RNandψ2:U2→RNbe two local parametrizations of Matx. Then we have x/equalxψ1(t1)/equalxψ2(t2)for some t1∈U1andt2∈U2. Do Dψ1(t1)and Dψ2(t2)have the same column spaces? In other words, is the tangent space TxMwell-defined? We will answer this question in the affirmative by characterizing tangent vectors to Min language that does not refer to local parametrizations. Namely, we will prove that all tangent vectors to Mare velocity vectors of paths in M. By a pathin a manifold MinRNwe simply mean a path in RNwhich happens to be contained in M, i.e. a smooth map cfrom an open interval ItoRNwith the property that c(t)is in Mfor all t∈I. /six.taboldstyle./one.taboldstyle/zero.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. LetMbe an n-manifold in RN. Let x0∈Mand let v∈RN. Then v is tangent to Matx0if and only if there exists a path c:(−ε,ε)→Mwith the properties c(0)/equalxx0andc′(0)/equalxv. P/r.sc/o.sc/o.sc/f.sc. Letψ:U→RNbe a local parametrization of Matx0, i.e. an embedding with the property that ψ(U)/equalxM∩Vfor some open subset VofRNcontaining x0. Let t0be the unique point in Usatisfyingψ(t0)/equalxx0. Suppose vis tangent to Matx0, i.e.v∈Tx0M. Then by definition v/equalxDψ(t0)(u)for some u∈Rn. Define c(h)/equalxψ(t0+hu). Then cis a path in Mpassing through c(0)/equalxψ(t0)/equalxx0. By the chain rule (see Example B./four.taboldstyle) we have c′(0)/equalxDψ(t0)u/equalxv. Thus vis the velocity vector at x0of some path cinMpassing through x0. Conversely, assume that v/equalxc′(0)for some path c:(−ε,ε)→Msatisfying c(0)/equalxx0. Can we find a vector u∈Rnsuch that Dψ(t0)u/equalxv? By Lemma /six.taboldstyle./one.taboldstyle/one.taboldstyle below, after replacing Uand Vwith smaller open sets if necessary, the map ψhas a smooth left inverse, i.e. a smooth map φ:V→Usatisfyingφ◦ψ(t)/equalxtfor all t∈U. Ifx∈M∩V, then x/equalxψ(t)for some t∈U, so x/equalxψ(t)/equalxψ(φ(ψ(t)))/equalxψ(φ(x)). Ifhis sufficiently small, then c(h)is contained in M∩V, and hence c(h)/equalx ψ(φ(c(h))). Differentiating this identity with respect to hath/equalx0gives c′(0)/equalxDψ(φ(0))D(φ◦c)(0)/equalxDψ(t0)u, with u/equalxD(φ◦c)(0). This shows that c′(0)is in Dψ(t0)(Rn)/equalxTx0M. QED The following technical result, which is a consequence of th e implicit function theorem, says that embeddings have smooth left inverses, at least if one suitably restricts the domain and the range. /six.taboldstyle./one.taboldstyle/one.taboldstyle. L/e.sc/m.sc/m.sc/a.sc. LetUbe an open subset of Rn, letVbe an open subset of RN, and let ψ:U→Vbe an embedding. Let t0∈Uandx0/equalxψ(t0). Then there exist an open subset ˜UofUcontaining t0, an open subset ˜VofVcontaining x0, and a smooth map φ:˜V→˜U with the property that φ(ψ(t))/equalxtfor all t∈˜U. /eight.taboldstyle/zero.taboldstyle /six.taboldstyle. MANIFOLDS P/r.sc/o.sc/o.sc/f.sc. Leta1,a2,...,anbe the columns of Dψ(t0). Asψis an embedding, these columns are independent, so we can complete them to a ba sisa1,a2,...,an, b1,b2,...,bkofRN, where k/equalxN−n. For t∈Uands∈Rkdefine ˜ψ(t,s)/equalx ψ(t)+/summationtextk j/equalx1sjbj. Then ˜ψis a map from U×RktoRNand its Jacobi matrix at (t0,0) is D˜ψ(t0,0)/equalx/parenleftBig a1a2··· anb1b2··· bk/parenrightBig . The columns of this matrix form a basis of RN, and therefore it is invertible. By the inverse function theorem, Theorem B./seven.taboldstyle, there exist an open subset ˜UofU containing t0and an open subset WofRkcontaining 0such that ˜V/equalx˜ψ(˜U×W)is an open subset of Vand the map ˜ψ:˜U×W→˜Vhas a smooth inverse ˜φ:˜V→˜U×W. Letπ:˜U×W→˜Vbe the map defined by π(t,s)/equalxt, and letφ/equalxπ◦˜φ. Then for allt∈˜Uwe have φ(ψ(t))/equalxπ◦˜φ◦˜ψ(t,0)/equalxπ(t,0)/equalxt, soφ:˜V→˜Uis a left inverse of ψ:˜U→˜V. QED /six.taboldstyle./two.taboldstyle. The regular value theorem Our definition of the notion of a manifold, Definition /six.taboldstyle./three.taboldstyle, is based on embed- dings, which are an “explicit” way of describing manifolds. However, embeddings can be hard find in practice. Instead, manifolds are often giv en “implicitly”, by a system of mequations in Nunknowns, φ1(x1,..., xN)/equalxc1, φ2(x1,..., xN)/equalxc2, ... φm(x1,..., xN)/equalxcm. Here theφi’s are smooth functions presumed to be defined on some common o pen subset UofRN. Writing in the usual way x/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAx1 x2 ... xN/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA, φ (x)/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAφ1(x) φ2(x) ... φm(x)/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA,c/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAc1 c2 ... cm/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA, we can abbreviate this system to a single equation φ(x)/equalxc. For a fixed vector c∈Rmwe denote the solution set by φ−1(c)/equalx{x∈U|φ(x)/equalxc} and call it the level set or the fibreofφatc, or the preimage ofcunderφ. Ifφis a linear map, the system of equations is inhomogeneous linear and we know from linear algebra that the solution set is an affine subspace of RN. The dimension of this affine subspace is N−m, provided that φhas rank m(i.e. has mindependent columns). We can generalize this idea to nonlinear equation s as follows. We say thatc∈Rmis aregular value ofφif the Jacobi matrix Dφ(x):RN→Rmhas rank m for all x∈φ−1(c). A vector that is not a regular value is called a singular value. (As an extreme special case, if φ−1(c)is empty, then cis automatically a regular value.) /six.taboldstyle./two.taboldstyle. THE REGULAR VALUE THEOREM /eight.taboldstyle/one.taboldstyle The following result is the most useful criterion for a set to be a manifold. (However, it does not apply to every manifold. In other words , it is a sufficient but not a necessary criterion.) The proof rests on the following important fact from linear algebra, nullity (A)+ rank (A)/equalxl, valid for any k×l-matrix A. Here the rank is the number of independent columns ofA(in other words the dimension of the column space A(Rl)) and the nullity is the number of independent solutions of the homogeneous eq uation Ax/equalx0(in other words the dimension of the nullspace ker(A)). /six.taboldstyle./one.taboldstyle/two.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc (/r.sc/e.sc/g.sc/u.sc/l.sc/a.sc/r.sc /v.sc/a.sc/l.sc/u.sc/e.sc /t.sc/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc). LetUbe open in RNand letφ:U→Rm be a smooth map. Suppose that cis a regular value of φand that M/equalxφ−1(c)is nonempty. Then Mis a manifold in RNof codimension m. Its tangent space at xis the nullspace of Dφ(x), TxM/equalxker(Dφ(x)). P/r.sc/o.sc/o.sc/f.sc. Letx∈M. Then Dφ(x)has rank mand so has mindependent columns. After relabelling the coordinates on RNwe may assume the last mcolumns are independent and therefore constitute an invertible m×m-submatrix AofDφ(x). Let us put n/equalxN−m. Identify RNwith Rn×Rmand correspondingly write anN-vector as a pair (u,v)with uan-vector and vanm-vector. Also write x/equalx(u0,v0). Now refer to Appendix B./four.taboldstyleand observe that the submatrix Ais nothing but the “partial” Jacobian Dvφ(u0,v0). This matrix being invertible, by the implicit function theorem, Theorem B./six.taboldstyle, there exist open neighbourhoods U ofu0inRnand Vofv0inRmsuch that for each u∈Uthere exists a unique v/equalxf(u)∈Vsatisfyingφ(u,f(u))/equalxc. The map f:U→VisC∞. In other words M∩(U×V)/equalxgraph (f)is the graph of a smooth map. We conclude from Example /six.taboldstyle./six.taboldstylethat M∩(U×V)is an n-manifold, namely the image of the embedding ψ:U→RNgiven byψ(u)/equalx(u,f(u)). Since U×Vis open in RNand the above argument is valid for every x∈M, we see that Mis an n-manifold. To compute TxMnote thatφ(ψ(u))/equalxc, a constant, for all u∈U. Hence Dφ(ψ(u))Dψ(u)/equalx0 by the chain rule. Plugging in u/equalxu0gives Dφ(x)Dψ(u0)/equalx0. The tangent space TxMis by definition the column space of Dψ(u0), so every tangent vector vtoMatxis of the form v/equalxDψ(u0)afor some a∈Rn. Therefore Dφ(x)v/equalxDφ(x)Dψ(u0)a/equalx0, i.e. TxM⊆ker(Dφ(x)). The tangent space TxM isn-dimensional (because the ncolumns of Dψ(u0)are independent) and so is the nullspace of Dφ(x)(because nullity (Dφ(x))/equalxN−m/equalxn). Hence TxM/equalx ker(Dφ(x)). QED The case of one single equation ( m/equalx1) is especially important. Then Dφis a single row vector and its transpose is the gradient of φ:DφT/equalxgrad (φ). It has rank 1atxif and only if it is nonzero, i.e. at least one of the partials o fφdoes not vanish at x. The solution set of a scalar equation φ(x)/equalxcis known as a level hypersurface . Level hypersurfaces, especially level curves, occur freq uently in all kinds of applications. For example, isotherms in weatherch arts and contour lines in topographical maps are types of level curves. /eight.taboldstyle/two.taboldstyle /six.taboldstyle. MANIFOLDS /six.taboldstyle./one.taboldstyle/three.taboldstyle. C/o.sc/r.sc/o.sc/l.sc/l.sc/a.sc/r.sc/y.sc (/l.sc/e.sc/v.sc/e.sc/l.sc /h.sc/y.sc/p.sc/e.sc/r.sc/s.sc/u.sc/r.sc/f.sc/a.sc/c.sc/e.sc/s.sc). LetUbe open in RNand letφ:U→R be a smooth function. Suppose that M/equalxφ−1(c)is nonempty and that grad (φ)(x)/nequal0 for all xinM. Then Mis a manifold in RNof codimension 1. Its tangent space at xis the orthogonal complement of grad (φ)(x), TxM/equalxgrad (φ)(x)⊥. /six.taboldstyle./one.taboldstyle/four.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetU/equalxR2andφ(x,y)/equalxx y. The level curves of φare hyperbolas in the plane and the gradient is grad (φ)(x)/equalx(y,x). The diagram below shows a few level curves as well as the gradient vector fi eld, which as you can see is perpendicular to the level curves. xy The gradient vanishes only at the origin, so φ(0)/equalx0is the only singular value of φ. By Corollary /six.taboldstyle./one.taboldstyle/three.taboldstylethis means that φ−1(c)is a1-manifold for c/nequal0. The fibreφ−1(0) is the union of the two coordinate axes, which has a self-inte rsection and so is not a manifold. However, the set φ−1(0)\{0}is a1-manifold since the gradient is nonzero outside the origin. Think of this diagram as a topographical map representing the surface z/equalxφ(x,y)shown below. The level curves of φare the contour lines of the surface, obtained by intersecting the surface with hori zontal planes at different heights. As explained in Appendix B./two.taboldstyle, the gradient points in the direction of steepest ascent. Where the contour lines self-intersect th e surface has a “mountain /six.taboldstyle./two.taboldstyle. THE REGULAR VALUE THEOREM /eight.taboldstyle/three.taboldstyle pass” or saddle point. e1e2e3 /six.taboldstyle./one.taboldstyle/five.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. A more interesting example of an equation in two variables is φ(x,y)/equalxx3+y3−3x y/equalxc. Here grad (φ)(x)/equalx3(x2−y,y2−x), sograd (φ) vanishes at the origin and at (1,1). The corresponding values of φare0, resp.−1, which are the singular values of φ. xy The level “curve” φ−1(−1)is not a curve at all, but consists of the single point (1,1). Hereφhas a minimum and the surface z/equalxφ(x,y)has a “valley”. The level curve φ−1(0)has a self-intersection at the origin, which corresponds to a saddle point on the surface. These features are also clearly visible in th e surface itself, which is shown in Example /six.taboldstyle./six.taboldstyle. /six.taboldstyle./one.taboldstyle/six.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetU/equalxRNandφ(x)/equalx/bardblx/bardbl2. Then grad (φ)(x)/equalx2x, so as in Example /six.taboldstyle./one.taboldstyle/four.taboldstylegrad (φ)vanishes only at the origin 0, which is contained in φ−1(0). So again any c/nequal0is a regular value of φ. Clearly,φ−1(c)is empty for c<0. For c>0,φ−1(c)is an N−1-manifold, the sphere of radius√cinRN. The tangent /eight.taboldstyle/four.taboldstyle /six.taboldstyle. MANIFOLDS space to the sphere at xis the set of all vectors perpendicular to grad (φ)(x)/equalx2x. In other words, TxM/equalxx⊥/equalx{y∈RN|y·x/equalx0}. Finally, 0is a singular value (the absolute minimum) of φandφ−1(0)/equalx{0}is not anN−1-manifold. (It happens to be a 0-manifold, though, just like the singular fibreφ−1(−1)in Example /six.taboldstyle./one.taboldstyle/five.taboldstyle. So if cis a singular value, you cannot be certain thatφ−1(c)isnota manifold. However, even if a singular fibre happens to be a manifold, it is often of the “wrong” dimension.) Here is an example of a manifold given by two equations ( m/equalx2). /six.taboldstyle./one.taboldstyle/seven.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Defineφ:R4→R2by φ(x)/equalx/parenleftBigg x2 1+x2 2 x1x3+x2x4/parenrightBigg . Then Dφ(x)/equalx/parenleftBigg 2x12x20 0 x3 x4x1x2/parenrightBigg . Ifx1/nequal0the first and third columns of Dφ(x)are independent, and if x2/nequal0the second and fourth columns are independent. On the other hand , ifx1/equalxx2/equalx0, Dφ(x)has rank 1andφ(x)/equalx0. This shows that the origin 0inR2is the only singular value of φ. Therefore, by the regular value theorem, for every nonzero vector cthe setφ−1(c)is a two-manifold in R4. For instance, M/equalxφ−1/parenleftbig1 0/parenrightbigis a two-manifold. Note that Mcontains the point x/equalx(1,0,0,0). Let us find a basis of the tangent space TxM. Again by the regular value theorem, this tangent space is equal to the nullspace of Dφ(x)/equalx/parenleftBigg 2 0 0 0 0 0 1 0/parenrightBigg , which is equal the set of all vectors ysatisfying y1/equalxy3/equalx0. A basis of TxMis therefore given by the standard basis vectors e2ande4. The orthogonal group. We now come to a more sophisticated example of a manifold determined by a large system of equations. Recall t hat an n×n-matrix Aisorthogonal ifATA/equalxI. This means that the columns (and also the rows) of A are perpendicular to one another and have length 1. (So they form an ortho normal basis of Rn—note the regrettable inconsistency in the terminology.) T he orthogonal matrices form a group under matrix multiplication: every or thogonal matrix Ais invertible with inverse A−1/equalxAT, the identity matrix Iis orthogonal, and the product of orthogonal matrices is orthogonal. This group is called the orthogonal group and denoted by O(n). /six.taboldstyle./one.taboldstyle/eight.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. The orthogonal group O(n)is a manifold of dimension1 2n(n−1). The tangent space to O(n)at the identity matrix is the space of antisymmetric n×n-matrices. P/r.sc/o.sc/o.sc/f.sc. This is an application of the regular value theorem. We start by noting thatO(n)/equalxφ−1(I), whereφis defined by φ(A)/equalxATA. /six.taboldstyle./two.taboldstyle. THE REGULAR VALUE THEOREM /eight.taboldstyle/five.taboldstyle The domain of the map φisV/equalxRn×n, the vector space of all n×n-matrices. We can regardφas a map from Vto itself, but then the identity matrix Iis not a regular value! To ensure that Iis a regular value we must restrict the range of φ. This is done by observing that (ATA)T/equalxATA, so ATAis a symmetric matrix. In other words, if we let W/equalx{C∈V|C/equalxCT}be the linear subspace of Vconsisting of all symmetric matrices, then we can regard φas a map φ:V−→W. We will show that Iis a regular value of this map. To do this we need to compute the total derivative of φ. For every matrix A∈Vthe total derivative at Ais a linear map Dφ(A):V→W, which can be computed by using Lemma B./one.taboldstyle. This lemma says that for every B∈Vthe result of applying the linear map Dφ(A)toBis the directional derivative of φatAalong B: Dφ(A)B/equalxlim t→01 t(φ(A+tB)−φ(A)) /equalxlim t→01 t(ATA+tATB+tBTA+t2BTB−ATA) /equalxATB+BTA. We need to show that for all A∈O(n)the linear map Dφ(A):V→Wis surjective. This amounts to showing that, given any orthogonal Aand any symmetric C, the equation ATB+BTA/equalxC (/six.taboldstyle./one.taboldstyle) is solvable for B. Here is a trick for guessing a solution: observe that C/equalx1 2(C+CT) and first try to solve ATB/equalx1 2C. Left multiplying both sides by Aand using AAT/equalxI gives B/equalx1 2AC. We claim that B/equalx1 2ACis a solution of equation ( /six.taboldstyle./one.taboldstyle). Indeed, ATB+BTA/equalxAT1 2AC+1 2CTATA/equalx1 2(C+CT)/equalxC. By the regular value theorem this proves that O(n)is a manifold. You will be asked to prove the remaining assertions in Exercise /six.taboldstyle./one.taboldstyle/two.taboldstyle. QED IfAis an orthogonal matrix, then det(ATA)/equalxdet(I)/equalx1, so(det(A))2/equalx1, i.e. det(A)/equalx±1. The special orthogonal group orrotation group is SO(n)/equalx{A∈O(n)|det(A)/equalx1}. IfAand Bare elements of O(n)that are not in SO(n), then AB∈O(n)and det(AB)/equalxdet(A)det(B)/equalx1, so AB∈SO(n). An example of an orthogonal matrix which is not a rotation is a reflection matrix, such as A0/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA−1 0··· 0 0 1··· 0 ............ 0 0··· 1/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA. Every orthogonal matrix Awhich is not in SO(n)can be written as A/equalxA0Cfor a unique C∈SO(n), namely C/equalxA0A. Thus we see that O(n)is a union of two disjoint pieces, O(n)/equalxSO(n)∪{A0C|C∈SO(n)}, each of which is a manifold of dimension1 2n(n−1). /eight.taboldstyle/six.taboldstyle /six.taboldstyle. MANIFOLDS In two dimenson the rotation group is the group SO(2)of all matrices of the form /parenleftBigg cosθ−sinθ sinθcosθ/parenrightBigg . As a manifold SO(2)is a copy of the unit circle S1. The rotation group in three dimensions SO(3)is a three-dimensional manifold, which is a little harder to vizualize. Every rotation can be parametrized by a vector x∈R3, namely as the rotation about the axis spanned by xthrough an angle of /bardblx/bardblradians. But SO(3) is not the same as R3, because many different vectors represent the same rotation . Every rotation can be represented by a vector of length ≤π, so let us restrict x to the (solid) ball Bof radiusπabout the origin. This gets rid of most of the ambiguity, except for the fact that two antipodal points on t he boundary sphere represent the same rotation (because rotation through πabout a given axis is the same as rotation through −π). We conclude that SO(3)is the manifold obtained by identifying opposite points on the boundary of B. This is the three-dimensional projective space . Exercises /six.taboldstyle./one.taboldstyle.This is a continuation of Exercise /one.taboldstyle./one.taboldstyle. Defineψ:R→R2byψ(t)/equalx(t−sint,1− cost). Show that ψis one-to-one. Determine all tfor whichψ′(t)/equalx0. Prove that ψ(R)is not a manifold at these points. /six.taboldstyle./two.taboldstyle. Leta∈(0,1)be a constant. Prove that the map ψ:R→R2given byψ(t)/equalx (t−asint,1−acost)is an embedding. (This becomes easier if you first show that t−asint is an increasing function of t.) Graph the curve defined by ψ. /six.taboldstyle./three.taboldstyle.Prove that the map ψ:R→R2given byψ(t)/equalx1 2(et+e−t,et−e−t)is an embedding. Conclude that M/equalxψ(R)is a1-manifold. Graph the curve M. Compute the tangent line to Mat(1,0)and try to find an equation for M. /six.taboldstyle./four.taboldstyle.LetIbe the open interval (−1,∞)and letψ:I→R2be the map ψ(t)/equalx(3at/(1 + t3,3at2/(1+t3)), where ais a nonzero constant. Show that ψis one-to-one and that ψ′(t)/nequal0 for all t∈I. Isψan embedding and is ψ(I)a manifold? (Observe that ψ(I)is a portion of the curve studied in Exercise /one.taboldstyle./two.taboldstyle.) /six.taboldstyle./five.taboldstyle.Defineψ:R→R2by ψ(t)/equalx/parenleftbig−f(t),f(t)/parenrightbigift≤0,/parenleftbigf(t),f(t)/parenrightbigift≥0, where fis the function given in Exercise B./three.taboldstyle. Show that ψis smooth, one-to-one and that its inverseψ−1:ψ(R)→Ris continuous. Sketch the image of ψ. Isψ(R)a manifold? /six.taboldstyle./six.taboldstyle.Define a map ψ:R2→R4byψ(t1,t2)/equalx/parenleftbigt3 1,t2 1t2,t1t2 2,t3 2/parenrightbig. (i) Show that ψis one-to-one. (ii) Show that Dψ(t)is one-to-one for all t/nequal0. (iii) Let Ube the punctured plane R2\{0}. Show that ψ:U→R4is an embedding. Conclude that ψ(U)is a two-manifold in R4. (iv) Find a basis of the tangent plane to ψ(U)at the point ψ(1,1). /six.taboldstyle./seven.taboldstyle. (i) Let Ube an open subset of Rnand letψ:U→RNbe a smooth map. Thetangent map ofψis the map Tψ:U×Rn→R2Ndefined by Tψ(t,u)/equalx/parenleftbigψ(t),Dψ(t)u/parenrightbigfort∈Uandu∈Rn. Prove that the tangent map of an embed- ding is an embedding. EXERCISES /eight.taboldstyle/seven.taboldstyle (ii) Let Mbe an n-manifold in RN. The tangent bundle ofMis the subset TMofR2N defined by TM/equalx{(x,v)∈R2N|v∈TxM}. Prove that the tangent bundle of Mis a2n-manifold. /six.taboldstyle./eight.taboldstyle.LetMbe the set of points in R2given by the equation (x2+y2)2+y2−x2/equalx0. (i) Show that M\{(0,0)}is a1-manifold. (ii) Determine the points where Mhas horizontal or vertical tangent lines. (iii) Sketch M. (Start by finding the intersection points of Mwith an arbitrary line through the origin, y/equalxax.) (iv) Is Ma manifold at (0,0)? Explain. /six.taboldstyle./nine.taboldstyle.Letφ:Rn\{0}→Rbe a homogeneous function of degree pas defined in Exercise B./six.taboldstyle. Assume that φis smooth and that p/nequal0. Show that 0is the only possible singular value ofφ. (Use the result of Exercise B./six.taboldstyle.) Conclude that, if nonempty, φ−1(c)is an n−1-manifold forc/nequal0. /six.taboldstyle./one.taboldstyle/zero.taboldstyle. Letφ(x)/equalxa1x2 1+a2x2 2+···+anx2n, where the aiare nonzero constants. Determine the regular and singular values of φ. For n/equalx3sketch the level surface φ−1(c)for a regular value c. (You have to distinguish between a few different cases.) /six.taboldstyle./one.taboldstyle/one.taboldstyle. Show that the trajectories of the Lotka-Volterra system of E xercise /one.taboldstyle./one.taboldstyle/one.taboldstyleare one- dimensional manifolds. /six.taboldstyle./one.taboldstyle/two.taboldstyle. LetVbe the vector space of n×n-matrices and let Wbe its linear subspace consisting of all symmetric matrices. (i) Prove that dim(V)/equalxn2anddim(W)/equalx1 2n(n+ 1). (Exhibit explicit bases of V and W, and count the number of elements in each basis.) (ii) Compute the dimension of the orthogonal group O(n)and show that its tangent space at the identity matrix Iis the set of all antisymmetric n×n-matrices. (Use the regular value theorem, which says that the dimension of O(n)isdim(V)− dim(W)and that the tangent space at the identity matrix is the kerne l ofDφ(I), whereφ:V→Wis defined by φ(A)/equalxATA. See the proof of Theorem /six.taboldstyle./one.taboldstyle/eight.taboldstyle.) /six.taboldstyle./one.taboldstyle/three.taboldstyle. LetVbe the vector space of n×n-matrices. (i) The general linear group is the subset of Vdefined by GL(n)/equalx{A∈V|det(A)/nequal0}. Show that GL(n)is a manifold. What is its dimension? (ii) Defineφ:V→Rbyφ(A)/equalxdet(A). Show that Dφ(A)B/equalxn/summationdisplay i/equalx1det(a1,a2,..., ai−1,bi,ai+1,..., an), where a1,a2,...,anandb1,b2,...,bndenote the column vectors of A, resp. B. (Apply Lemma B./one.taboldstylefor the derivative and use the multilinearity of the determi - nant.) (iii) The special linear group is the subset of Vdefined by SL(n)/equalx{A∈V|det(A)/equalx1}. Show that SL(n)is a manifold. What is its dimension? (iv) Show that for A/equalxI, the identity matrix, we have Dφ(A)B/equalx/summationtextn i/equalx1bi,i/equalxtr(B), thetrace ofB. Conclude that the tangent space to SL(n)atIis the set of traceless matrices, i.e. matrices Bsatisfying tr(B)/equalx0. /eight.taboldstyle/eight.taboldstyle /six.taboldstyle. MANIFOLDS /six.taboldstyle./one.taboldstyle/four.taboldstyle. (i) Let Wbe punctured 4-space R4\{0}and defineφ:W→Rby φ(x)/equalxx1x4−x2x3. Show that 0is a regular value of φ. (ii) Let Abe a real 2×2-matrix. Show that rank (A)/equalx1if and only if det(A)/equalx0and A/nequal0. (iii) Let Mbe the set of 2×2-matrices of rank 1. Show that Mis a three-dimensional manifold. (iv) Compute TAM, where A/equalx/parenleftbig1 1 0 0/parenrightbig. /six.taboldstyle./one.taboldstyle/five.taboldstyle. Defineφ:R4→R2byφ(x)/equalx(x1+x2+x3x4,x1x2x3+x4). (i) Show that Dφ(x)has rank 2unless xis of the form (t−2,t−2,t,t−3)for some t/nequal0. (Row reduce the matrix Dφ(x)to compute its rank.) (ii) Show that M/equalxφ−1(0)is a2-manifold (where 0is the origin in R2). (iii) Find a basis of the tangent space TxMfor all x∈Mwith x3/equalx0. (The answer depends on x.) /six.taboldstyle./one.taboldstyle/six.taboldstyle. LetUbe an open subset of Rnand letφ:U→Rmbe a smooth map. Let Mbe the manifold φ−1(c), where cis a regular value of φ. Let f:U→Rbe a smooth function. A point x∈Mis called a critical point for the restricted function f|MifD f(x)v/equalx0for all tangent vectors v∈TxM. Prove that x∈Mis critical for f|Mif and only if there exist numbers λ1, λ2,...,λmsuch that grad (f)(x)/equalxλ1grad (φ1)(x)+λ2grad (φ2)(x)+···+λmgrad (φm)(x). (Use the characterization of TxMgiven by the regular value theorem.) /six.taboldstyle./one.taboldstyle/seven.taboldstyle. Find the critical points of the function f(x,y,z)/equalx−x+ 2y+ 3zover the circle C given by x2+y2+z2/equalx1, x+z/equalx0. Where are the maxima and minima of f|C? /six.taboldstyle./one.taboldstyle/eight.taboldstyle (/e.sc/i.sc/g.sc/e.sc/n.sc/v.sc/e.sc/c.sc/t.sc/o.sc/r.sc/s.sc /v.sc/i.sc/a.sc /c.sc/a.sc/l.sc/c.sc/u.sc/l.sc/u.sc/s.sc). LetA/equalxATbe a symmetric n×n-matrix and define f:Rn→Rbyf(x)/equalxx·Ax. Let Mbe the unit sphere {x∈Rn|/bardblx/bardbl/equalx1}. (i) Calculate grad (f)(x). (ii) Show that x∈Mis a critical point of f|Mif and only if xis an eigenvector for A of length 1. (iii) Given an eigenvector xof length 1, show that f(x)is the corresponding eigenvalue ofx. CHAPTER /seven.taboldstyle Differential forms on manifolds /seven.taboldstyle./one.taboldstyle. First definition There are several different ways to define differential forms o n manifolds. In this section we present a practical, workaday definition. A more theoretical approach is taken in Section /seven.taboldstyle./two.taboldstyle. LetMbe an n-manifold in RNand let us first consider what we might mean by a0-form or smooth function on M. A function f:M→Ris simply an assignment of a unique number f(x)to each point xinM. For instance, Mcould be the surface of the earth and fcould represent temperature at a given time, or height above sea level. But how would we define such a function to be di fferentiable? The difficulty here is that if xis in Mandejis one of the standard basis vectors, the straight line x+tejmay not be contained in M, so we cannot form the limit ∂f/∂xj/equalxlim t→0(f(x+tej)−f(x))/t. Here is one way out of this difficulty. Because Mis a manifold there exist open sets UiinRnand embeddings ψi:Ui→RNsuch that the images ψi(Ui)cover M: M/equalx/uniontext iψi(Ui). (Here iranges over some unspecified, possibly infinite, index set.) For each iwe define a function fi:Ui→Rbyfi(t)/equalxf(ψi(t)), i.e. fi/equalxψ∗ i(f). We call fithelocal representative offrelative to the embedding ψi. (For instance, if M is the earth’s surface, fis temperature, and ψiis a map of New York State, then fi represents a temperature chart of NY.) Since fiis defined on the open subset Uiof Rn, it makes sense to ask whether its partial derivatives exist . We say that fisCkif each of the local representatives fiisCk. Now suppose that xis in the overlap of two charts. Then we have two indices iand jand vectors t∈Uiandu∈Ujsuch that x/equalxψi(t)/equalxψj(u). Then we must have f(x)/equalxf(ψi(t))/equalxf(ψj(u)), sofi(t)/equalxfj(u). Alsoψi(t)/equalxψj(u)implies t/equalxψ−1 i◦ψj(u)and therefore fj(u)/equalxfi/parenleftbigψ−1 i◦ψj(u)/parenrightbig. This identity must hold for all u∈Ujsuch thatψj(u)∈ψi(Ui), i.e. for all uin ψ−1 j(ψi(Ui)). We can abbreviate this by saying that fj/equalx(ψ−1 i◦ψj)∗(fi) onψ−1 j(ψi(Ui)). This is a consistency condition on the functions fiimposed by the fact that they are pullbacks of a single function fdefined everywhere on M. The map ψ−1 i◦ψjis often called a change of coordinates ortransition map , and the consistency condition is also known as the transformation law for the local representatives fi. (Pursuing the weather chart analogy, it expresses nothing but the obvious fact that where the maps of New York and Pennsylva nia overlap, the corresponding two temperature charts must show the same temperatures.) Conversely, the collection of all local representatives fidetermines f, because we /eight.taboldstyle/nine.taboldstyle /nine.taboldstyle/zero.taboldstyle /seven.taboldstyle. DIFFERENTIAL FORMS ON MANIFOLDS have f(x)/equalxfi(ψ−1 i(x))ifx∈ψi(Ui). (That is to say, if we have a complete set of weather charts for the whole world, we know the temperature e verywhere.) Following this cue we formulate the following definition. /seven.taboldstyle./one.taboldstyle. D/e.sc/f.sc/i.sc/n.sc/i.sc/t.sc/i.sc/o.sc/n.sc. Adifferential form of degree k, or simply a k-form ,αonMis a collection of k-formsαionUisatisfying the transformation law αj/equalx(ψ−1 i◦ψj)∗(αi) (/seven.taboldstyle./one.taboldstyle) onψ−1 j(ψi(Ui)). We callαithelocal representative ofαrelative to the embedding ψiand denote it by αi/equalxψ∗ i(α). The collection of all k-forms on Mis denoted by Ωk(M). This definition is rather indirect, but it works really well i f a specific atlas for the manifold Mis known. Definition /seven.taboldstyle./one.taboldstyleis particularly tractible if Mis the image of a single embedding ψ:U→RN. In that case the compatibility relation ( /seven.taboldstyle./one.taboldstyle) is vacuous and a k-formαonMis determined by one single representative, a k-form ψ∗(α)onU. Sometimes it is useful to write the transformation law ( /seven.taboldstyle./one.taboldstyle) in components. Ap- pealing to Theorem /three.taboldstyle./one.taboldstyle/three.taboldstylewe see that ( /seven.taboldstyle./one.taboldstyle) is equivalent to the following requirement: if αi/equalx/summationdisplay IfIdtI andαj/equalx/summationdisplay JgJdtJ are two local representatives for α, then gJ/equalx/summationdisplay I(ψ−1 i◦ψj)∗/parenleftbigfIdet(D(ψ−1 i◦ψj)I,J)/parenrightbig. onψ−1 j(ψi(Ui)). Just like forms on Rn, forms on a manifold can be added, multiplied, differ- entiated and integrated. For example, suppose αis ak-form andβanl-form on M. Supposeαi, resp.βi, is the local representative of α, resp.β, relative to an embedding ψi:Ui→M. Then we define the product γ/equalxαβby settingγi/equalxαiβi. To see that this definition makes sense, we check that the form sγisatisfy the transformation law ( /seven.taboldstyle./one.taboldstyle): γj/equalxαjβj/equalx(ψ−1 i◦ψj)∗(αi)(ψ−1 i◦ψj)∗(βi)/equalx(ψ−1 i◦ψj)∗(αiβi)/equalx(ψ−1 i◦ψj)∗(γi). Here we have used the multiplicative property of pullbacks, Proposition /three.taboldstyle./one.taboldstyle/zero.taboldstyle(ii). Similarly, the exterior derivative of αis defined by setting (dα)i/equalxdαi. As before, let us check that the forms (dα)isatisfy the transformation law ( /seven.taboldstyle./one.taboldstyle): (dα)j/equalxdαj/equalxd(ψ−1 i◦ψj)∗(αi)/equalx(ψ−1 i◦ψj)∗(dαi)/equalx(ψ−1 i◦ψj)∗((dα)i), where we used Theorem /three.taboldstyle./one.taboldstyle/one.taboldstyle. /seven.taboldstyle./two.taboldstyle. Second definition This section presents some of the algebraic underpinnings o f the theory of differential forms. This branch of algebra, now called exterior oralternating algebra was invented by Grassmann in the mid-nineteenth century and is a prerequisite for much of the more advanced literature on the subject. /seven.taboldstyle./two.taboldstyle. SECOND DEFINITION /nine.taboldstyle/one.taboldstyle Covectors. A covector is a little dinosaur that eats vectors and spits ou t num- bers, in a linear way. The formal definition goes as follows. Let Vbe a vector space over the real numbers, for example Rnor a linear subspace of Rn. Acovector , ordual vector , or linear functional , is a linear map from VtoR. /seven.taboldstyle./two.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetV/equalxC0([a,b],R), the collection of all continuous real-valued functions on a closed and bounded interval [a,b]. A linear combination of con- tinuous functions is continuous, so Vis a vector space. Define µ(f)/equalx/integraltextb af(x)dx. Thenµ(c1f1+c2f2)/equalxc1µ(f1)+c2µ(f2)for all functions f1,f2∈Vand all scalars c1,c2, soµis a linear functional on V. The collection of all covectors on Vis denoted by V∗and called the dualofV. The dual is a vector space in its own right: if µ1andµ2are in V∗we defineµ1+µ2 and cµ1by setting (µ1+µ2)(v)/equalxµ1(v)+µ2(v)and(cµ1)(v)/equalxcµ1(v)for all v∈V. For the next example, recall that if Ais an m×n-matrix and xann-vector, then Axis an m-vector, and the map which sends xtoAxis linear. Moreover, every linear map from RntoRmis of this form for a unique matrix A. /seven.taboldstyle./three.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. A covector on Rnis a linear map from RntoR/equalxR1and is therefore given by a 1×n-matrix, which is nothing but a row vector. Thus (Rn)∗ is the space of row n-vectors. A row vector y“eating” a column vector xmeans multiplying the two, which results in a number: yx/equalx/parenleftBig y1y2··· yn/parenrightBig/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAx1 x2 ... xn/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA/equalxn/summationdisplay i/equalx1yixi. Now suppose that Vis a vector space of finite dimension nand choose a basis b1,b2,...,bnofV. Then every vector b∈Vcan be written in a unique way as a linear combination/summationtext jcjbj. Define a covector βi∈V∗byβi(b)/equalxci. In other words,βiis determined by the rule βi(bj)/equalxδi,j, where δi,j/equalx1ifi/equalxj, 0ifi/nequalj is the Kronecker delta . We callβithei-th coordinate function . /seven.taboldstyle./four.taboldstyle. L/e.sc/m.sc/m.sc/a.sc. The coordinate functions β1,β2,...,βnform a basis of V∗. It follows thatdim(V∗)/equalxn/equalxdim(V). P/r.sc/o.sc/o.sc/f.sc. Letβ∈V∗. We need to prove that βcan be written as a linear combina- tionβ/equalx/summationtextn i/equalx1ciβiwith unique coefficients ci. First we prove uniqueness. Assuming /nine.taboldstyle/two.taboldstyle /seven.taboldstyle. DIFFERENTIAL FORMS ON MANIFOLDS thatβcan be expressed as β/equalx/summationtextn i/equalx1ciβi, we can apply both sides to the vector bjto obtain β(bj)/equalxn/summationdisplay i/equalx1ciβi(bj)/equalxn/summationdisplay i/equalx1ciδi,j/equalxcj. (/seven.taboldstyle./two.taboldstyle) Socj/equalxβ(bj)is the only possible choice for the coefficient cj. This argument establishes the uniqueness of the coefficients. Moreover, it tells us what the co- efficients should be, which helps us prove that they exist. Nam ely, let us define β′/equalx/summationtextn i/equalx1β(bi)βi. Then by equation ( /seven.taboldstyle./two.taboldstyle),β′(bj)/equalxβ(bj)for all j, soβ′/equalxβ, and thereforeβ/equalx/summationtextn i/equalx1β(bi)βi. This proves that β1,β2,...,βnconstitute a basis of V∗. The cardinality of the basis is n, sodim(V∗)/equalxn. QED The basis{β1,β2,...,β n}ofV∗is said to be dualto the basis{b1,b2,..., bn}of V. /seven.taboldstyle./five.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Let{b1,b2,..., bn}be a basis of Rn. What is the dual basis {β1,β2,...,β n}of(Rn)∗? Theβi’s are row vectors determined by the equations βibj/equalxδi,j. These equations can be written as a single matrix equation: letBbe then×n-matrix with columns b1,b2,...,bnand let Abe the n×n-matrix with rowsβ1,β2,...,βn; then AB/equalxI. Therefore Ais the inverse of B. In other words, βiis the i-th row of B−1. As a special case consider the standard basis {e1,..., en}. Then B/equalxI, soA/equalxI, and the dual basis of (Rn)∗is{eT 1,eT 2,..., eT n}. Dual bases come in handy when writing the matrix of a linear ma p. Let L:V→Wbe a linear map between vector spaces Vand W. To write the matrix ofLwe need to start by picking a basis b1,b2,...,bnofVand a basis c1,c2,..., cmofW. Then for each j/equalx1,2,...,nthe vector Lbjcan be expanded uniquely in terms of the c’s:Lbj/equalx/summationtextm i/equalx1li,jci. The m×nnumbers li,jmake up the matrix of L relative to the two bases of Vand W. /seven.taboldstyle./six.taboldstyle. L/e.sc/m.sc/m.sc/a.sc. Letγ1,γ2,...,γn∈W∗the dual basis of c1,c2,...,cn. Then the (i,j)-th matrix element of a linear map L:V→Wis equal to li,j/equalxγi(Lbj). P/r.sc/o.sc/o.sc/f.sc. We have Lbj/equalx/summationtextm k/equalx1lk,jck, so γi(Lbj)/equalxm/summationdisplay k/equalx1lk,jγi(ck)/equalxm/summationdisplay k/equalx1lk,jδi,k/equalxli,j, that is to say li,j/equalxγi(Lbj). QED 1-Forms on Rnre-examined. LetUbe an open subset of Rn. Recall that a vector field on Uis a smooth map F:U→Rn. A1-form is a type of object “dual” to a vector field. Formally, a 1-form orcovector field onUis defined as a smooth map α:U→(Rn)∗. This means that αis a row vector α/equalx(f1f2···fn) whose entries are smooth functions on U. The form is called constant if the entries f1,...,fnare constant. By definition dxiis the constant 1-form dxi/equalxeT i/equalx(0···0 1 0···0), (/seven.taboldstyle./three.taboldstyle) /seven.taboldstyle./two.taboldstyle. SECOND DEFINITION /nine.taboldstyle/three.taboldstyle the transpose of ei, the i-th standard basis vector of Rn. Every 1-form can thus be written as α/equalx(f1f2···fn)/equalxn/summationdisplay i/equalx1fidxi. Using this formalism we can write for any smooth function gonU dg/equalxn/summationdisplay i/equalx1∂g ∂xidxi/equalx/parenleftbigg∂g ∂x1∂g ∂x2···∂g ∂xn/parenrightbigg , which is simply the Jacobi matrix D gofg! (This is the reason that many authors use the notation dgfor the Jacobi matrix.) In what sense does the row vector dxirepresent an “infinitesimal increment” along the xi-axis? Let v∈Rnbe the velocity vector of a path c(t)at time t. 0c(t)v In an infinitesimal time interval ∆tthe position changes to c(t+∆t)≈c(t)+∆tv, so the infinitesimal displacement is ∆tv. The xi-coordinate changes by an amount ∆tvi/equalx∆t dx i(v). We conclude that the number dxi(v)represents the rate of change of the i-th coordinate along the path per unit time. Multilinear algebra. Multilinear algebra is needed to make sense of differen- tial forms of higher degree. LetVbe a vector space and let Vkdenote the Cartesian product V×···× V (ktimes). Thus an element of Vkis a k-tuple (v1,v2,..., vk)of vectors in V. A k-multilinear function on Vis a function µ:Vk→Rwhich is linear in each vector, i.e. µ(v1,v2,..., cvi+c′v′ i,..., vk)/equalxcµ(v1,v2,..., vk)+c′µ(v1,v2,..., v′ i,..., vk) for all scalars c,c′and all vectors v1,v2,...,vi,v′ i,...,vk. /seven.taboldstyle./seven.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetV/equalxRnand letµ(x,y)/equalxx·y, the inner product of xandy. Thenµis bilinear (i.e. 2-multilinear). /seven.taboldstyle./eight.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetV/equalxR4,k/equalx2. The function µ(v,w)/equalxv1w2−v2w1+v3w4− v4w3is bilinear on R4. /seven.taboldstyle./nine.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetV/equalxRn,k/equalxn. It follows from Corollary /three.taboldstyle./six.taboldstylethat the determinant det(v1,v2,..., vn)is an n-multilinear function on Rn. Ak-multilinear function is alternating orantisymmetric if it has the alternating property, µ(v1,..., vj,..., vi,..., vk)/equalx−µ(v1,..., vi,..., vj,..., vk) for all v1,v2,...,vkinV. More generally, if µis alternating, then for any permu- tationσ∈Skwe have µ(vσ(1),..., vσ(k))/equalxsign(σ)µ(v1,..., vk). /nine.taboldstyle/four.taboldstyle /seven.taboldstyle. DIFFERENTIAL FORMS ON MANIFOLDS /seven.taboldstyle./one.taboldstyle/zero.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. The inner product of Example /seven.taboldstyle./seven.taboldstyleis bilinear, but it is not alter- nating. Indeed it is symmetric :y·x/equalxx·y. The bilinear function of Example /seven.taboldstyle./eight.taboldstyleis alternating, and so is the determinant function of Example /seven.taboldstyle./nine.taboldstyle. Here is a useful trick for producing alternating k-multilinear functions starting from kcovectorsµ1,µ2,...,µk∈V∗. The (wedge) product is the function µ1µ2···µk:Vk→R defined by µ1µ2···µk(v1,v2,..., vk)/equalxdet/parenleftbigµi(vj)/parenrightbig 1≤i,j≤k. (The determinant on the right is a k×k-determinant.) It follows from the mul- tilinearity and the alternating property of the determinan t thatµ1µ2···µkis an alternating k-multilinear function. Some authors denote the wedge produ ct by µ1∧µ2∧···∧µkto distinguish it from other products, such as the tensor pro duct defined in Exercise /seven.taboldstyle./six.taboldstyle. The collection of all alternating k-multilinear functions is denoted by Ak(V). For any k,k-multilinear functions can be added and scalar-multiplied just like ordinary linear functions, so the set Ak(V)forms a vector space. Fork/equalx1the alternating property is vacuous, so an alternating 1-multilinear function is nothing but a linear function. Thus A1(V)/equalxV∗. A0-multilinear function is by convention just a number. Thus A0(V)/equalxR. There is a nice way to construct a basis of the vector space Ak(V)starting from a basis{b1,..., bn}ofV. The idea is to take wedge products of dual basis vectors. Let{β1,...,β n}be the corresponding dual basis of V∗. Let I/equalx(i1,i2,..., ik)be an increasing multi-index, i.e. 1≤i1<i2<···<ik≤n. Write βI/equalxβi1βi2···βik∈Ak(V), bI/equalx(bi1,bi2,..., bik)∈Vk. /seven.taboldstyle./one.taboldstyle/one.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetV/equalxR3with standard basis {e1,e2,e3}. The dual basis of (R3)∗is{dx1,dx2,dx3}. Let k/equalx2and I/equalx(1,2),J/equalx(2,3). Then dxI(eI)/equalx/barex/barex/barex/barex/barexdx1(e1)dx1(e2) dx2(e1)dx2(e2)/barex/barex/barex/barex/barex/equalx/barex/barex/barex/barex/barex1 0 0 1/barex/barex/barex/barex/barex/equalx1, dxI(eJ)/equalx/barex/barex/barex/barex/barexdx1(e2)dx1(e3) dx2(e2)dx2(e3)/barex/barex/barex/barex/barex/equalx/barex/barex/barex/barex/barex0 0 1 0/barex/barex/barex/barex/barex/equalx0, dxJ(eI)/equalx/barex/barex/barex/barex/barexdx2(e1)dx2(e2) dx3(e1)dx3(e2)/barex/barex/barex/barex/barex/equalx/barex/barex/barex/barex/barex0 1 0 0/barex/barex/barex/barex/barex/equalx0, dxJ(eJ)/equalx/barex/barex/barex/barex/barexdx2(e2)dx2(e3) dx3(e2)dx3(e3)/barex/barex/barex/barex/barex/equalx/barex/barex/barex/barex/barex1 0 0 1/barex/barex/barex/barex/barex/equalx1. This example generalizes as follows. For multi-indices Iand Jlet us define a generalized Kronecker delta δI,Jby δI,J/equalx1ifI/equalxJ, 0ifI/nequalJ. /seven.taboldstyle./one.taboldstyle/two.taboldstyle. L/e.sc/m.sc/m.sc/a.sc. LetIandJbe increasing multi-indices of degree k. ThenβI(bJ)/equalxδI,J. /seven.taboldstyle./two.taboldstyle. SECOND DEFINITION /nine.taboldstyle/five.taboldstyle P/r.sc/o.sc/o.sc/f.sc. LetI/equalx(i1,..., ik)and J/equalx(j1,..., jk). Then βI(bJ)/equalxdet/parenleftbigβir(bjs)/parenrightbig 1≤r,s≤k/equalx/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barexδi1,j1... δ i1,jk...... δil,j1... δ il,jk...... δik,j1... δ ik,jk/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex. IfI/equalxJ, then this matrix is the identity k×k-matrix, soβI(bJ)/equalx1. IfI/nequalJ, then there is some i∈Iwhich is not in J, say i/equalxil, which causes all entries in the l-th row of the matrix to vanish. Hence its determinant is 0, and therefore βI(bJ)/equalx0. QED We need one further technical result before showing that the functionsβIare a basis of Ak(V). /seven.taboldstyle./one.taboldstyle/three.taboldstyle. L/e.sc/m.sc/m.sc/a.sc. Letβ∈Ak(V). Supposeβ(bI)/equalx0for all increasing multi-indices I of degree k. Thenβ/equalx0. P/r.sc/o.sc/o.sc/f.sc. The assumption implies β(bi1,..., bik)/equalx0 (/seven.taboldstyle./four.taboldstyle) forallmulti-indices (i1,..., ik), because of the alternating property. We need to show thatβ(v1,v2,..., vk)/equalx0for arbitrary vectors v1,v2,...,vk. We can expand theviusing the basis: v1/equalxa1,1b1+a1,2b2+···+a1,kbk, v2/equalxa2,1b1+a2,2b2+···+a2,kbk, ... vk/equalxak,1b1+ak,2b2+···+ak,kbk. Therefore by multilinearity β(v1,v2,..., vk)/equalxk/summationdisplay i1/equalx1···k/summationdisplay ik/equalx1a1,i1a2,i2···ak,ikβ(bi1,bi2,..., bik). Each term in the right-hand side is 0by equation ( /seven.taboldstyle./four.taboldstyle). QED /seven.taboldstyle./one.taboldstyle/four.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. LetVbe an n-dimensional vector space with basis {b1,b2,..., bn}. Let{β1,β2,...,β n}be the corresponding dual basis of V∗. Then the alternating k- multilinear functions βI/equalxβi1βi2···βik, where Iranges over the set of all increasing multi-indices of degree k, form a basis of Ak(V). Hence dim(Ak(V))/equalx/parenleftbign k/parenrightbig. P/r.sc/o.sc/o.sc/f.sc. The proof is closely analogous to that of Lemma /seven.taboldstyle./four.taboldstyle. Letβ∈Ak(V). We need to write βas a linear combination β/equalx/summationtext IcIβI. Assuming for the moment that this is possible, we apply both sides to the k-tuple of vectors bJ. Using Lemma /seven.taboldstyle./one.taboldstyle/two.taboldstylewe obtain β(bJ)/equalx/summationdisplay IcIβI(bJ)/equalx/summationdisplay IcIδI,J/equalxcJ. SocJ/equalxβ(bJ)is the only possible choice for the coefficient cJ. To show that this choice of coefficients works, let us define β′/equalx/summationtext Iβ(bI)βI. Then for all increasing multi-indices Iwe haveβ(bI)−β′(bI)/equalxβ(bI)−β(bI)/equalx0. Applying Lemma /seven.taboldstyle./one.taboldstyle/three.taboldstyle /nine.taboldstyle/six.taboldstyle /seven.taboldstyle. DIFFERENTIAL FORMS ON MANIFOLDS toβ−β′we findβ−β′/equalx0. In other words, β/equalx/summationtext Iβ(bI)βI. We have proved that theβIform a basis. The dimension of Ak(V)is equal to the cardinality of the basis, which is/parenleftbign k/parenrightbig. QED /seven.taboldstyle./one.taboldstyle/five.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetV/equalxRnwith standard basis {e1,..., en}. The dual basis of (Rn)∗is{dx1,..., dxn}. (See ( /seven.taboldstyle./three.taboldstyle) and Example /seven.taboldstyle./five.taboldstyle.) Therefore Ak(V)has a basis consisting of all k-multilinear functions of the form dxI/equalxdxi1dxi2···dxik, with 1≤i1<···<ik≤n. Hence a general alternating k-multilinear function µ onRnlooks like µ/equalx/summationdisplay IaIdxI, with aIconstant. By Lemma /seven.taboldstyle./one.taboldstyle/two.taboldstyle,µ(eJ)/equalx/summationtext IaIdxI(eJ)/equalx/summationtext IaIδI,J/equalxaJ, so the coefficient aIis equal toµ(eI). k-Forms on Rnre-examined. LetUbe an open subset of Rn. We define a k-formαonUto be a smooth map α:U→Ak(Rn). This means that αcan be written as α/equalx/summationdisplay IfIdxI, where the coefficients fIare smooth functions on U. The value of αatx∈U is denoted by αx, so that we have αx/equalx/summationtext IfI(x)dxIfor all x∈U. For each xthe objectαxis an element of Ak(Rn), that is to say a k-multilinear function onRn. So for any k-tuple vI/equalx(v1,v2,..., vk)of vectors in Rnthe expression αx(vI)/equalxαx(v1,v2,..., vk)is a number. Example /seven.taboldstyle./one.taboldstyle/five.taboldstylegives us a useful formula for the coefficients fI, namely fI/equalxα(eI)(which is to be interpreted as fI(x)/equalxαx(eI) for all x). Pullbacks re-examined. In the light of this new definition we can give a fresh interpretation of a pullback. This will be useful in our stud y of forms on manifolds. LetUand Vbe open subsets of Rn, resp. Rm, andφ:U→Va smooth map. For a k-formα∈Ωk(V)define the pullback φ∗(α∈Ωk(U)by φ∗(α)x(v1,v2,..., vk)/equalxαφ(x)(Dφ(x)v1,Dφ(x)v2,..., Dφ(x)vk). Let us check that this formula agrees with the old definition. We writeα/equalx/summationtext IfIdyI, where the fIare smooth functions on V, andφ∗(α)/equalx/summationtext JgJdxJ, where the gJare smooth functions on U. What is the relationship between gJand fI? We use the formula gJ/equalxφ∗(α)(eJ), our new definition of pullback and the definition of the wedge product to obtain gJ(x)/equalxφ∗(α)x(eJ)/equalxαφ(x)(Dφ(x)ej1,Dφ(x)ej2,..., Dφ(x)ejk) /equalx/summationdisplay IfI(φ(x))dyI(Dφ(x)ej1,Dφ(x)ej2,..., Dφ(x)ejk) /equalx/summationdisplay Iφ∗(fI)(x)det/parenleftbigdyir(Dφ(x)ejs)/parenrightbig 1≤r,s≤k. By Lemma /seven.taboldstyle./six.taboldstylethe number dyir(Dφ(x)ejs)is the irjs-matrix entry of the Jacobi ma- trixDφ(x)(with respect to the standard basis e1,e2,...,enofRnand the standard basis e1,e2,...,emofRm). In other words, gJ(x)/equalx/summationtext Iφ∗(fI)(x)det(DφI,J(x)). This EXERCISES /nine.taboldstyle/seven.taboldstyle formula is identical to the one in Theorem /three.taboldstyle./one.taboldstyle/three.taboldstyleand therefore our new definition agrees with the old! Forms on manifolds. LetMbe an n-dimensional manifold in RN. For each point xinMthe tangent space TxMis an n-dimensional linear subspace of RN. The book [ BT/eight.taboldstyle/two.taboldstyle ] describes a k-form on Mas an animal that inhabits the world M, eats k-tuples of tangent vectors, and spits out numbers. Formally , adifferential form of degree kor a k-formαonMis a choice of an alternating k-multilinear map αxon the vector space TxM, one for each x∈M. This alternating map αxis required to depend smoothly on xin the following sense. Let ψ:U→RNbe a local parametrization of Matx. The tangent space at xis then TxM/equalxDψ(t)(Rn), where t∈Uis chosen such that ψ(t)/equalxx. The pullback ofαunder the local parametrization ψis defined by ψ∗(α)t(v1,v2,..., vk)/equalxαψ(t)(Dψ(t)v1,Dψ(t)v2,..., Dψ(t)vk). Thenψ∗(α)is ak-form on U, an open subset of Rn, soψ∗(α)/equalx/summationtext IfIdtIfor certain functions fIdefined on U. We will require the functions fIto be smooth. (The formψ∗(α)/equalx/summationtext IfIdtIis the local representative ofαrelative to the embedding ψ introduced in Section /seven.taboldstyle./one.taboldstyle.) To recapitulate: /seven.taboldstyle./one.taboldstyle/six.taboldstyle. D/e.sc/f.sc/i.sc/n.sc/i.sc/t.sc/i.sc/o.sc/n.sc. Ak-formαonMis a choice, for each x∈M, of an alternating k-multilinear map αxonTxM, which depends smoothly on x. We can calculate the local representative ψ∗(α)of a k-formαfor any lo- cal parametrization ψ:U→RNofM. Suppose we had two different local parametrizations ψi:Ui→RNandψj:Uj→RNofMatx. Then the local expressions αi/equalxψ∗ i(α)andαj/equalxψ∗ j(α)forαare related by the formula αj/equalx(ψ−1 i◦ψj)∗(αi). This is identical to the transformation law ( /seven.taboldstyle./one.taboldstyle), which shows that Definitions /seven.taboldstyle./one.taboldstyle and/seven.taboldstyle./one.taboldstyle/six.taboldstyleof differential forms on a manifold are equivalent. /seven.taboldstyle./one.taboldstyle/seven.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetMbe a one-dimensional manifold in RN. Let us choose an orientation (“direction”) on M. A tangent vector to Mispositive if it points in the same direction as the orientation and negative if it points in the opposite direction. Define a 1-formαonMas follows. For x∈Mand a tangent vector v∈TxMput αx(v)/equalx/bardblv/bardblifvis positive, −/bardblv/bardblifvis negative. The formαis the element of arc length ofM. We shall see in Chapter /eight.taboldstylehow to generalize it to higher-dimensional manifolds and in Chapt er/nine.taboldstylehow to use it to calculate arc lengths and volumes. Exercises /seven.taboldstyle./one.taboldstyle.The vectors e1+e2ande1−e2form a basis of R2. What is the dual basis of (R2)∗? /seven.taboldstyle./two.taboldstyle.Let{b1,b2,..., bn}be a basis of Rn. Show that{bT 1,bT 2,..., bT n}is the correspond- ing dual basis of (Rn)∗if and only if the basis is orthonormal. /seven.taboldstyle./three.taboldstyle. Letµbe a k-multilinear function on a vector space V. Suppose that µsatisfies µ(v1,v2,..., vk)/equalx0whenever two of the vectors v1,v2,...,vkare equal, i.e. vi/equalxvjfor some pair of distinct indices i/nequalj. Prove that µis alternating. /nine.taboldstyle/eight.taboldstyle /seven.taboldstyle. DIFFERENTIAL FORMS ON MANIFOLDS /seven.taboldstyle./four.taboldstyle.Show that the bilinear function µof Example /seven.taboldstyle./eight.taboldstyleis equal to dx1dx2+dx3dx4. /seven.taboldstyle./five.taboldstyle.The wedge product is a generalization of the cross product to arbitrary dimensions in the sense that x×y/equalx/parenleftbig∗(xT∧yT)/parenrightbigT for all x,y∈R3. Prove this formula. (Interpretation: xandyare column vectors, xTandyT are row vectors, xT∧yTis a2-form on R3,∗(xT∧yT)is a1-form, i.e. a row vector. So both sides of the formula represent column vectors.) /seven.taboldstyle./six.taboldstyle. LetVbe a vector space and let µ1,µ2,...,µk∈V∗be covectors. Their tensor product is the function µ1⊗µ2⊗···⊗µk:Vk→R defined by µ1⊗µ2⊗···⊗µk(v1,v2,..., vk)/equalxµ1(v1)µ2(v2)···µk(vk). Show thatµ1⊗µ2⊗···⊗µkis ak-multilinear function. /seven.taboldstyle./seven.taboldstyle.Letµ:Vk→Rbe a k-multilinear function. Define a new function Alt(µ):Vk→R by Alt(µ)(v1,v2,..., vk)/equalx1 k!/summationdisplay σ∈Sksign(σ)µ(vσ(1),vσ(2),..., vσ(k)). Prove the following. (i)Alt(µ)is an alternating k-multilinear function. (ii)Alt(µ)/equalxµifµis alternating. (iii)Alt(Alt(µ))/equalxAlt(µ)for all k-multilinear µ. (iv) Letµ1,µ2,...,µk∈V∗. Then µ1µ2···µk/equalxk! Alt(µ1⊗µ2⊗···⊗µk). /seven.taboldstyle./eight.taboldstyle. Show that det(v1,v2,..., vn)/equalxdx1dx2···dxn(v1,v2,..., vn)for all vectors v1, v2,...,vn∈Rn. In short, det/equalxdx1dx2···dxn. /seven.taboldstyle./nine.taboldstyle.LetVand Wbe vector spaces and L:V→Wa linear map. Show that L∗(λµ)/equalx L∗(λ)L∗(µ)for all covectors λ,µ∈W∗. CHAPTER /eight.taboldstyle Volume forms /eight.taboldstyle./one.taboldstyle. n-Dimensional volume in RN Theparallelepiped spanned by nvectors a1,a2,...,aninRNis the set of all linear combinations/summationtextn i/equalx1ciai, where the coefficients cirange over the unit interval [0,1]. This is the same as the definition given in Section /three.taboldstyle./one.taboldstyle, except that here we allow the number of vectors nto be different from the dimension N. (Think of a parallelogram in three-space.) We will need a formula fo r the volume of a parallelepiped. If n<Nthere is no coherent way of defining an orientation on alln-parallelepipeds in RN, so this volume will be not an oriented but an absolute volume. (The reason is that for n<Nann-dimensional parallelepiped in RNcan be rotated onto its mirror image through the extra dimension s. This is impossible forn/equalxN.) It turns out that n-dimensional volume in RN, like the determinant, can be characterized by a few reasonable axioms. /eight.taboldstyle./one.taboldstyle. D/e.sc/f.sc/i.sc/n.sc/i.sc/t.sc/i.sc/o.sc/n.sc. An(absolute) n-dimensional Euclidean volume function on RNis a function vol n:RN×RN×···× RN/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipupright ntimes→R with the following properties: (i) homogeneity: vol n(a1,a2,..., cai,..., an)/equalx|c|vol n(a1,a2,..., an) for all scalars cand all vectors a1,a2,...,an; (ii) invariance under shear transformations: vol n(a1,..., ai+caj,..., aj,..., an)/equalxvol n(a1,..., ai,..., aj,..., an) for all scalars cand all pairs of indices i/nequalj; (iii) invariance under Euclidean motions: vol n(Qa1,Qa2,..., Qan)/equalxvol n(a1,a2,..., an) for all orthogonal matrices Q; (iv) normalization: vol n(e1,e2,..., en)/equalx1. We shall shortly see that these axioms uniquely determine th en-dimensional volume function. /eight.taboldstyle./two.taboldstyle. L/e.sc/m.sc/m.sc/a.sc. Leta1,a2,...,anbe vectors in RN. (i)vol n(a1,a2,..., an)/equalx0if the vectors a1,a2,...,anare linearly dependent. (ii)vol n(a1,a2,..., an)/equalx/bardbla1/bardbl/bardbla2/bardbl···/bardbl an/bardblifa1,a2,...,anare orthogonal vectors. /nine.taboldstyle/nine.taboldstyle /one.taboldstyle/zero.taboldstyle/zero.taboldstyle /eight.taboldstyle. VOLUME FORMS P/r.sc/o.sc/o.sc/f.sc. (i) Assume a1,a2,...,anare linearly dependent. For simplicity suppose a1is a linear combination of the other vectors, a1/equalx/summationtextn i/equalx2ciai. By repeatedly applying Axiom ( ii) we get vol n(a1,a2,..., an)/equalxvol n/parenleftbiggn/summationdisplay i/equalx2ciai,a2,..., an/parenrightbigg /equalxvol n/parenleftbiggn/summationdisplay i/equalx3ciai,a2,..., an/parenrightbigg /equalx···/equalxvol n(0,a2,..., an). Now by Axiom ( i), vol n(0,a2,..., an)/equalxvol n(00,a2,..., an)/equalx0 vol n(0,a2,..., an)/equalx0, which proves property ( i). (ii) Suppose a1,a2,...,anare orthogonal. First assume they are nonzero. Then we can define qi/equalx/bardblai/bardbl−1ai. The vectors q1,q2,...,qnare ortho normal . Complete them to an orthonormal basis q1,q2,...,qn,qn+1,...,qNofRN. Let Qbe the matrix whose i-th column is qi. Then Qis orthogonal and Qei/equalxqi. Therefore vol n(a1,a2,..., an)/equalx/bardbla1/bardbl/bardbla2/bardbl···/bardbl an/bardblvol n(q1,q2,..., qn) by Axiom ( i) /equalx/bardbla1/bardbl/bardbla2/bardbl···/bardbl an/bardblvol n(Qe1,Qe2,..., Qen) /equalx/bardbla1/bardbl/bardbla2/bardbl···/bardbl an/bardblvol n(e1,e2,..., en) by Axiom ( iii) /equalx/bardbla1/bardbl/bardbla2/bardbl···/bardbl an/bardbl by Axiom ( iv), which proves part ( ii) if all aiare nonzero. If one of the aiis0, the vectors a1,a2,..., anare dependent, so then the statement follows from part ( i). QED A special case of Lemma /eight.taboldstyle./two.taboldstyle(i) is the following obervation: if n>Nthen every seta1,a2,...,anofnvectors in RNis dependent, so vol n(a1,a2,..., an)/equalx0. This makes sense: a degenerate parallelepiped spanned by three v ectors in the plane has three-dimensional volume equal to 0. This brings us to the volume formula. We can form an N×n-matrix Aout of the column vectors a1,a2,...,an. It does not make sense to take det(A)because A is not square, unless n/equalxN. However, the Gram matrix ATAofAis square and we can take itsdeterminant. /eight.taboldstyle./three.taboldstyle. L/e.sc/m.sc/m.sc/a.sc. We have det(ATA)≥0for all N×n-matrices A, and det(ATA)/equalx0if and only if the columns of Aare dependent. P/r.sc/o.sc/o.sc/f.sc. The inner product of two vectors uandvcan be written as u·v/equalxuTv. In particular, (ATAv)·v/equalx(ATAv)Tv/equalxvTATAv/equalx(Av)·(Av)≥0 (/eight.taboldstyle./one.taboldstyle) for all vectors v∈Rn. The Gram matrix is symmetric and therefore, by the spectral theorem, has an eigenbasis consisting of real eigenvectors . Ifvis an eigenvector ofATAwith eigenvalue λ, thenλv·v/equalx(ATAv)·v, which is nonnegative by (/eight.taboldstyle./one.taboldstyle). Henceλ≥0: all eigenvalues of the Gram matrix are nonnegative, and therefore its determinant is non-negative. If the columns o fAare dependent, then Ahas a nontrivial nullspace, so Av/equalx0for some nonzero v. Hence ATAv/equalx0, so the columns of ATAare dependent as well, so det(ATA)/equalx0. Conversely, if det(ATA)/equalx0then ATAhas a nontrivial nullspace, so ATAw/equalx0for some nonzero /eight.taboldstyle./one.taboldstyle.n-DIMENSIONAL VOLUME IN RN/one.taboldstyle/zero.taboldstyle/one.taboldstyle w. Therefore (Aw)·(Aw)/equalx0by (/eight.taboldstyle./one.taboldstyle), i.e. Aw/equalx0, so the columns of Aare dependent. QED It follows from the lemma that the formula in the next theorem makes sense. /eight.taboldstyle./four.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. There exists a unique n-dimensional volume function on RN. It is given by the following formula: vol n(a1,a2,..., an)/equalx/radicalbig det(ATA) for all vectors a1,a2,...,aninRN, where Ais the N×n-matrix whose i-th column is ai. P/r.sc/o.sc/o.sc/f.sc. The existence is proved by checking that the function/radicalbig det(ATA)sat- isfies the axioms of Definition /eight.taboldstyle./one.taboldstyle. You will be asked to do this in Exercise /eight.taboldstyle./two.taboldstyle. The uniqueness is proved by verifying that the formula holds, wh ich we proceed to do now. First assume that a1,a2,...,anare dependent. Then vol n(a1,a2,..., an)/equalx0by Lemma /eight.taboldstyle./two.taboldstyle(i) and det(ATA)/equalx0by Lemma /eight.taboldstyle./three.taboldstyle, so the formula holds in this case. Next consider a sequence of independent vectors a1,a2,...,an. Recall that such a sequence can be transformed into an orthonormal seque ncea⊥ 1,a⊥ 2,...,a⊥ n by the Gram-Schmidt process. This works as follows: let b1/equalx0and for i>1letbi be the orthogonal projection of aionto the span of a1,a2,...,ai−1; then a⊥ i/equalxai−bi. (See illustration below.) Each biis a linear combination of a1,a2,...,ai−1, so by repeated applications of Axiom ( ii) we get vol n(a1,a2,..., an)/equalxvol n(a⊥ 1+b1,a⊥ 2+b2,..., a⊥ n+bn) /equalxvol n(a⊥ 1,a⊥ 2+b2,..., a⊥ n+bn) /equalxvol n(a⊥ 1,a⊥ 2,..., a⊥ n+bn) ... /equalxvol n(a⊥ 1,a⊥ 2,..., a⊥ n) /equalx/bardbla⊥ 1/bardbl/bardbla⊥ 2/bardbl···/bardbl a⊥ n/bardbl, (/eight.taboldstyle./two.taboldstyle) where the last equality follows from Lemma /eight.taboldstyle./two.taboldstyle(ii). The Gram-Schmidt process can be expressed in matrix form by letting qibe the normalized vector qi/equalxa⊥ i//bardbla⊥ i/bardbl and QtheN×n-matrix with columns q1,q2,...,qn. Then we have the QR- decomposition A/equalxQR, where Ris an n×n-matrix of the form R/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA/bardbla⊥ 1/bardbl ∗ ∗ ··· ∗ 0/bardbla⊥ 2/bardbl ∗ ··· ∗ 0 0/bardbla⊥ 3/bardbl ··· ∗ .........∗ 0 0··· 0/bardbla⊥ n/bardbl/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA. Since Qis orthogonal, ATA/equalxRTQTQR/equalxRTR, and therefore det(ATA)/equalxdet(RTR)/equalx/bardbla⊥ 1/bardbl2/bardbla⊥ 2/bardbl2···/bardbla⊥ n/bardbl2. Comparing this with ( /eight.taboldstyle./two.taboldstyle) gives the desired conclusion. QED The Gram-Schmidt process transforms a sequence of nindependent vectors a1,a2,...,aninto an orthogonal sequence a⊥ 1,a⊥ 2,...,a⊥ n. (The horizontal “floor” /one.taboldstyle/zero.taboldstyle/two.taboldstyle /eight.taboldstyle. VOLUME FORMS represents the plane spanned by a1anda2.) The parallelepiped spanned by the a’s has the same volume as the rectangular block spanned by the a⊥’s. a1a2a3 a1=a⊥ 1a2a3 b2 b3a⊥ 2a⊥ 3 a1a2a3 a⊥ 1a⊥ 2a⊥ 3 /eight.taboldstyle./five.taboldstyle. C/o.sc/r.sc/o.sc/l.sc/l.sc/a.sc/r.sc/y.sc. vol n(a1,a2,..., an)≥0for all vectors a1,a2,...,aninRN. Forn/equalxNTheorem /eight.taboldstyle./four.taboldstylegives the following result. /eight.taboldstyle./six.taboldstyle. C/o.sc/r.sc/o.sc/l.sc/l.sc/a.sc/r.sc/y.sc. Leta1,a2,...,anbe vectors in Rnand let Abe the n×n-matrix whose i-th column is ai. Then vol n(a1,a2,..., an)/equalx|det(A|. P/r.sc/o.sc/o.sc/f.sc. Since Ais square, we have det(ATA)/equalxdet(AT)det(A)/equalx(det(A))2 by Theorem /three.taboldstyle./seven.taboldstyle(iii) and therefore vol n(a1,a2,..., an)/equalx/radicalbig (det(A))2/equalx|det(A)|by Theorem /eight.taboldstyle./four.taboldstyle. QED /eight.taboldstyle./two.taboldstyle. Orientations Oriented vector spaces. You are probably familiar with orientations of vector spaces of dimension ≤3. An orientation of a point is a sign, positive or negative. +− An orientation of a line is a direction, an arrow pointing eit her way. An orientation of a plane is a direction of rotation, clockwi se versus counterclock- wise. /eight.taboldstyle./two.taboldstyle. ORIENTATIONS /one.taboldstyle/zero.taboldstyle/three.taboldstyle An orientation of a three-dimensional space is a “handednes s” convention, left hand versus right hand. These notions can be generalized as follows. Let Vbe an n-dimensional vector space over the real numbers. A frame orordered basis ofVis an n-tuple (b1,b2,..., bn) consisting of vectors b1,b2,...,bnwhich form a basis of V. In other words, a frame is a basis together with a specified ordering among the basis v ectors. An oriented frame ofVis an ordered n+ 1-tuple B/equalx(b1,b2,..., bn;ε)consisting of a frame (b1,b2,..., bn)together with a sign ε/equalx±1. Suppose that B/equalx(b1,b2,..., bn;ε) andB′/equalx(b′ 1,b′ 2,..., b′ n;ε′)are two oriented frames of V. Then we can write b′ j/equalx/summationtext jai,jbiandbj/equalx/summationtext ja′ i,jb′ iwith unique coefficients ai,jand a′ i,j. The n×n- matrices A/equalx(ai,j)andA′/equalx(a′ i,j)satisfy AA′/equalxA′A/equalxIand are therefore invertible. In particular, the determinant of Ais nonzero. If ε′/equalxsign(det(A))εwe say that the oriented frames BandB′define the same orientation ofV. Ifε′/equalx−sign(det(A))ε we say that the oriented frames BandB′define opposite orientations . For instance, if (b′ 1,b′ 2,..., b′ n)/equalx(b2,b1,..., bn), then A/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA0 1 0... 0 1 0 0... 0 0 0 1... 0 ............... 0 0 0... 1/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA, sodet(A)/equalx−1. Hence the oriented frames (b2,b1,..., bn; 1)and (b1,b2,..., bn; 1) define opposite orientations, while (b2,b1,..., bn; 1)and (b1,b2,..., bn;−1)de- fine the same orientation. We know now what it means for two bases to have “the same orient ation”, but what about the concept of an orientation itself? We define the orientation of V determined by the oriented frame Bto be the collection of all oriented frames that have the same orientation as B. (This is analogous to the definition of the number 29as being the collection of all sets that contain twenty-nine elements.) The orien- tation determined by B/equalx(b1,b2,..., bn;ε)is denoted by [B]or[b1,b2,..., bn;ε]. So ifBandB′define the same orientation then [B]/equalx[B′]. If they define opposite orientations we write [B]/equalx−[B′]. There are two possible orientations of V. An oriented vector space is a vector space together with a specified orientation. This preferred orientation is then called positive . /eight.taboldstyle./seven.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Thestandard orientation onRnis the orientation [e1,..., en; 1], where (e1,..., en)is the standard ordered basis. We shall always use this orien ta- tion on Rn. /one.taboldstyle/zero.taboldstyle/four.taboldstyle /eight.taboldstyle. VOLUME FORMS Maps and orientations. LetVand Wbe oriented vector spaces of the same dimension and let L:V→Wbe an invertible linear map. Choose a posi- tively oriented frame (b1,b2,..., bn;ε)ofV. Because Lis invertible, the n+ 1- tuple (Lb1,Lb2,..., Lbn;ε)is an oriented frame of W. If this frame is positively, resp. negatively, oriented we say that Lisorientation-preserving , resp. orientation- reversing . This definition does not depend on the choice of the basis, fo r if (b′ 1,b′ 2,..., b′ n;ε′)is another positively oriented frame of V, then b′ i/equalx/summationtext jai,jbj withε′/equalxsign(det(ai,j))ε. Therefore Lb′ i/equalxL/parenleftbig/summationtext jai,jbj/parenrightbig/equalx/summationtext jai,jLbj, and hence the two oriented frames (Lb1,Lb2,..., Lbn;ε)and (Lb′ 1,Lb′ 2,..., Lb′ n;ε′)ofW determine the same orientation of W. Oriented manifolds. Now let Mbe a manifold in RN. We define an orientation ofMto be a choice of an orientation for each tangent space TxMwhich varies continuously over M. “Continuous” means that for every x∈Mthere exists a local parametrization ψ:U→RNofMatxwith the property that Dψy:Rn→TyM preserves the orientation for all y∈W. (Here Rnis equipped with its standard orientation.) A manifold is orientable if it possesses an orientation; it is oriented if a specific orientation has been chosen. Hypersurfaces. The case of a smooth hypersurface, a manifold of codimension 1, is particularly instructive. A unit normal vector field on a manifold MinRNis a smooth function n:M→RNsuch that n(x)⊥TxMand/bardbln(x)/bardbl/equalx1for all x∈M. /eight.taboldstyle./eight.taboldstyle. P/r.sc/o.sc/p.sc/o.sc/s.sc/i.sc/t.sc/i.sc/o.sc/n.sc. A smooth hypersurface in RNis orientable if and only if it possesses a unit normal vector field. P/r.sc/o.sc/o.sc/f.sc. LetMbe a smooth hypersurface in RNand put n/equalxdim(M)/equalxN−1. Suppose Mpossesses a unit normal vector field. Let b1,b2,...,bnbe a basis of TxMfor some x∈M. Then n(x),b1,b2,...,bnis a basis of RN, because n(x)⊥bi for all i. Chooseε/equalx±1such that (n(x),b1,b2,..., bn;ε)is a positively oriented frame of RN. Then we call (b1,b2,..., bn;ε)a positively oriented frame of TxM. This defines an orientation on M, called the orientation induced by the normal vector field n. Conversely, let us suppose that Mis an oriented smooth hypersurface in Rn. For each x∈Mthe tangent space TxMisn-dimensional, so its orthogonal complement (TxM)⊥is a line. There are therefore precisely two vectors of lengt h1which are perpendicular to TxM. We can pick a preferred unit normal vector as follows. Let(b1,b2,..., bn;ε)be a positively oriented frame of TxM. The positive unit normal vector is that unit normal vector n(x)that makes (n(x),b1,b2,..., bn;ε) a positively oriented frame of Rn. In Exercise /eight.taboldstyle./seven.taboldstyleyou will be asked to check that n(x)depends smoothly on x. In this way we have produced a unit normal vector field on M. QED /eight.taboldstyle./nine.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Let us regard RN−1as the linear subspace of RNspanned by the first N−1standard basis vectors e1,e2,...,eN−1. The standard orientations on RN and on RN−1are[e1,e2,..., eN; 1], resp. [e1,e2,..., eN−1; 1]. What is the positive unit normal ntoRN−1? According to the proof of Proposition /eight.taboldstyle./eight.taboldstylewe must choose nin such a way that [n,e1,e2,..., eN−1; 1)/equalx[e1,e2,..., eN; 1]. /eight.taboldstyle./three.taboldstyle. VOLUME FORMS /one.taboldstyle/zero.taboldstyle/five.taboldstyle The only two possibilities are n/equalxρeNwithρ/equalx±1. By parts ( i) and ( iv) of Exercise /eight.taboldstyle./five.taboldstylewe have [n,e1,e2,..., eN−1; 1]/equalx(−1)N+1[e1,e2,..., eN−1,n; 1] /equalx(−1)N+1[e1,e2,..., eN−1,ρeN; 1]/equalx(−1)N+1ρ[e1,e2,..., eN−1,eN; 1], so we want (−1)N+1ρ/equalx1. We conclude that the positive unit normal to RN−1in RNis(−1)N+1eN. The positive unit normal on an oriented smooth hypersurface MinRNcan be regarded as a map nfrom Minto the unit sphere SN−1, which is often called the Gauss map ofM. The unit normal enables one to distinguish between two side s of M: the direction of nis “out” or “up”; the opposite direction is “in” or “down”. For this reason orientable hypersurfaces are often called two-sided , whereas the nonorientable ones are called one-sided . Let us show that a hypersurface given by a single equation is always orientable. /eight.taboldstyle./one.taboldstyle/zero.taboldstyle. P/r.sc/o.sc/p.sc/o.sc/s.sc/i.sc/t.sc/i.sc/o.sc/n.sc. LetUbe open in RNand letφ:U→Rbe a smooth function. Letcbe a regular value of φ. Then the smooth hypersurface φ−1(c)has a unit normal vector field given by n(x)/equalxgrad (φ)(x)//bardblgrad (φ)(x)/bardbland is therefore orientable. P/r.sc/o.sc/o.sc/f.sc. The regular value theorem tells us that M/equalxφ−1(c)is a smooth hyper- surface in RN(if nonempty), and also that TxM/equalxker(Dφ(x))/equalxgrad (φ)(x)⊥. The function n(x)/equalxgrad (φ)(x)//bardblgrad (φ)(x)/bardbltherefore defines a unit normal vector field on M. Appealing to Proposition /eight.taboldstyle./eight.taboldstylewe conclude that Mis orientable. QED /eight.taboldstyle./one.taboldstyle/one.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Takingφ(x)/equalx/bardblx/bardbl2and c/equalxr2we obtain that the sphere of radius rabout the origin is orientable. The unit normal is n(x)/equalxgrad (φ)(x)//bardblgrad (φ)(x)/bardbl/equalxx//bardblx/bardbl. /eight.taboldstyle./three.taboldstyle. Volume forms Now let Mbe an oriented n-manifold in RN. Choose an atlas of Mconsisting of local parametrizations ψi:Ui→RNwith the property that Dψi(t):Rn→TxMis orientation-preserving for all t∈Ui. The volume form µM, also denoted by µ, is the n-form on Mwhose local representative relative to the embedding ψiis defined by µi/equalxψ∗ i(µ)/equalx/radicalBig det(Dψi(t)TDψi(t))dt1dt2···dtn. Theorem /eight.taboldstyle./four.taboldstyletells us that the square-root factor measures the volume of t hen- dimensional parallelepiped in the tangent space TxMspanned by the columns of Dψi(t), the Jacobi matrix of ψiatt. Hence you should think of µas measuring the volume of infinitesimal parallelepipeds inside M. /eight.taboldstyle./one.taboldstyle/two.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. For any oriented n-manifold MinRNthe volume form µMis a well-defined n-form. P/r.sc/o.sc/o.sc/f.sc. To show that µis well-defined we need to check that its local represen- tatives satisfy the transformation law ( /seven.taboldstyle./one.taboldstyle). So let us put φ/equalxψ−1 i◦ψjand substitute t/equalxφ(u)intoµi. Since each of the embeddings ψiis orientation-preserving, we have det(Dφ)>0. Hence by Theorem /three.taboldstyle./one.taboldstyle/four.taboldstylewe have φ∗(dt1dt2···dtn)/equalxdet(Dφ(u)du1du2···dun/equalx|det(Dφ(u))|du1du2···dun. /one.taboldstyle/zero.taboldstyle/six.taboldstyle /eight.taboldstyle. VOLUME FORMS Therefore φ∗(µi)/equalx/radicalBig det/parenleftbigDψi(φ(u))TDψi(φ(u))/parenrightbig|det(Dφ(u))|du1du2···dun /equalx/radicalBig det(Dφ(u))Tdet/parenleftbigDψi(φ(u))TDψi(φ(u))/parenrightbigdet(Dφ(u))du1du2···dun /equalx/radicalBig det/parenleftbig(Dψi(φ(u))Dφ(u))TDψi(φ(u))Dφ(u)/parenrightbigdu1du2···dun /equalx/radicalBig det((Dψj(u))TDψj(u))du1du2···dun/equalxµj, where in the second to last identity we applied the chain rule . QED Forn/equalx1the volume form is usually called the element of arc length , for n/equalx2, theelement of surface area , and for n/equalx3, the volume element . Traditionally these are denoted by ds,dA, and dV, respectively. Don’t be misled by these old-fashioned notations: volume forms are seldom exact! Another thing to r emember is that the volume form µMdepends on the embedding of MintoRN. It changes if we dilate or shrink or otherwise deform M. /eight.taboldstyle./one.taboldstyle/three.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetUbe an open subset of Rn. Recall from Example /six.taboldstyle./four.taboldstylethat U is a manifold covered by a single embedding, namely the ident ity mapψ:U→U, ψ(x)/equalxx. Then det(DψTDψ)/equalx1, so the volume form on Uis simply dt1dt2···dtn, the ordinary volume form on Rn. /eight.taboldstyle./one.taboldstyle/four.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetIbe an interval in the real line and f:I→Ra smooth function. Let M⊆R2be the graph of f. By Example /six.taboldstyle./six.taboldstyleMis a1-manifold in R2. Indeed, Mis the image of the embedding ψ:I→R2given byψ(t)/equalx(t,f(t)). Let us give Mthe orientation induced by the embedding ψ, i.e. “from left to right”. What is the element of arc length of M? Let us compute the pullback ψ∗(µ), a 1-form on I. We have Dψ(t)/equalx/parenleftBigg 1 f′(t)/parenrightBigg , Dψ(t)TDψ(t)/equalx/parenleftBig 1f′(t)/parenrightBig/parenleftBigg 1 f′(t)/parenrightBigg /equalx1 +f′(t)2, soψ∗(µ)/equalx/radicalbig det(Dψ(t)TDψ(t))dt/equalx/radicalbig 1 +f′(t)2dt. The next result can be regarded as an alternative definition o fµM. It is perhaps more intuitive, but it requires familiarity with Section /seven.taboldstyle./two.taboldstyle. /eight.taboldstyle./one.taboldstyle/five.taboldstyle. P/r.sc/o.sc/p.sc/o.sc/s.sc/i.sc/t.sc/i.sc/o.sc/n.sc. LetMbe an oriented n-manifold in RN. Let x∈Mandv1, v2,...,vn∈TxM. Then the volume form of Mis given by µM,x(v1,v2,..., vn) /equalxvol n(v1,v2,..., vn)if(v1,v2,..., vn; 1)is a positively oriented frame , −vol n(v1,v2,..., vn)if(v1,v2,..., vn; 1)is a negatively oriented frame , 0 ifv1,v2,...,vnare linearly dependent , i.e.µM,x(v1,v2,..., vn)is the oriented volume of the n-dimensional parallelepiped in TxM spanned by v1,v2,...,vn. P/r.sc/o.sc/o.sc/f.sc. For each xinMand n-tuple of tangent vectors v1,v2,...,vnatxlet ωx(v1,v2,..., vn)be the oriented volume of the block spanned by these nvectors. This defines an n-formωonMand we must show that ω/equalxµM. Let Ube an open subset of Rnandψ:U→RNan orientation-preserving local parametrization of /eight.taboldstyle./three.taboldstyle. VOLUME FORMS /one.taboldstyle/zero.taboldstyle/seven.taboldstyle M. Choose t∈Usatisfyingψ(t)/equalxx. Let us calculate the n-formψ∗(ω)onU. We haveψ∗(ω)/equalxg dt 1dt2···dtnfor some function g. By Lemma /seven.taboldstyle./one.taboldstyle/two.taboldstylethis function is given by g(t)/equalxψ∗(ωx)(e1,e2,..., en)/equalxωx(Dψ(t)e1,Dψ(t)e2,..., Dψ(t)en), where in the second equality we used the definition of pullbac k. Then the tuple (Dψ(t)e1,Dψ(t)e2,..., Dψ(t)en; 1)is a positively oriented frame of TxM and, moreover, are the columns of the matrix Dψ(t), so by Theorem /eight.taboldstyle./four.taboldstylethey span a positive volume of magnitude/radicalbig det(Dψ(t)TDψ(t)). This shows that g/equalx/radicalbig det(DψTDψ)and therefore ψ∗(ω)/equalx/radicalBig det(DψTDψ)dt1dt2···dtn. Thusψ∗(ω)is equal to the local representative of µMwith respect to the embedding ψ. Since this holds for all embeddings ψ, we haveω/equalxµM. QED Volume form of a hypersurface. Recall the vector-valued forms dx/equalx/parenlefttpA/parenleftexA/parenleftexA /parenleftbtAdx1 ... dxN/parenrighttpA/parenrightexA/parenrightexA /parenrightbtAand∗dx/equalx/parenlefttpA/parenleftexA/parenleftexA /parenleftbtA∗dx1 ... ∗dxN/parenrighttpA/parenrightexA/parenrightexA /parenrightbtA onRN, which were introduced in Section /two.taboldstyle./five.taboldstyle. We will use these forms to give a convenient expression for the volume form on a hypersurface . Let Mbe an oriented hypersurface in RN. Let nbe the positive unit normal vector field on Mand let F be any vector field on M, i.e. a smooth map F:M→RN. Then the inner product F·nis a function defined on M. It measures the component of Forthogonal to M. The product (F·n)µMis an n-form on M, where n/equalxdim(M)/equalxN−1. On the other hand we have the n-form∗(F·dx)/equalxF·∗dx. /eight.taboldstyle./one.taboldstyle/six.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. On the hypersurface Mwe have F·∗dx/equalx(F·n)µM. F/i.sc/r.sc/s.sc/t.sc /p.sc/r.sc/o.sc/o.sc/f.sc. This proof is short but requires familiarity with the materi al in Section /seven.taboldstyle./two.taboldstyle. Let x∈M. Let us make an orthogonal change of coordinates in RN in such a way that (e1,e2...,eN−1; 1)is a positively oriented frame of TxM. Then, according to Example /eight.taboldstyle./nine.taboldstyle, the positive unit normal at xis given by n(x)/equalx(−1)N+1eN and the volume form satisfies µM,x(e1,..., eN−1)/equalx1. Writing F/equalx/summationtextN i/equalx1Fiei, we have F(x)·n(x)/equalx(−1)N+1FN(x). On the other hand F·∗dx/equalx/summationdisplay i(−1)i+1Fidx1···/hatwiderdxi···dxN, and therefore (F·∗dx)(e1,..., eN−1)/equalx(−1)N+1FN. This proves that (F·∗dx)x(e1,..., eN−1)/equalx(F(x)·n(x))µM(e1,..., eN−1), which implies (F·∗dx)x/equalx(F(x)·n(x))µM. Since this equality holds for every x∈M, we find F·∗dx/equalx(F·n)µM. QED S/e.sc/c.sc/o.sc/n.sc/d.sc /p.sc/r.sc/o.sc/o.sc/f.sc. Choose a local parametrization ψ:U→RNofMatx. Let t∈U be the point satisfying ψ(t)/equalxx. As a preliminary step in the proof we are going to replace the embedding ψwith a new one enjoying a particularly nice property. /one.taboldstyle/zero.taboldstyle/eight.taboldstyle /eight.taboldstyle. VOLUME FORMS Let us change the coordinates on RNin such a way that (e1,e2...,eN−1; 1)is a positively oriented frame of TxM. Then at xthe positive unit normal is given by n(x)/equalx(−1)N+1eN. Since the columns of the Jacobi matrix Dψ(t)are independent, there exist unique vectors a1,a2,...,aN−1inRN−1such that Dψ(t)ai/equalxeifori/equalx1, 2,...,N−1. These vectors aiare independent, because the eiare independent. Therefore the (N−1)×(N−1)-matrix Awith i-th column vector equal to aiis invertible. Put ˜U/equalxA−1(U),˜t/equalxA−1tand ˜ψ/equalxψ◦A. Then ˜Uis open in RN−1, ˜ψ(˜t)/equalxx,˜ψ:˜U→RNis an embedding with ˜ψ(˜U)/equalxψ(U), and D˜ψ(˜t)/equalxDψ(t)◦DA(˜t/equalxDψ(t)◦A by the chain rule. Therefore the i-th column vector of D˜ψ(˜t)is D˜ψ(˜t)ei/equalxDψ(t)Aei/equalxDψ(t)ai/equalxei (/eight.taboldstyle./three.taboldstyle) fori/equalx1,2,...,N−1. (On the left eidenotes the i-th standard basis vector in RN−1, on the right it denotes the i-th standard basis vector in RN.) In other words, the Jacobi matrix of ˜ψat˜tis the (N−1)×N-matrix D˜ψ(˜t)/equalx/parenleftBigg IN−1 0/parenrightBigg , where IN−1is the (N−1)×(N−1)identity matrix and 0denotes a row consisting ofN−1zeros. Let us now calculate ˜ψ∗/parenleftbig(F·n)µM/parenrightbigand ˜ψ∗(F·∗dx)at the point ˜t. Writing F·n/equalx/summationtextN i/equalx1Finiand using the definition of µMwe get ˜ψ∗/parenleftbig(F·n)µM/parenrightbig/equalx/parenleftbiggN/summationdisplay i/equalx1˜ψ∗(Fini)/parenrightbigg/radicalBig det(D˜ψTD˜ψ)d˜t1d˜t2···d˜tN−1. From formula ( /eight.taboldstyle./three.taboldstyle) we have det(D˜ψ(˜t)TD˜ψ(˜t))/equalx1. So evaluating this expression at the point ˜tand using n(x)/equalx(−1)N+1eNwe get /parenleftbig˜ψ∗(F·n)µM/parenrightbig ˜t/equalx(−1)N+1FN(x)d˜t1d˜t2···d˜tN−1. From F·∗dx/equalx/summationtextN i/equalx1(−1)i+1Fidx1dx2···/hatwiderdxi···dxNwe get ˜ψ∗(F·∗dx)/equalxN/summationdisplay i/equalx1(−1)i+1˜ψ∗(Fi)d˜ψ1d˜ψ2···/hatwidestd˜ψi···d˜ψN. From formula ( /eight.taboldstyle./three.taboldstyle) we see∂˜ψi(˜t)/∂˜tj/equalxδi,jfor1≤i,j≤N−1and∂˜ψN(˜t)/∂˜tj/equalx0 for1≤j≤N−1. Therefore /parenleftbig˜ψ∗(F·∗dx)/parenrightbig ˜t/equalx(−1)N+1FN(x)d˜t1d˜t2···d˜tN−1. We conclude that/parenleftbig˜ψ∗(F·n)µM/parenrightbig ˜t/equalx/parenleftbig˜ψ∗(F·∗dx)/parenrightbig ˜t, in other words/parenleftbig(F·n)µM/parenrightbig x/equalx (F·∗dx)x. Since this holds for all x∈Mwe have F·∗dx/equalx(F·n)µM. QED This theorem gives insight into the physical interpretatio n of N−1-forms on RN. Think of the vector field Fas representing the flow of a fluid or gas. The direction of the vector Findicates the direction of the flow and its magnitude measures the strength of the flow. Then Theorem /eight.taboldstyle./one.taboldstyle/six.taboldstylesays that the N−1-form F·∗dxmeasures, for any unit vector ninRN, the amount of fluid per unit of time passing through a hyperplane of unit volume perpendicular t on. We call F·∗dx thefluxof the vector field F. EXERCISES /one.taboldstyle/zero.taboldstyle/nine.taboldstyle Another application of the theorem is the following formula for the volume form on a hypersurface. The formula provides a heuristic int erpretation of the vector-valued form ∗dx: ifnis a unit vector in RN, then the scalar-valued N−1-form n·∗dxmeasures the volume of an infinitesimal N−1-dimensional parallelepiped perpendicular to n. /eight.taboldstyle./one.taboldstyle/seven.taboldstyle. C/o.sc/r.sc/o.sc/l.sc/l.sc/a.sc/r.sc/y.sc. Letnbe the unit normal vector field and µMthe volume form of the oriented hypersurface M. Then µM/equalxn·∗dx. P/r.sc/o.sc/o.sc/f.sc. SetF/equalxnin Proposition /eight.taboldstyle./one.taboldstyle/six.taboldstyle. Then F·n/equalx1because/bardbln/bardbl/equalx1. QED /eight.taboldstyle./one.taboldstyle/eight.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Suppose the hypersurface Mis given by an equation φ(x)/equalxc, where cis a regular value of a function φ:U→R, with Uopen in Rn. Then by Proposition /eight.taboldstyle./one.taboldstyle/zero.taboldstyle Mhas a unit normal n/equalxgrad (φ)//bardblgrad (φ)/bardbl. The volume form is thereforeµ/equalx/bardblgrad (φ)/bardbl−1grad (φ)·∗dx. In particular, if Mis the sphere of radius Rabout the origin in Rn, then n(x)/equalxx/R, soµM/equalxR−1x·∗dx. Exercises /eight.taboldstyle./one.taboldstyle.Letaandbbe vectors in RN. (i) Deduce from Theorem /eight.taboldstyle./four.taboldstylethat the area of the parallelogram spanned by aandb is given by/bardbla/bardbl/bardblb/bardblsinφ, whereφis the angle between aandb(which is taken to lie between 0andπ). (ii) Show that for N/equalx3we have/bardbla/bardbl/bardblb/bardblsinφ/equalx/bardbla×b/bardbl. (Consider det(a,b,a×b).) /eight.taboldstyle./two.taboldstyle.Check that the function voln(a1,a2,..., an)/equalx/radicalbig det(ATA)satisfies the axioms of Definition /eight.taboldstyle./one.taboldstyle. /eight.taboldstyle./three.taboldstyle.Letu1,u2,...,ukandv1,v2,...,vlbe vectors in RNsatisfying ui·vj/equalx0fori/equalx1, 2,...,kand j/equalx1,2,...,l. (“The u’s are perpendicular to the v’s.”) Prove that volk+l(u1,u2,..., uk,v1,v2,..., vl)/equalxvolk(u1,u2,..., uk)voll(v1,v2,..., vl). /eight.taboldstyle./four.taboldstyle.Leta1,a2,...,anbe real numbers, let c/equalx/radicalBig 1 +/summationtextn i/equalx1a2 iand let u1/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA1 0 ... 0 a1/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA,u2/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA0 1 ... 0 a2/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA, ..., un/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA0 0 ... 1 an/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA,un+1/equalx1 c/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA−a1 −a2 ... −an 1/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA be vectors in Rn+1. (i) Deduce from Exercise /eight.taboldstyle./three.taboldstylethat voln(u1,u2,..., un)/equalxvoln+1(u1,u2,..., un,un+1). (ii) Prove that /barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex1 +a2 1a1a2 a1a3... a1an a2a11 +a2 2a2a3... a2an a3a1 a3a21 +a2 3... a3an ............... ana1 ana2 ana3... 1 +a2n/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/barex/equalx1 +n/summationdisplay i/equalx1a2 i. /one.taboldstyle/one.taboldstyle/zero.taboldstyle /eight.taboldstyle. VOLUME FORMS /eight.taboldstyle./five.taboldstyle. LetVbe an n-dimensional vector space with basis b1,b2,...,bn. Letε/equalx±1. Prove the following identities concerning orientations of V. (i)[b1,..., cbi,..., bn;ε]/equalxsign(c)[b1,..., bi,..., bn;ε]for all nonzero scalars c. (ii)[b1,b2,..., bn;ε]/equalxε[b1,b2,..., bn; 1]. (iii)[b1,b2,..., bn;−ε]/equalx−[b1,b2,..., bn;ε]. (iv)/bracketleftbigbσ(1),bσ(2),..., bσ(n);ε/bracketrightbig/equalxsign(σ)[b1,b2,..., bn;ε]for all permutations σin Sn. /eight.taboldstyle./six.taboldstyle. A frame (b1,b2,..., bn)ofRnisorthonormal if the vectors b1,b2,...,bnform an orthonormal basis of Rn. Denote the set of orthonormal frames by Fnand the set of positively oriented orthonormal frames by F+n. (i) Explain how to identify Fnwith the orthogonal group O(n)and F+nwith the special orthogonal group SO(n). Conclude that Fnand F+nare manifolds of dimension1 2n(n−1). (Use Theorem /six.taboldstyle./one.taboldstyle/eight.taboldstyle). (ii) Explain how to identify F+ 3with the configuration space M0discussed in Exam- ple/one.taboldstyle./seven.taboldstyleand conclude that M0is a three-dimensional projective space. (See the discussion following Theorem /six.taboldstyle./one.taboldstyle/eight.taboldstyle). /eight.taboldstyle./seven.taboldstyle.Show that the unit normal vector field n:M→Rndefined in the proof of Proposi- tion/eight.taboldstyle./eight.taboldstyleis smooth. (Compute nin terms of an orientation-preserving local parametrizati on ψofM.) /eight.taboldstyle./eight.taboldstyle.LetUbe open in Rnand let f:U→Rbe a smooth function. Let ψ:U→Rn+1be the embedding ψ(x)/equalx(x,f(x))and let M/equalxψ(U), the graph of f. We define an orientation onMby requiring ψto be orientation-preserving. Deduce from Exercise /eight.taboldstyle./four.taboldstylethat the volume form of Mis given byψ∗(µM)/equalx/radicalBig 1 +/bardblgrad f(x)/bardbl2dx1dx2···dxn. /eight.taboldstyle./nine.taboldstyle.LetM/equalxgraph (f)be the oriented hypersurface of Exercise /eight.taboldstyle./eight.taboldstyle. (i) Show that the positive unit normal vector field on Mis given by n/equalx(−1)n+1 /radicalBig 1 +/bardblgrad (f)(x)/bardbl2/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA∂f/∂x1 ∂f/∂x2 ... ∂f/∂xn −1/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA. (ii) Derive the formula ψ∗(µM)/equalx/radicalBig 1 +/bardblgrad f(x)/bardbl2dx1dx2···dxnof Exercise /eight.taboldstyle./eight.taboldstyle from Corollary /eight.taboldstyle./one.taboldstyle/seven.taboldstyleby substituting xn+1/equalxf(x1,x2,..., xn). (You must replace Nwith n+ 1in Corollary /eight.taboldstyle./one.taboldstyle/seven.taboldstyle.) /eight.taboldstyle./one.taboldstyle/zero.taboldstyle. Letψ:U→RNbe an embedding of an open subset UofRnintoRN. Let Mbe the image of ψand letµbe the volume form of M. Now let Rbe a nonzero number, let ψR be the embedding ψR(t)/equalxRψ(t), letMRbe the image of ψR, and letµRbe the volume form ofMR. Show that ψ∗ R(µR)/equalxRnψ∗(µ). (Useψ∗(µ)/equalx/radicalBig det(Dψi(t)TDψi(t))dt1dt2···dtn.) CHAPTER /nine.taboldstyle Integration and Stokes’ theorem for manifolds In this chapter we will see how to integrate an n-form over an oriented n- manifold. In particular, by integrating the volume form we fi nd the volume of the manifold. We will also discuss a version of Stokes’ theorem f or manifolds. This requires the slightly more general notion of a manifold with boundary. /nine.taboldstyle./one.taboldstyle. Manifolds with boundary The notion of a spherical earth developed in classical Greec e around the time of Plato and Aristotle. Older cultures (and also Western cultu re until the rediscovery of Greek astronomy in the late Middle Ages) visualized the ea rth as a flat disc surrounded by an ocean or a void. A closed disc is not a manifol d, because no neighbourhood of a point on the edge is the image of an open sub set of R2under an embedding. Rather, it is a manifold with boundary, a notio n which is defined as follows. The n-dimensional half-space is Hn/equalx{x∈Rn|xn≥0}. Theboundary ofHnis∂Hn/equalx{x∈Rn|xn/equalx0}and its interior isint(Hn)/equalx{x∈ Rn|xn>0}. /nine.taboldstyle./one.taboldstyle. D/e.sc/f.sc/i.sc/n.sc/i.sc/t.sc/i.sc/o.sc/n.sc. Ann-dimensional manifold with boundary (orn-manifold with boundary ) inRNis a subset MofRNsuch that for each x∈Mthere exist •an open subset V⊆RNcontaining x, •an open subset U⊆Rn, •and an embedding ψ:U→RNsatisfyingψ(U∩Hn)/equalxM∩V. You should compare this definition carefully with Definition /six.taboldstyle./three.taboldstyleof a manifold. If x/equalxψ(t)with t∈∂Hn, then xis aboundary point ofM. The boundary ofMis the set of all boundary points and is denoted by ∂M. Its complement M\∂Mis the interior ofMand is denoted by int(M). Definition /nine.taboldstyle./one.taboldstyledoes not rule out the possibility that the boundary of an n- manifold with boundary might be empty! If the boundary of Mis empty, then M is a manifold in the sense of Definition /six.taboldstyle./three.taboldstyle. If the boundary ∂Mis nonempty, then ∂Mis an n−1-dimensional manifold. The interior int(M)is an n-manifold. The most obvious example of an n-manifold with boundary is the half-space Hnitself. Its boundary is the hyperplane ∂Hn, which is a copy of Rn−1, and its interior is the open half-space {x∈Rn|xn>0}. Here is a more interesting type of example, which generalizes the graph of a function. /nine.taboldstyle./two.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetU′be an open subset of Rn−1and let f:U′→Rbe a smooth function. Put U/equalxU′×Rand write elements of Uas/parenleftbigx y/parenrightbigwith xinU′and yinR. /one.taboldstyle/one.taboldstyle/one.taboldstyle /one.taboldstyle/one.taboldstyle/two.taboldstyle /nine.taboldstyle. INTEGRATION AND STOKES’ THEOREM FOR MANIFOLDS Theregion above the graph or the supergraph of fis the set consisting of all/parenleftbigx y/parenrightbiginU such that y≥f(x). ∂M=graph fM xy We assert that the supergraph is an n-manifold with boundary, whose boundary is exactly the graph of f. We will prove this by describing it as the image of a single embedding. Define ψ:U→Rnby ψ/parenleftBigg t u/parenrightBigg /equalx/parenleftBigg t f(t)+u/parenrightBigg . As in Example /six.taboldstyle./two.taboldstyleone verifies that ψis an embedding, using the fact that Dψ/parenleftBigg t u/parenrightBigg /equalx/parenleftBigg In−10 D f(t)1/parenrightBigg , where 0is the origin in Rn−1. By definition the image M/equalxψ(U∩Hn)is therefore ann-manifold in Rnwith boundary ∂M/equalxψ(U∩∂Hn). What are Mand∂M? A point/parenleftbigx y/parenrightbigis in Mif and only if it is of the form /parenleftBigg x y/parenrightBigg /equalxψ/parenleftBigg t u/parenrightBigg /equalx/parenleftBigg t f(t)+u/parenrightBigg for some/parenleftbigt u/parenrightbiginU∩Hn. Since Hnis given by u≥0, this is equivalent to x∈U′ and y≤f(x). Thus Mis exactly the supergraph. On ∂Hnwe have u/equalx0, so∂Mis given by the equality y/equalxf(x), i.e.∂Mis the graph. /nine.taboldstyle./three.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Iff:U′→Rmis a vector-valued map one cannot speak about the region “above” the graph, but one can do the following. Ag ain put U/equalxU′×R. LetN/equalxn+m−1and think of RNas the set of vectors/parenleftbigx y/parenrightbigwith xinRn−1andyin Rm. Defineψ:U→RNby ψ/parenleftBigg t u/parenrightBigg /equalx/parenleftBigg t f(t)+uem/parenrightBigg This time we have Dψ(t)/equalx/parenleftBigg In−1 0 D f(t)em/parenrightBigg /nine.taboldstyle./one.taboldstyle. MANIFOLDS WITH BOUNDARY /one.taboldstyle/one.taboldstyle/three.taboldstyle and againψis an embedding. Therefore M/equalxψ(U∩Hn)is an n-manifold in RN with boundary ∂M/equalxψ(U∩∂Hn). This time Mis the set of points/parenleftbigx y/parenrightbigof the form /parenleftBigg x y/parenrightBigg /equalx/parenleftBigg t f(t)+uem/parenrightBigg with t∈U′and u≥0. Hence Mis the set of points/parenleftbigx y/parenrightbigwhere xis in U′and where ysatisfies m−1equalities and one inequality: y1/equalxf1(x),y2/equalxf2(x), ..., ym−1/equalxfm−1(x),ym≥fm(x). Again∂Mis given by y/equalxf(x), so∂Mis the graph of f. Here is an extension of the regular value theorem, Theorem /six.taboldstyle./one.taboldstyle/two.taboldstyle, to manifolds with boundary. /nine.taboldstyle./four.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc (/r.sc/e.sc/g.sc/u.sc/l.sc/a.sc/r.sc /v.sc/a.sc/l.sc/u.sc/e.sc /t.sc/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc /f.sc/o.sc/r.sc /m.sc/a.sc/n.sc/i.sc/f.sc/o.sc/l.sc/d.sc/s.sc /w.sc/i.sc/t.sc/h.sc /b.sc/o.sc/u.sc/n.sc /d.sc/a.sc/r.sc/y.sc). LetUbe open in RNand letφ:U→Rmbe a smooth map. Let Mbe the set of xinRNsatisfying φ1(x)/equalxc1, φ 2(x)/equalxc2, ..., φ m−1(x)/equalxcm−1, φ m(x)≥cm. Suppose that c/equalx(c1,c2,..., cm)is a regular value of φand that Mis nonempty. Then Mis a manifold in RNof codimension m−1and with boundary ∂M/equalxφ−1(c). We will not spell out the proof, which is similar to that of The orem /six.taboldstyle./one.taboldstyle/two.taboldstyle. The statement remains true if we replace “ ≥” with “≤”, as one sees by replacing φwith −φ. /nine.taboldstyle./five.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetU/equalxRn,m/equalx1andφ(x)/equalx/bardblx/bardbl2. The set given by the inequalityφ(x)≤1is then the closed unit ball {x∈Rn| /bardblx/bardbl ≤ 1}. Since grad (φ)(x)/equalx2x, any nonzero value is a regular value of φ. Hence the ball is an n-manifold in Rn, whose boundary is φ−1(1), the unit sphere Sn−1. If more than one inequality is involved, singularities ofte n arise. A simple example is the closed quadrant in R2given by the pair of inequalities x≥0and y≥0. This is not a manifold with boundary because its edge has a sh arp angle at the origin. Similarly, a closed square is not a manifold with boundary. However, one can show that a set given by a pair of inequalitie s of the form a≤f(x)≤b, where aand bare both regular values of a function f, is a manifold with boundary. For instance, the spherical shell {x∈Rn|R1≤/bardblx/bardbl≤R2} is an n-manifold whose boundary is a union of two concentric sphere s. /one.taboldstyle/one.taboldstyle/four.taboldstyle /nine.taboldstyle. INTEGRATION AND STOKES’ THEOREM FOR MANIFOLDS Other examples of manifolds with boundary are the pair of pants , a2-manifold whose boundary consists of three closed curves, and the Möbius band shown in Chapter /one.taboldstyle. The Möbius band is a nonorientable manifold with boundary. We will not give a proof of this fact, but you can convince yourself that it is true by trying to paint the two sides of a Mö bius band in different colours. Ann-manifold with boundary contained in Rn(i.e. of codimension 0) is often called a domain . For instance, a closed ball is a domain in Rn. Thetangent space to a manifold with boundary Mat a point xis defined in the usual way: choose a local parametrization ψofMatxand put TxM/equalxDψ(t)(Rn). As in the case of a manifold, the tangent space does not depend on the choice of the embedding ψ. At boundary points we can distinguish between three differe nt types of tangent vectors. Suppose xis a boundary point of Mand let v∈TxMbe a tangent vector. Then v/equalxDψ(t)ufor a unique vector u∈Rn. We say that the tangent vector v points inward ifun>0, is tangent to ∂Mifun/equalx0, points outward ifun<0. The tangent space to the boundary at xis Tx∂M/equalxDψ(t)(∂Hn). The above picture of the pair of pants shows some tangent vect ors at boundary points that are tangent to the boundary or outward pointing. Orienting the boundary. LetMbe an oriented manifold with boundary. The orientation of Mgives rise to an orientation of the boundary ∂Mby a method very similar to the one which produces an orientation of a hyp ersurface. (See Proposition /eight.taboldstyle./eight.taboldstyle.) Namely, for x∈∂Mwe let n(x)∈TxMbe the unique outward pointing tangent vector of length 1which is orthogonal to Tx∂M. This defines theunit outward pointing normal vector field non∂M. Let b1,b2,...,bn−1be a basis of Tx∂M. Then n(x),b1,b2,...,bn−1is a basis of TxM. Chooseε/equalx±1 such that (n(x),b1,b2,..., bn−1;ε)is a positively oriented frame of TxM. Then we /nine.taboldstyle./two.taboldstyle. INTEGRATION OVER ORIENTABLE MANIFOLDS /one.taboldstyle/one.taboldstyle/five.taboldstyle define (b1,b2,..., bn−1;ε)to be a positively oriented frame of Tx∂M. The resulting orientation of ∂Mis called the induced orientation . /nine.taboldstyle./six.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Consider the upper half-space Hnwith its standard orientation [e1,..., en; 1]. At each point of ∂Hnthe outward pointing normal is −en. Since [−en,e1,e2,..., en−1; 1]/equalx−[en,e1,e2,..., en−1; 1] /equalx−(−1)n−1[e1,e2,..., en−1,en; 1]/equalx[e1,e2,..., en−1,en;(−1)n], the induced orientation on ∂Hnis[e1,e2,..., en−1;(−1)n]. You may wonder why we didn’t get rid of the (−1)nby adopting a different convention for the orientation of the boundary. The justific ation for our convention is that it avoids the need for sign corrections in the stateme nt of Stokes’ theorem, Theorem /nine.taboldstyle./nine.taboldstyle. /nine.taboldstyle./two.taboldstyle. Integration over orientable manifolds As we saw in Chapter /five.taboldstyle, a form of degree ncan be integrated over a chain of dimension n. The integral does not change if we reparametrize the chain i n an orientation-preserving manner. This suggests the possi bility of integrating an n-form over an oriented n-manifold. One might try to do this by breaking up the manifold into n-chains and then integrating over each of the chains, but tha t turns out to be not so easy. Instead we shall employ the simpler meth od of breaking the differential form into small pieces and then integrating eac h of the pieces. In the remainder of this section M⊆RNdenotes an n-manifold with boundary andαdenotes a differential form on M. Support. Thesupport ofαis defined as the set of all points xinMwith the property that for every open ball Baround xthere is a y∈B∩Msuch thatαy/nequal0. The support of αis denoted by supp (α). Ifαxis nonzero, then xis in the support ofα(because we can take y/equalxxfor all B). But for xto be in the support it is not necessary for αxto be nonzero; we only need to be able to find points arbitraril y close to xwhereαis nonzero. In other words, xisnotin the support if and only if there exists a ball Baround xsuch thatαy/equalx0for all y∈B∩M. /nine.taboldstyle./seven.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetM/equalxRandαi/equalxfidx, where fiis one of the following smooth functions. (i)f1(x)/equalxsinx. This function has infinitely many zeroes, but they are all isolated: sinπk/equalx0, but siny/nequal0foryclose to but distinct from πk. Thus supp (α1)/equalxR. (ii) f2is a nonzero polynomial function. Again f2has isolated zeroes, so supp (α2)/equalxR. (iii) f3(x)/equalxexp(−1/x)forx>0and f3(x)/equalx0forx≤0. (This function is similar to the function of Exercise B./three.taboldstyle.) We have f3(x)>0for all x>0. It follows that x∈supp (α3)for all x≥0. On the other hand, negative x are not in the support, so supp (α3)/equalx[0,∞). (iv) f4(x)/equalxf3(x−a)f3(b−x), where a<bare constants. We have f4(x)>0 fora<x<band f4(x)/equalx0forx<aand x>b. Hence supp (α4)/equalx[a,b]. /nine.taboldstyle./eight.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. The volume form µofMis nowhere 0, sosupp (µ)/equalxM. /one.taboldstyle/one.taboldstyle/six.taboldstyle /nine.taboldstyle. INTEGRATION AND STOKES’ THEOREM FOR MANIFOLDS Partitions of unity. Chopping differential forms into “little pieces” requires a device known as a partition of unity. Let ψi:Ui→RNbe an atlas of M, where i runs over an indexing set I. Apartition of unity subordinate to the atlas is a collection of smooth functions λi:M→Rwith the following properties: (i)λi≥0for all i; (ii)supp (λi)is contained in ψi(Ui)for all i; (iii) for every x∈Mthere exists a ball Baround xwith the property that supp (λi)∩Bis empty for all but finitely many i∈I; (iv)/summationtext i∈Iλi/equalx1. Condition ( iv) says that the functions λiadd up to the constant function 1; it is in this sense that they “partition” the “unit” function. Toget her with the positivity condition ( i) this implies that every λitakes values between 0an1. Condition (ii) expresses that λiis “small” in another sense as well: for every point xofM which is not contained in the coordinate patch ψi(Ui)the function λivanishes identically in a neighbourhood of x. Condition ( iii) is imposed to ensure that even if the indexing set Iis infinite the sum in condition ( iv) is finite at every point and is a well-defined smooth function. It is a very useful technical fact that partitions of unity ex ist subordinate to any atlas of M. See Chapter /three.taboldstyle of the book [ Spi/seven.taboldstyle/one.taboldstyle ] for a proof, or see Exercise /nine.taboldstyle./three.taboldstylefor a special case. Defining the integral. From now on we assume that Mis oriented and that α is of degree n/equalxdim(M). Moreover, we assume that the support of αis a compact set. (A subset of RNis called compact if it is closed and bounded; see Appendix A./two.taboldstyle. The compactness assumption is made to ensure that the integr al ofαis a proper integral and therefore converges. For instance, if we let M/equalxRandαione of the 1-forms of Example /nine.taboldstyle./seven.taboldstyle, then onlyα4has a well-defined integral over M. Also note that the support of αis certainly compact if the manifold Mitself is compact.) Step/one.taboldstyle. Assume there exists an orientation-preserving local par ametrization ψ:U→RNofMwith the property that the support of αis contained in ψ(U∩Hn). Then we define the integral of αover Mby /integraldisplay Mα/equalx/integraldisplay U∩Hnψ∗(α). The right-hand side is well-defined because the integrand is of the form ψ∗(α)/equalx g dt 1dt2···dtn, where gis a smooth function on U∩Hnwhich vanishes outside a compact subset. Moreover, the integral does not depend on t he choice of ψ: if ψ′:U′→RNis another orientation-preserving local parametrization ofMsuch that supp (α)is contained in ψ′(U′∩Hn), then, letting ζ/equalxψ−1◦ψ′, we have ψ◦ζ/equalxψ′, so /integraldisplay U′∩Hn(ψ′)∗α/equalx/integraldisplay U′∩Hn(ψ◦ζ)∗(α)/equalx/integraldisplay U′∩Hnζ∗(ψ∗(α))/equalx/integraldisplay U∩Hnψ∗(α), where the last step uses Theorem /five.taboldstyle./one.taboldstyleand the fact that ζpreserves the orientation. Step/two.taboldstyle. In the general case we choose an atlas of Mconsisting of orientation- preserving local parametrizations ψi:Ui→RN, and we choose a partition of unity subordinate to this atlas consisting of functions λi:M→R. Letαi/equalxλiα. Thenαi is an n-form with support contained in ψi(Ui∩Hn), so its integral is well-defined /nine.taboldstyle./two.taboldstyle. INTEGRATION OVER ORIENTABLE MANIFOLDS /one.taboldstyle/one.taboldstyle/seven.taboldstyle by step /one.taboldstyle. Moreover,/summationtext i∈Iαi/equalx/summationtext i∈Iλiα/equalx(/summationtext i∈Iλi)α/equalxα. We now define the integral ofαby/integraldisplay Mα/equalx/summationdisplay i∈I/integraldisplay Mαi/equalx/summationdisplay i∈I/integraldisplay Ui∩Hnψ∗ i(αi). (/nine.taboldstyle./one.taboldstyle) The most important property of the integral is the following version of Stokes’ theorem, which can be viewed as a parametrization-independ ent version of Theo- rem/five.taboldstyle./one.taboldstyle/one.taboldstyle. /nine.taboldstyle./nine.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc (S/t.sc/o.sc/k.sc/e.sc/s.sc’ /t.sc/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc /f.sc/o.sc/r.sc /m.sc/a.sc/n.sc/i.sc/f.sc/o.sc/l.sc/d.sc/s.sc). Letαbe an n−1-form with compact support on an oriented n-manifold with boundary M. Give the boundary ∂Mthe induced orientation. Then/integraldisplay Mdα/equalx/integraldisplay ∂Mα. P/r.sc/o.sc/o.sc/f.sc. Step/one.taboldstyle. Suppose M/equalxHn. Then we can write α/equalxn/summationdisplay i/equalx1gidt1dt2···/hatwidedti···dtn for certain smooth functions gidefined on Hn. We have dα/equalxn/summationdisplay i/equalx1(−1)i+1∂gi ∂tidt1dt2···dtn. The support of αis a compact subset of Hnand so is enclosed in a box of the shape [a1,b1]×[a2,b2]×···× [an−1,bn−1]×[0,c]. (/nine.taboldstyle./two.taboldstyle) Therefore /integraldisplay Hndα/equalxn/summationdisplay i/equalx1(−1)i+1/integraldisplayc 0/integraldisplaybn−1 an−1···/integraldisplayb2 a2/integraldisplayb1 a1∂gi ∂tidt1dt2···dtn. The coefficients giofαare smooth functions on Hnwhich vanish outside the box (/nine.taboldstyle./two.taboldstyle). In particular the giand their partial derivatives vanish along all the walls of the box except possibly the “floor” [a1,b1]×[a2,b2]×···× [an−1,bn−1]×{0}. Hence, by the fundamental theorem of calculus, /integraldisplaybi ai∂gi ∂tidti/equalxgi(t1,..., bi,..., tn)−gi(t1,..., ai,..., tn)/equalx0 fori≤n−1, while /integraldisplayc 0∂gn ∂tndtn/equalxgn(t1,..., tn−1,c)−gn(t1,..., tn−1,0)/equalx−gn(t1,..., tn−1,0). Hence /integraldisplay Hndα/equalx(−1)n/integraldisplaybn−1 an−1···/integraldisplayb2 a2/integraldisplayb1 a1gn(t1,..., tn−1,0)dt1dt2···dtn−1/equalx/integraldisplay ∂Hnα, where the sign (−1)nis accounted for by Example /nine.taboldstyle./six.taboldstyle, which says that the orienta- tion of∂Hn/equalxRn−1is(−1)ntimes the standard orientation of Rn−1. /one.taboldstyle/one.taboldstyle/eight.taboldstyle /nine.taboldstyle. INTEGRATION AND STOKES’ THEOREM FOR MANIFOLDS Step/two.taboldstyle. In the general case we choose an atlas of Mconsisting of orientation- preserving local parametrizations ψi:Ui→RN, and a subordinate partition of unity consisting of functions λi:M→R. Letαi/equalxλiα. Then /integraldisplay Mdα/equalx/summationdisplay i∈I/integraldisplay Ui∩Hnψ∗ i(dα)/equalx/summationdisplay i∈I/integraldisplay Ui∩Hndψ∗ i(α)/equalx/summationdisplay i∈I/integraldisplay Ui∩∂Hnψ∗ i(α)/equalx/integraldisplay ∂Mα, where the first and last equalities follow from the definition (/nine.taboldstyle./one.taboldstyle) of the integral, and the third equality uses step /one.taboldstyle. QED We conclude this section by considering a few special cases o f the integral. Let Ma compact oriented manifold in RN. The volume ofMisvol(M)/equalx/integraltext Mµ, where µis the volume form of M. (Ifdim(M)/equalx1, resp. 2, we speak of the arc length , resp. surface area ofM.) The integral of a function fonMis defined as/integraltext Mfµ. The mean oraverage offis the number ¯f/equalx(vol(M))−1/integraltext Mfµ. The centroid orbarycentre ofM is the point ¯xinRnwhose i-th coordinate is the mean value of xiover M, i.e. ¯xi/equalx1 vol(M)/integraldisplay Mxiµ. The volume form depends on the embedding of MintoRN, so the volume and barycentre of Mdepend on the embedding as well. /nine.taboldstyle./three.taboldstyle. Gauss and Stokes Stokes’ theorem, Theorem /nine.taboldstyle./nine.taboldstyle, contains as special cases the integral theorems of vector calculus. These classical results involve a vecto r field F/equalx/summationtextn i/equalx1Fiei defined on an open subset UofRn. As discussed in Section /two.taboldstyle./five.taboldstyle, to this vector field corresponds a 1-formα/equalxF·dx/equalx/summationtextn i/equalx1Fidxi, which we can think of as the work done by the force Falong an infinitesimal line segment dx. We will now derive the classical integral theorems by applying Theorem /nine.taboldstyle./nine.taboldstyleto one-dimensional, resp. n-dimensional, resp. two-dimensional manifolds Mcontained in U. Fundamental theorem of calculus. IfFis conservative, F/equalxgrad (g)for a function g, thenα/equalxgrad (g)·dx/equalxdg. IfMis a compact oriented 1-manifold with boundary in Rn, then/integraltext Mdg/equalx/integraltext ∂Mgby Theorem /nine.taboldstyle./nine.taboldstyle. The boundary consists of two points aandbifMis connected. If the orientation of Mis “from atob”, then aacquires a minus and ba plus. Stokes’ theorem therefore gives the fundamental theorem of calculus in Rn, /integraldisplay MF·dx/equalxg(b)−g(a). If we interpret Fas a force acting on a particle travelling along M, then−gstands for the potential energy of the particle in the force field. Th us the potential energy of the particle decreases by the amount of work done. Gauss’ divergence theorem. We have ∗α/equalxF·∗dx and d∗α/equalxdiv(F)dx1dx2···dxn. IfZis a oriented hypersurface in Rnwith positive unit normal n, then∗α/equalx(F·n)µZ onZby Theorem /eight.taboldstyle./one.taboldstyle/six.taboldstyle. In this situation it is best to think of Fas the flow vector field of a fluid, where the direction of F(x)gives the direction of the flow at a point EXERCISES /one.taboldstyle/one.taboldstyle/nine.taboldstyle xand the magnitude /bardblF(x)/bardblgives the mass of the amount of fluid passing per unit time through a hypersurface of unit area placed at xperpendicular to the vector F(x). Then∗αdescribes the amount of fluid passing per unit time and per uni t area through the hypersurface Z. For this reason the n−1-form∗αis also called thefluxofF, and its integral over Zthetotal flux through Z. Now let Z/equalx∂M, the boundary of a compact domain MinRn. Applying Stokes’ theorem to Mand d∗αwe get/integraltext Md∗α/equalx/integraltext ∂M∗α. Written in terms of the vector field Fthis is Gauss’ divergence theorem, /integraldisplay Mdiv(F)dx1dx2···dxn/equalx/integraldisplay ∂M(F·n)µ∂M. Thus the total flux out of the hypersurface ∂Mis the integral of div(F)over M. If the fluid is incompressible (e.g. most liquids) then this for mula leads to the inter- pretation of the divergence of F(or equivalently d∗α) as a measure of the sources or sinks of the flow. Thus div(F)/equalx0for an incompressible fluid without sources or sinks. If the fluid is a gas and if there are no sources or sink s then div(F)(x)>0 (resp.<0) indicates that the gas is expanding (resp. being compresse d) at x. Classical version of Stokes’ theorem. Next let Mbe a compact two-dimen- sional oriented surface with boundary and let us rewrite Sto kes’ theorem/integraltext Mdα/equalx/integraltext ∂Mαin terms of the vector field F. The right-hand side represents the work ofFdone around the boundary curve(s) of M, which is not necessarily 0ifFis not conservative. The left-hand side has a nice interpretat ion if n/equalx3. Then ∗dα/equalxcurl(F)·dx, sodα/equalxcurl(F)·∗dx. Hence if nis the positive unit normal of the surface MinR3, then dα/equalxcurl(F)·nµMonM. In this way we get the classical formula of Stokes,/integraldisplay Mcurl(F)·nµM/equalx/integraldisplay ∂MF·dx. In other words, the total flux of curl(F)through the surface Mis equal to the work done by Faround the boundary curves of M. This formula shows that curl(F), or equivalently∗dα, can be regarded as a measure of the vorticity of the vector fie ld. Exercises /nine.taboldstyle./one.taboldstyle. LetUbe an open subset of Rnand let f,g:U→Rbe two smooth functions satisfying f(x)<g(x)for all xinU. Let Mbe the set of all pairs (x,y)such that xinUand f(x)≤y≤g(x). (i) Show directly from the definition that Mis a manifold with boundary. (Use two embeddings to cover M.) What is the dimension of Mand what are the boundary and the interior? (ii) Draw a picture of MifUis the open unit disc given by x2+y2<1and f(x,y)/equalx −/radicalBig 1−x2−y2and g(x,y)/equalx2−x2−y2. (iii) Give an example showing that Mis not necessarily a manifold with boundary if the condition f(x)<g(x)fails. /nine.taboldstyle./two.taboldstyle.Letαandβbe differential forms on a manifold M. Show that supp (αβ)is contained insupp (α)∩supp (β). Give an example to show that we can have supp (αβ)/nequalsupp (α)∩ supp (β). /one.taboldstyle/two.taboldstyle/zero.taboldstyle /nine.taboldstyle. INTEGRATION AND STOKES’ THEOREM FOR MANIFOLDS /nine.taboldstyle./three.taboldstyle.LetI1/equalx[a1,b1)and I2/equalx(a2,b2]be two half-open intervals, where a1<a2<b1< b2, and let M/equalxI1∪I2/equalx[a1,b2]. Show that there exist two smooth functions λ1,λ2:M→R with the following properties: (i) λ1(x)≥0andλ2(x)≥0for all x∈M; (ii)λ1+λ2/equalx1; (iii)supp (λ1)⊆I1andsupp (λ2)⊆I2. (Use the function of Example /nine.taboldstyle./seven.taboldstyle(iii) as a building block. First show there exist smooth functions χ1,χ2:M→Rwith properties (i), (iii), and property (ii)’ χ1(x)+χ2(x)>0for all x∈M.) /nine.taboldstyle./four.taboldstyle.Letψ:(a,b)→Rnbe an embedding. Then M/equalxψ((a,b))is a smooth 1-manifold. Let us call the direction of the tangent vector ψ′(t)positive; this defines an orientation of M. Letµbe the element of arc length of M. (i) Show that ψ∗(µ)/equalx/bardblψ′(t)/bardbldt/equalx/radicalBig ψ′ 1(t)2+ψ′ 2(t)2+···+ψ′n(t)2dt, where t denotes the coordinate on R. Conclude that the arc length (“volume”) of Mis/integraltextb a/bardblψ′(t)/bardbldt. (ii) Compute the arc length of the astroid x/equalxcos3t,y/equalxsin3t, where t∈[0,π/2]. (iii) Consider a plane curve given in polar coordinates by an equation r/equalxf(θ). Show that its element of arc length is/radicalBig f′(θ)2+f(θ)2dθ. (Apply the result of part ( i) toψ(θ)/equalx/parenleftbigf(θ)cosθ,f(θ)sinθ/parenrightbig.) (iv) Compute the arc length of the cardioid given by r/equalx1 + cosθ. /nine.taboldstyle./five.taboldstyle. (i) Letα/equalxx dy−y dx and let Mbe a compact domain in the plane R2. Show that/integraltext ∂Mαis twice the surface area of M. (ii) Apply the observation of part ( i) to find the area enclosed by the astroid x/equalxcos3t, y/equalxsin3t. (iii) Letα/equalxx·∗dxand let Mbe a compact domain in Rn. Show that/integraltext ∂Mαis a constant times the volume of M. What is the value of the constant? /nine.taboldstyle./six.taboldstyle. Write the divergence theorem for the vector field F/equalx−cxnenonRn, where cis a positive constant. Deduce Archimedes’ Law: the buoyant fo rce exerted on a submerged body is equal to the weight of the displaced fluid. E´υρηκα ! /nine.taboldstyle./seven.taboldstyle.LetR1≥R2≥0be constants. Define a 3-cube c: [0,R2]×[0,2π]×[0,2π]→R3by c/parenlefttpA/parenleftexA /parenleftbtAr θ1 θ2/parenrighttpA/parenrightexA /parenrightbtA/equalx/parenlefttpA/parenleftexA /parenleftbtA(R1+rcosθ2)cosθ1 (R1+rcosθ2)sinθ1 rsinθ2/parenrighttpA/parenrightexA /parenrightbtA. (i) Sketch the image of c. (ii) Let x1,x2,x3be the standard coordinates on R3. Compute c∗(dx1),c∗(dx2), c∗(dx3)and c∗(dx1dx2dx3). (iii) Find the volume of the solid parametrized by c. (iv) Find the surface area of the boundary of this solid. /nine.taboldstyle./eight.taboldstyle.LetMbe a compact domain in Rn. Let fand gbe smooth functions on M. The Dirichlet integral offand gisD(f,g)/equalx/integraltext Mgrad (f)·grad (g)µ, whereµ/equalxdx1dx2···dxn is the volume form on M. (i) Show that d f(∗dg)/equalxgrad (f)·grad (g)µ. (ii) Show that d∗dg/equalx(∆g)µ, where∆g/equalx/summationtextn i/equalx1∂2g/∂x2 i. (iii) Deduce from parts ( i)–(ii) that d(f(∗dg))/equalx(grad (f)·grad (g)+f∆g)µ. (iv) Let nbe the outward pointing unit normal vector field on ∂M. Write∂g/∂nfor the directional derivative (D g)n/equalxgrad (g)·n. Show that /integraldisplay ∂Mf(∗dg)/equalx/integraldisplay ∂Mf∂g ∂nµ∂M. EXERCISES /one.taboldstyle/two.taboldstyle/one.taboldstyle (v) Deduce from parts ( iii) and ( iv) Green’s formula, /integraldisplay ∂Mf∂g ∂nµ∂M/equalxD(f,g)+/integraldisplay M(f∆g)µ. (vi) Deduce Green’s symmetric formula, /integraldisplay ∂M/parenleftbigg f∂g ∂n−g∂f ∂n/parenrightbigg µ∂M/equalx/integraldisplay M(f∆g−g∆f)µ. /nine.taboldstyle./nine.taboldstyle.In this problem we will calculate the volume of a ball and a sph ere in Euclidean space. Let B(R)be the closed ball of radius Rabout the origin in Rn. Then its boundary S(R)/equalx∂B(R)is the sphere of radius R. Put Vn(R)/equalxvoln(B(R))andAn(R)/equalxvoln−1(S(R)). Also put Vn/equalxVn(1)and An/equalxAn(1). (i) Deduce from Corollary /eight.taboldstyle./one.taboldstyle/seven.taboldstylethat the volume form on S(R)is the restriction of ν toS(R), whereνis as in Exercise /two.taboldstyle./two.taboldstyle/one.taboldstyle. Conclude that An(R)/equalx/integraltext S(R)ν. (ii) Show that Vn(R)/equalxRnVnand An(R)/equalxRn−1An. (Substitute y/equalxRxin the volume forms of B(R)and S(R).) (iii) Let f: [0,∞)→Rbe a continuous function. Define g:Rn→Rbyg(x)/equalxf(/bardblx/bardbl). Use Exercise /two.taboldstyle./two.taboldstyle/one.taboldstyle(ii) to prove that /integraldisplay B(R)g dx 1dx2···dxn/equalx/integraldisplayR 0f(r)An(r)dr/equalxAn/integraldisplayR 0f(r)rn−1dr. (iv) Show that/parenleftbigg/integraldisplay∞ −∞e−r2dr/parenrightbiggn /equalxAn/integraldisplay∞ 0e−r2rn−1dr. (Take f(r)/equalxe−r2in part ( iii) and let R→∞ .) (v) Using Exercises B./one.taboldstyle/five.taboldstyle andB./one.taboldstyle/six.taboldstyle conclude that An/equalx2πn 2 Γ/parenleftbign 2/parenrightbig,whence A2m/equalx2πm (m−1)!and A2m+1/equalx2m+1πm 1·3·5···(2m−1). (vi) By taking f(r)/equalx1in part ( iii) show that An/equalxnVnand An(R)/equalx∂Vn(R)/∂R. (vii) Deduce that Vn/equalxπn 2 Γ/parenleftbign 2+ 1/parenrightbig,whence V2m/equalxπm m!and V2m+1/equalx2m+1πm 1·3·5···(2m+ 1). (viii) Complete the following table. (Conventions: a space of negative dimension is empty; the volume of a zero-dimensional manifold is its numb er of points.) n 0 1 2 3 4 5 Vn(R) πR2 4 3πR3 An(R) 2πR (ix) Find limn→∞An,limn→∞Vnandlimn→∞(An+1/An). Use Stirling’s formula, limx→∞Γ(x+ 1)ex xx+1 2/equalx√ 2π. CHAPTER /one.taboldstyle/zero.taboldstyle Applications to topology In this chapter, to avoid endless repetitions it will be conv enient to make a slight change in terminology. By a “manifold” we will now mea n a “manifold with boundary”. It is understood that the boundary of a “manifold with boundary” may be empty. If we specifically require the boundary to be emp ty, we will speak of a “manifold without boundary”. /one.taboldstyle/zero.taboldstyle./one.taboldstyle. Brouwer’s fixed point theorem LetMbe a manifold (with boundary). A retraction ofMonto a subset Ais a smooth map φ:M→Asuch thatφ(x)/equalxxfor all xinA. For instance, let Mbe the punctured unit ball in n-space, M/equalx{x∈Rn|0</bardblx/bardbl≤1}. Then the normalization map φ(x)/equalxx//bardblx/bardblis a retraction of Monto its boundary A/equalx∂M, the unit sphere. The following theorem says that a retracti on onto the boundary is impossible if Mis compact and orientable. /one.taboldstyle/zero.taboldstyle./one.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. LetMbe a compact orientable manifold with nonempty boundary. Then there does not exist a retraction from Monto∂M. P/r.sc/o.sc/o.sc/f.sc. Supposeφ:M→∂Mwas a retraction. Let us choose an orientation of Mand equip∂Mwith the induced orientation. Let β/equalxµ∂Mbe the volume form on the boundary (relative to some embedding of MintoRN). Letα/equalxφ∗(β)be its pullback to M. Let ndenote the dimension of M. Note that βis an n−1-form on the n−1-manifold∂M, so dβ/equalx0. Therefore dα/equalxdφ∗(β)/equalxφ∗(dβ)/equalx0and hence by Stokes’ theorem 0/equalx/integraltext Mdα/equalx/integraltext ∂Mα. Butφis a retraction onto ∂M, so the restriction of φto∂Mis the identity map and therefore α/equalxβon∂M. Thus 0/equalx/integraldisplay ∂Mα/equalx/integraldisplay ∂Mβ/equalxvol(∂M)/nequal0, which is a contradiction. Therefore φdoes not exist. QED This brings us to one of the oldest results in topology. Suppo sefis a map from a set Xinto itself. An element xofXis afixed point offiff(x)/equalxx. /one.taboldstyle/zero.taboldstyle./two.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc (B/r.sc/o.sc/u.sc/w.sc/e.sc/r.sc’/s.sc /f.sc/i.sc/x.sc/e.sc/d.sc /p.sc/o.sc/i.sc/n.sc/t.sc /t.sc/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc). Every smooth map from the closed unit ball into itself has at least one fixed point. P/r.sc/o.sc/o.sc/f.sc. LetM/equalx{x∈Rn|/bardblx/bardbl≤1}be the closed unit ball. Suppose f:M→M was a smooth map without fixed points. Then f(x)/nequalxfor all x. For each xin the ball consider the half-line starting at f(x)and pointing in the direction of x. This /one.taboldstyle/two.taboldstyle/three.taboldstyle /one.taboldstyle/two.taboldstyle/four.taboldstyle /one.taboldstyle/zero.taboldstyle. APPLICATIONS TO TOPOLOGY half-line intersects the unit sphere ∂Min a unique point that we shall call φ(x), as in the following picture. xf(x) φ(x)y f(y)φ(y) This defines a smooth map φ:M→∂M. Ifxis in the unit sphere, then φ(x)/equalxx, soφis a retraction of the ball onto its boundary, which contradi cts Theorem /one.taboldstyle/zero.taboldstyle./one.taboldstyle. Therefore fmust have a fixed point. QED This theorem can be stated imprecisely as saying that after s tirring a cup of coffee at least one molecule must return to its original posit ion. Brouwer originally stated his result for arbitrary continuous maps. This more g eneral statement can be derived from Theorem /one.taboldstyle/zero.taboldstyle./two.taboldstyleby an argument from analysis which shows that every continuous map is homotopic to a smooth map. (See Section /one.taboldstyle/zero.taboldstyle./two.taboldstylefor the definition of homotopy.) The theorem also remains valid if the closed ba ll is replaced by a closed cube or a similar shape. /one.taboldstyle/zero.taboldstyle./two.taboldstyle. Homotopy Definition and first examples. Suppose that φ0andφ1are two maps from a manifold Mto a manifold Nand thatαis a form on N. What is the relationship between the pullbacks φ∗ 0(α)andφ∗ 1(α)? There is a reasonable answer to this question ifφ0can be “smoothly deformed” into φ1. (See Theorems /one.taboldstyle/zero.taboldstyle./eight.taboldstyleand/one.taboldstyle/zero.taboldstyle./one.taboldstyle/zero.taboldstyle below.) The notion of a smooth deformation can be defined form ally as follows. The mapsφ0andφ1arehomotopic if there exists a smooth map φ:M×[0,1]→N such thatφ(x,0)/equalxφ0(x)andφ(x,1)/equalxφ1(x)for all xinM. The mapφis called a homotopy . Instead of φ(x,t)we often write φt(x). Then each φtis a map from M toN. We can think of φtas a family of maps parametrized by tin the unit interval that interpolates between φ0andφ1, or as a one-second “movie” that at time 0 starts atφ0and at time 1ends up atφ1. /one.taboldstyle/zero.taboldstyle./three.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetM/equalxN/equalxRnandφ0(x)/equalxx(identity map) and φ1(x)/equalx0 (constant map). Then φ0andφ1are homotopic. A homotopy is given by φ(x,t)/equalx (1−t)x. This homotopy collapses Euclidean space onto the origin by moving each point radially inward at a speed equal to its distance to the o rigin. There are many other homotopies from φ0toφ1, such as (1−t)2xand (1−t2)x. We can also interchange the roles of φ0andφ1: ifφ0(x)/equalx0andφ1(x)/equalxx, then we find a homotopy by reversing time (playing the movie backwards), φ(x,t)/equalxtx. /one.taboldstyle/zero.taboldstyle./four.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetM/equalxNbe the punctured Euclidean space Rn\{0}and let φ0(x)/equalxx(identity map) and φ1(x)/equalxx//bardblx/bardbl(normalization map). Then φ0and φ1are homotopic. A homotopy is given for instance by φ(x,t)/equalxx//bardblx/bardbltor by φ(x,t)/equalx(1−t)x+tx//bardblx/bardbl. Either of these homotopies collapses the punctured /one.taboldstyle/zero.taboldstyle./two.taboldstyle. HOMOTOPY /one.taboldstyle/two.taboldstyle/five.taboldstyle Euclidean space onto the unit sphere by smoothly stretching or shrinking each vector until it has length 1. /one.taboldstyle/zero.taboldstyle./five.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. A manifold Mis said to be contractible if there exists a point x0inMsuch that the constant map φ0(x)/equalxx0is homotopic to the identity map φ1(x)/equalxx. A specific homotopy φ:M×[0,1]→Mfromφ0toφ1is acontraction of Monto x0. (Perhaps “expansion” would be a more accurate term, a “cont raction” being the result of replacing twith 1−t.) Example /one.taboldstyle/zero.taboldstyle./three.taboldstyleshows that Rnis contractible onto the origin. (In fact it is contractible onto any point x0by means of a contraction given by a very similar formula.) The same formula shows that an open or closed ball around the origin is contractible. We shall see in Theor em/one.taboldstyle/zero.taboldstyle./one.taboldstyle/nine.taboldstyle that punctured n-space Rn\{0}isnotcontractible. Homotopy of paths. IfMis an interval [a,b]and Nany manifold, then maps from MtoNare nothing but paths (parametrized curves) in N. A homotopy of paths can be visualized as a piece of string moving through th e manifold N. a b Nφ Homotopy of loops. Aloopin a manifold Nis a smooth map from the unit circle S1into N. This can be visualized as a thin rubber band sitting in N. A homotopy of loops φ:S1×[0,1]→Ncan be pictured as a rubber band floating through Nfrom time 0until time 1. S1Nφ /one.taboldstyle/zero.taboldstyle./six.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Consider the two loops φ0,φ1:S1→R2in the plane given by φ0(x)/equalxxandφ1(x)/equalxx+/parenleftbig2 0/parenrightbig. A homotopy of loops is given by shifting φ0to the right,φt(x)/equalxx+/parenleftbig2t 0/parenrightbig. What if we regard φ0andφ1as loops in the punctured /one.taboldstyle/two.taboldstyle/six.taboldstyle /one.taboldstyle/zero.taboldstyle. APPLICATIONS TO TOPOLOGY plane R2\{0}? Clearly the homotopy φdoes not work, because it moves the loop through the forbidden point 0. (E.g.φt(x)/equalx0forx/equalx/parenleftbig−1 0/parenrightbigand t/equalx1/2.) In fact, however you try to move φ0toφ1you get stuck at the origin, so it seems intuitively clear that there exists no homotopy of loops from φ0toφ1in the punctured plane. This is indeed the case, as we shall see in Example /one.taboldstyle/zero.taboldstyle./one.taboldstyle/three.taboldstyle . The homotopy formula. The product M×[0,1]is often called the cylinder with base M. The two maps defined by ι0(x)/equalx(x,0)andι1(x)/equalx(x,1)send Mto the bottom, resp. the top of the cylinder. A homotopy ι:M×[0,1]→M×[0,1] between these maps is given by the identity map ι(x,t)/equalx(x,t). (“Slide the bottom to the top at speed 1.”) ι0ι4/10ι1 base cylinder IfMis an open subset of Rn, ak-formαon the cylinder can be written as α/equalx/summationdisplay IfI(x,t)dxI+/summationdisplay JgJ(x,t)dt dx J, with Irunning over increasing multi-indices of degree kand Jover increasing multi-indices of degree k−1. (Here we write the dtin front of the dx’s because that is more convenient in what follows.) The cylinder operator turns forms on the cylinder into forms on the base lowering the degree by 1, κ:Ωk(M×[0,1])→Ωk−1(M), by taking the piece of αinvolving dtand integrating it over the unit interval, κ(α)/equalx/summationdisplay J/parenleftbigg/integraldisplay1 0gJ(x,t)dt/parenrightbigg dxJ. (In particular κ(α)/equalx0for anyαthat does not involve dt.) For a general manifold Mwe can write a k-form on the cylinder as α/equalxβ+dtγ, whereβandγare forms onM×[0,1](of degree kand k−1respectively) that do not involve dt. We then defineκ(α)/equalx/integraltext1 0dtγ. The following result will enable us to compare pullbacks of f orms under ho- motopic maps. It can be regarded as an application of Stokes’ theorem, but we shall give a direct proof. /one.taboldstyle/zero.taboldstyle./seven.taboldstyle. L/e.sc/m.sc/m.sc/a.sc (/c.sc/y.sc/l.sc/i.sc/n.sc/d.sc/e.sc/r.sc /f.sc/o.sc/r.sc/m.sc/u.sc/l.sc/a.sc). LetMbe a manifold. Then ι∗ 1(α)−ι∗ 0(α)/equalx κ(dα)+dκ(α)for all k-formsαonM×[0,1]. In short, ι∗ 1−ι∗ 0/equalxκd+dκ. /one.taboldstyle/zero.taboldstyle./two.taboldstyle. HOMOTOPY /one.taboldstyle/two.taboldstyle/seven.taboldstyle P/r.sc/o.sc/o.sc/f.sc. We write out the proof for an open subset MofRn. The proof for arbitrary manifolds is similar. It suffices to consider two ca ses:α/equalxf dx Iand α/equalxg dt dx J. Case /one.taboldstyle. Ifα/equalxf dx I, thenκ(α)/equalx0and dκ(α)/equalx0. Also dα/equalx∂f ∂tdt dx I+/summationdisplay i∂f ∂xidxidxI/equalx∂f ∂tdt dx I+terms not involving dt, so dκ(α)+κ(dα)/equalxκ(dα)/equalx/parenleftbigg/integraldisplay1 0∂f ∂t(x,t)dt/parenrightbigg dxI /equalx/parenleftbigf(x,1)−f(x,0)/parenrightbigdxI/equalxι∗ 1(α)−ι∗ 0(α). Case /two.taboldstyle. Ifα/equalxg dt dx J, thenι∗ 0(α)/equalxι∗ 1(α)/equalx0and dα/equalx/summationdisplay i∂g ∂xidxidt dx J/equalx−/summationdisplay i∂g ∂xidt dx idxJ, so κ(dα)/equalx−n/summationdisplay i/equalx1/parenleftbigg/integraldisplay1 0∂g ∂xi(x,t)dt/parenrightbigg dxidxJ. Alsoκ(α)/equalx/parenleftbigg/integraldisplay1 0g(x,t)dt/parenrightbigg dxJ, so dκ(α)/equalxn/summationdisplay i/equalx1∂ ∂xi/parenleftbigg/integraldisplay1 0g(x,t)dt/parenrightbigg dxidxJ/equalxn/summationdisplay i/equalx1/parenleftbigg/integraldisplay1 0∂g ∂xi(x,t)dt/parenrightbigg dxidxJ. Hence dκ(α)+κ(dα)/equalx0/equalxι∗ 1(α)−ι∗ 0(α). QED Now suppose we have a pair of maps φ0andφ1going from a manifold M to a manifold Nand thatφ:M×[0,1]→Nis a homotopy between φ0andφ1. ForxinMwe haveφ◦ι0(x)/equalxφ(x,0)/equalxφ0(x), in other words φ0/equalxφ◦ι0. Similarlyφ1/equalxφ◦ι1. Hence for any k-formαonNwe haveι∗ 0(φ∗(α))/equalxφ∗ 0(α)and ι∗ 1(φ∗(α))/equalxφ∗ 1(α). Applying the cylinder formula to the form φ∗(α)onM×[0,1] we see that the pullbacks φ∗ 0(α)andφ∗ 1(α)are related in the following manner. /one.taboldstyle/zero.taboldstyle./eight.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc (/h.sc/o.sc/m.sc/o.sc/t.sc/o.sc/p.sc/y.sc /f.sc/o.sc/r.sc/m.sc/u.sc/l.sc/a.sc). Letφ0,φ1:M→Nbe smooth maps from a manifold Mto a manifold Nand letφ:M×[0,1]→Nbe a homotopy from φ0toφ1. Thenφ∗ 1(α)−φ∗ 0(α)/equalxκφ∗(dα)+dκφ∗(α)for all k-formsαonN. In short, φ∗ 1−φ∗ 0/equalxκφ∗d+dκφ∗. In particular, if dα/equalx0we getφ∗ 1(α)/equalxφ∗ 0(α)+dκφ∗(α). /one.taboldstyle/zero.taboldstyle./nine.taboldstyle. C/o.sc/r.sc/o.sc/l.sc/l.sc/a.sc/r.sc/y.sc. Ifφ0,φ1:M→Nare homotopic maps between manifolds and α is a closed form on N, thenφ∗ 0(α)andφ∗ 1(α)differ by an exact form. This implies that if the degree of αis equal to the dimension of M,φ∗ 0(α)and φ∗ 1(α)have the same integral. /one.taboldstyle/two.taboldstyle/eight.taboldstyle /one.taboldstyle/zero.taboldstyle. APPLICATIONS TO TOPOLOGY /one.taboldstyle/zero.taboldstyle./one.taboldstyle/zero.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. LetMandNbe manifolds and let αbe a closed n-form on N, where n/equalxdim(M). Suppose Mis compact and oriented and without boundary. Let φ0andφ1 be homotopic maps from MtoN. Then/integraldisplay Mφ∗ 0(α)/equalx/integraldisplay Mφ∗ 1(α). P/r.sc/o.sc/o.sc/f.sc. By Corollary /one.taboldstyle/zero.taboldstyle./nine.taboldstyle,φ∗ 1(α)−φ∗ 0(α)/equalxdβfor an n−1-formβonM. Hence by Stokes’ theorem/integraldisplay M/parenleftbigφ∗ 1(α)−φ∗ 0(α)/parenrightbig/equalx/integraldisplay Mdβ/equalx/integraldisplay ∂Mβ/equalx0, because∂Mis empty. QED A/l.sc/t.sc/e.sc/r.sc/n.sc/a.sc/t.sc/i.sc/v.sc/e.sc /p.sc/r.sc/o.sc/o.sc/f.sc. Here is a proof based on Stokes’ theorem for the manifold M×[0,1]. The boundary of M×[0,1]consists of two copies of M, namely M×{1} and M×{0}, the first of which is counted with a plus sign and the second wi th a minus. Therefore, if φ:M×[0,1]→Nis a homotopy between φ0andφ1, 0/equalx/integraldisplay M×[0,1]φ∗(dα)/equalx/integraldisplay M×[0,1]dφ∗(α)/equalx/integraldisplay ∂(M×[0,1])φ∗(α) /equalx/integraldisplay Mφ∗ 1(α)−/integraldisplay Mφ∗ 0(α), so/integraltext Mφ∗ 1(α)/equalx/integraltext Mφ∗ 0(α). QED /one.taboldstyle/zero.taboldstyle./one.taboldstyle/one.taboldstyle. C/o.sc/r.sc/o.sc/l.sc/l.sc/a.sc/r.sc/y.sc. Homotopic loops in R2\{0}have the same winding number about the origin. P/r.sc/o.sc/o.sc/f.sc. LetMbe the circle S1,Nthe punctured plane R2\{0}, andα0the angle form (−y dx+x dy)/(x2+y2). Then a map φfrom MtoNis nothing but a loop in the punctured plane, and the integral/integraltext Mφ∗(α0)is2πtimes the winding number w(φ,0). (See Section /four.taboldstyle./three.taboldstyle.) Thus, ifφ0andφ1are homotopic loops in N, Theorem /one.taboldstyle/zero.taboldstyle./one.taboldstyle/zero.taboldstyle tells us that w(φ0,0)/equalxw(φ1,0). QED /one.taboldstyle/zero.taboldstyle./one.taboldstyle/two.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Unfolding the three self-intersections in the path picture d below does not affect its winding number. 0 0 /one.taboldstyle/zero.taboldstyle./one.taboldstyle/three.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. The two circles φ0andφ1of Example /one.taboldstyle/zero.taboldstyle./six.taboldstyle have winding number 1, resp. 0about the origin and therefore are not homotopic (as loops in the punctured plane). /one.taboldstyle/zero.taboldstyle./three.taboldstyle. CLOSED AND EXACT FORMS RE-EXAMINED /one.taboldstyle/two.taboldstyle/nine.taboldstyle /one.taboldstyle/zero.taboldstyle./three.taboldstyle. Closed and exact forms re-examined The homotopy formula throws light on our old question of when a closed form is exact, which we looked into in Section /two.taboldstyle./three.taboldstyle. The answer turns out to depend on the “shape” of the manifold on which the forms are de fined. On some manifolds all closed forms (of positive degree) are exact, o n others this is true only in certain degrees. Failure of exactness is typically detec ted by integrating over a submanifold of the correct dimension and finding a nonzero an swer. In a certain sense all obstructions to exactness are of this nature. It is to make these statements precise that de Rham developed his cohomology theory. We sha ll not develop this theory in detail, but study a few representative specia l cases. The matter is explored in [ Fla/eight.taboldstyle/nine.taboldstyle ] and at a more advanced level in [ BT/eight.taboldstyle/two.taboldstyle ]. 0-forms. A closed 0-form on a manifold is a smooth function fsatisfying d f/equalx0. This means that fis constant (on each connected component of M). If this constant is nonzero, then fis not exact (because forms of degree −1are by definition 0). So a closed 0-form is never exact (unless it is 0) for a rather uninteresting reason. 1-forms and simple connectivity. Let us now consider 1-forms on a manifold M. Theorem /four.taboldstyle./seven.taboldstylesays that the integral of an exact 1-form along a loop is 0. With a stronger assumption on the loop the same is true for arbitra ryclosed 1-forms. A loop c:S1→Misnull-homotopic if it is homotopic to a constant loop. The integral of a1-form along a constant loop is 0, so from Theorem /one.taboldstyle/zero.taboldstyle./one.taboldstyle/zero.taboldstyle (where we set the M of the theorem equal to S1) we get the following. /one.taboldstyle/zero.taboldstyle./one.taboldstyle/four.taboldstyle. P/r.sc/o.sc/p.sc/o.sc/s.sc/i.sc/t.sc/i.sc/o.sc/n.sc. Letcbe a null-homotopic loop in M. Then/integraltext cα/equalx0for all closed formsαonM. A manifold is simply connected if every loop in it is null-homotopic. /one.taboldstyle/zero.taboldstyle./one.taboldstyle/five.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. All closed 1-forms on a simply connected manifold are exact. P/r.sc/o.sc/o.sc/f.sc. Letαbe a closed 1-form and ca loop in M. Then cis null-homotopic, so/integraltext cα/equalx0by Proposition /one.taboldstyle/zero.taboldstyle./one.taboldstyle/four.taboldstyle . The result now follows from Theorem /four.taboldstyle./seven.taboldstyle. QED /one.taboldstyle/zero.taboldstyle./one.taboldstyle/six.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. The punctured plane R2\{0}is not simply connected, because it possesses a nonexact closed 1-form. (See Example /four.taboldstyle./six.taboldstyle.) In contrast it can be proved that for n≥3the sphere Sn−1and punctured n-space Rn\{0}are simply connected. Intuitively, the reason is that in two dimension s a loop that encloses the puncture at the origin cannot be crumpled up to a point wit hout getting stuck at the puncture, whereas in higher dimensions there is enoug h room to slide any loop away from the puncture and then squeeze it to a point. The Poincaré lemma. On a contractible manifold allclosed forms of positive degree are exact. /one.taboldstyle/zero.taboldstyle./one.taboldstyle/seven.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc (P/o.sc/i.sc/n.sc/c.sc/a.sc/r.sc/eacute.sc /l.sc/e.sc/m.sc/m.sc/a.sc). All closed forms of degree k≥1on a contractible manifold are exact. P/r.sc/o.sc/o.sc/f.sc. LetMbe a manifold and let φ:M×[0,1]→Mbe a contraction onto a point x0inM, i.e. a smooth map satisfying φ(x,0)/equalxx0andφ(x,1)/equalxxfor all x. Letαbe a closed k-form on Mwith k≥1. Thenφ∗ 1(α)/equalxαandφ∗ 0(α)/equalx0, so puttingβ/equalxκφ∗(α)we get dβ/equalxdκφ∗(α)/equalxφ∗ 1(α)−φ∗ 0(α)−κdφ∗(α)/equalxα. /one.taboldstyle/three.taboldstyle/zero.taboldstyle /one.taboldstyle/zero.taboldstyle. APPLICATIONS TO TOPOLOGY Here we used the homotopy formula, Theorem /one.taboldstyle/zero.taboldstyle./eight.taboldstyle, and the assumption that dα/equalx0. Hence dβ/equalxα. QED The proof provides us with a formula for the “antiderivative ”, namelyβ/equalx κφ∗(α), which can be made quite explicit in certain cases. /one.taboldstyle/zero.taboldstyle./one.taboldstyle/eight.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetMbeRnand letφ(x,t)/equalxtxbe the radial contraction. Let α/equalx/summationtext ifidxibe a1-form and let gbe the function κφ∗(α). Then φ∗(α)/equalx/summationdisplay ifi(tx)d(txi)/equalx/summationdisplay ifi(tx)(xidt+t dx i), so g/equalxκφ∗(α)/equalx/summationdisplay ixi/integraldisplay1 0fi(tx)dt. According to the proof of the Poincaré lemma the function gsatisfies dg/equalxα, provided that dα/equalx0. We checked this directly in Exercise /four.taboldstyle./three.taboldstyle. Another typical application of the Poincaré lemma is showin g that a manifold is not contractible by exhibiting a closed form that is not ex act. For example, the punctured plane R2\{0}is not contractible because it possesses a nonexact closed 1-form, namely the angle form. (See Example /four.taboldstyle./six.taboldstyle.) The generalized angle form is the n−1-formα0on punctured n-space Rn\{0}defined by α0/equalxx·∗dx /bardblx/bardbln. /one.taboldstyle/zero.taboldstyle./one.taboldstyle/nine.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. The generalized angle form α0is a closed but non-exact n−1-form on punctured n-space. Hence punctured n-space is not contractible. P/r.sc/o.sc/o.sc/f.sc. That dα0/equalx0follows from Exercise /two.taboldstyle./two.taboldstyle/zero.taboldstyle. The n−1-sphere M/equalxSn−1 has unit normal vector field x, so by Corollary /eight.taboldstyle./one.taboldstyle/seven.taboldstyle onMwe haveα0/equalxµ, the volume form. Hence/integraltext Mα0/equalxvol(M)/nequal0. On the other hand, suppose α0was exact,α0/equalxdβfor an n−1-formβ. Then /integraldisplay Mα0/equalx/integraldisplay Mdβ/equalx/integraldisplay ∂Mβ/equalx0 by Stokes’ theorem, Theorem /nine.taboldstyle./nine.taboldstyle. This is a contradiction, so α0is not exact. It now follows from the Poincaré lemma, Theorem /one.taboldstyle/zero.taboldstyle./one.taboldstyle/seven.taboldstyle , that Rn\{0}is not contractible. QED A similar argument gives the next result. /one.taboldstyle/zero.taboldstyle./two.taboldstyle/zero.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. Compact orientable manifolds without boundary of dimensio n≥1 are not contractible. P/r.sc/o.sc/o.sc/f.sc. LetMbe a compact orientable manifold of dimension n≥1. Let µ/equalxµMbe the volume form of M. Then/integraltext Mµ/equalxvol(M)/nequal0. On the other hand, ifMwas contractible, then µwould be exact by the Poicaré lemma, so µ/equalxdνand/integraltext Mµ/equalx/integraltext Mdν/equalx/integraltext ∂Mν/equalx0because Mhas no boundary. This is a contradiction, so Mis not contractible. QED /one.taboldstyle/zero.taboldstyle./three.taboldstyle. CLOSED AND EXACT FORMS RE-EXAMINED /one.taboldstyle/three.taboldstyle/one.taboldstyle In particular the unit sphere Snis not contractible for n≥1: it has a closed nonexact n-form. But how about forms of degree not equal to n−1? Without proof we state the following fact. /one.taboldstyle/zero.taboldstyle./two.taboldstyle/one.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. OnRn\{0}and on Sn−1every closed form of degree k/nequal1,n−1 is exact. For a compact oriented hypersurface without boundary Mcontained in Rn\{0} the integral w(M,0)/equalx1 vol n−1(Sn−1)/integraldisplay Mx·∗dx /bardblx/bardbln is called the winding number ofMabout the origin. It generalizes the winding number of a closed path in R2\{0}around the origin. It can be shown that the winding number in any dimension is always an integer. It prov ides a measure of how many times the hypersurface wraps around the origin. For instance, the proof of Theorem /one.taboldstyle/zero.taboldstyle./one.taboldstyle/nine.taboldstyle shows that the winding number of the n−1-sphere about the origin is 1. Contractibilityversus simple connectivity. Theorems /one.taboldstyle/zero.taboldstyle./one.taboldstyle/five.taboldstyle and/one.taboldstyle/zero.taboldstyle./one.taboldstyle/seven.taboldstyle suggest that the notions of contractibility and simple connectivit y are not independent. /one.taboldstyle/zero.taboldstyle./two.taboldstyle/two.taboldstyle. P/r.sc/o.sc/p.sc/o.sc/s.sc/i.sc/t.sc/i.sc/o.sc/n.sc. A contractible manifold is simply connected. P/r.sc/o.sc/o.sc/f.sc. Use a contraction to collapse any loop onto a point. Mx0 c1 Mx0 c1 Formally, let c1:S1→Mbe a loop,φ:M×[0,1]→Ma contraction of Monto x0. Putc(s,t)/equalxφ(c1(s),t). Then cis a homotopy between c1and the constant loop c0(t)/equalxφ(c1(s),0)/equalxx0positioned at x0. QED As mentioned in Example /one.taboldstyle/zero.taboldstyle./one.taboldstyle/six.taboldstyle , the sphere Sn−1and punctured n-space Rn\{0} are simply connected for n≥3, although it follows from Theorem /one.taboldstyle/zero.taboldstyle./one.taboldstyle/nine.taboldstyle that they are not contractible. Thus simple connectivity is weaker th an contractibility. The Poincaré conjecture. Not long after inventing the notion of homotopy Poincaré posed the following question. Let Mbe a compact three-dimensional manifold without boundary. Suppose Mis simply connected. Is Mhomeomorphic to the three-dimensional sphere? (This means: does there ex ist a bijective map M→S3which is continuous and has a continuous inverse?) This ques tion became (inaccurately) known as the Poincaré conjecture . It is famously difficult and was /one.taboldstyle/three.taboldstyle/two.taboldstyle /one.taboldstyle/zero.taboldstyle. APPLICATIONS TO TOPOLOGY the force that drove many of the developments in twentieth-c entury topology. It has an n-dimensional analogue, called the generalized Poincaré conjecture, which asks whether every compact n-dimensional manifold without boundary which is homotopy equivalent to Snis homeomorphic to Sn. We cannot go into this fascinating problem in any serious way, other than to report that it has now been completely solved. Strangely, the case n≥5of the generalized Poincaré conjecture conjecture was the easiest and was confirmed by S. Smale in /one.taboldstyle/nine.taboldstyle/six.taboldstyle /zero.taboldstyle. The case n/equalx4 was done by M. Freedman in /one.taboldstyle/nine.taboldstyle/eight.taboldstyle/two.taboldstyle. The case n/equalx3, the original version of the conjecture, turned out to be the hardest, but was finally confi rmed by G. Perelman in /two.taboldstyle/zero.taboldstyle/zero.taboldstyle/two.taboldstyle-/zero.taboldstyle/three.taboldstyle. For a discussion and references, see the paper [ Mil/zero.taboldstyle/three.taboldstyle ] listed in the bibliography. De Rham cohomology. The distinction between closed and exact differential forms on a manifold can be encoded in an invariant called de Rh am cohomology. To explain this we need to review a little set theory and linea r algebra. LetXbe a set. A binary relation ∼(i.e. a relation among pairs of elements) on Xis an equivalence relation if it is (i) reflexive: x∼x; (ii) symmetric: if x∼ythen y∼x; (iii) transitive: if x∼yand y∼zthen x∼z for all x,y,z∈X. /one.taboldstyle/zero.taboldstyle./two.taboldstyle/three.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. Consider the following binary relations. (i) Let X/equalxRwith the order relation ≤(“less than or equal to”). (ii) Let X/equalxZ. Fix n∈Z. Define x≡yifx−yis divisible by n(“congruence modulo n”). (iii) Let Xbe the set of all people. (a) Define x⊲⊳yifxis a blood relative of y. (b) Define x⌣yifxis a friend of y. (iv) Let Xbe the set of straight lines in the plane. Define x/bardblyifxis parallel toy. Then we have the following table. reflexive symmetric transitive (i) Y N Y (ii) Y Y Y (iiia) Y Y ? (iiib) N N N (iv) Y Y Y Whether relation ( iiia) is transitive is a matter of taste. In a broad sense we are all blood relatives of mitochondrial Eve, but perhaps this stre tches the definition too far. Only relations ( ii) and ( iv) are true equivalence relations. Let∼be an equivalence relation on a set X. The equivalence class ofx∈Xis the set of all y∈Xthat are equivalent to x. Notation: [x]/equalx{y∈X|y∼x}. Note that we have [x]/equalx[y]ifx∼y. Any element of an equivalence class is called a representative of that class. An equivalence class usually has many differen t /one.taboldstyle/zero.taboldstyle./three.taboldstyle. CLOSED AND EXACT FORMS RE-EXAMINED /one.taboldstyle/three.taboldstyle/three.taboldstyle representatives. The set of all equivalence classes is call ed the quotient ofXby the equivalence relation and denoted by X/∼. In Example /one.taboldstyle/zero.taboldstyle./two.taboldstyle/three.taboldstyle (ii) we write Z/nfor the quotient. An element of Z/nis a remainder class modulo n, [i]/equalx{i,i+n,i−n,i+n,i+ 2n,i−2n,...}/equalx{i+kn|k∈Z}. Example /one.taboldstyle/zero.taboldstyle./two.taboldstyle/three.taboldstyle (iv) comes up in projective geometry. The equivalence class of a line in the plane is a point at infinity in the plane. If one adds the points at infinity to the plane one arrives at the projective plane . /one.taboldstyle/zero.taboldstyle./two.taboldstyle/four.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetEbe a real vector space. Fix a linear subspace F. Vectors x, y∈Earecongruent modulo Fifx−y∈F. Congruence modulo Fis an equivalence relation on E. The equivalence class [x]ofxis the affine subspace through xparallel to F. The equivalence class [0]is equal to the linear subspace Fitself. We denote the quotient by E/Fand call it the quotient of EbyF. A basic fact is that E/Fis a vector space in its own right. The vector space operations are define d by [x] + [y]/equalx[x+y] and c[x]/equalx[cx]for all x,y∈Eand c∈R. One can check that these operations are well-defined and obey the axioms of a real vector space. The or igin of E/Fis the equivalence class [0], and the opposite of a class [x]is the class [−x]. We have the following special case of Example /one.taboldstyle/zero.taboldstyle./two.taboldstyle/four.taboldstyle in mind. Let Mbe a manifold andΩk(M)the vector space of all k-forms on M. Let Ebe the linear subspace ofΩk(M)consisting of all closed k-forms, E/equalx{α∈Ωk(M)|dα/equalx0}, and let Fbe the subspace of Econsisting of all exact k-forms, F/equalx{α∈Ωk(M)|α/equalxdβfor someβ∈Ωk−1(M)}. The quotient Hk DR(M)/equalxE/F/equalxclosed k-forms on M exact k-forms on M is the de Rham cohomology ofMin degree k. Elements of Hk DR(M)are equivalence classes [α]whereαis a closed k-form andα∼βifα−βis exact. Both vector spaces Eand Fare usually infinite-dimensional, but it may very well happe n that the quotient Hk DR(M)is finite-dimensional. For any n-manifold Mwe have Hk(M)/equalx0 ifk<0ork>n. The reason is simply that we don’t have any nonzero forms on M in degrees below 0or above n. More interesting are the following assertions. /one.taboldstyle/zero.taboldstyle./two.taboldstyle/five.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. IfMis contractible then Hk DR(M)/equalxRifk/equalx0, 0ifk≥1. P/r.sc/o.sc/o.sc/f.sc. This is a restatement of the Poincaré lemma, Theorem /one.taboldstyle/zero.taboldstyle./one.taboldstyle/seven.taboldstyle . QED /one.taboldstyle/zero.taboldstyle./two.taboldstyle/six.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. LetMbe a compact connected oriented n-manifold without bound- ary. Then Hn DR(M)is1-dimensional. A basis of Hn DR(M)is[µ], the class of the volume form of M. We shall not prove this theorem in general (except to note tha t the class [µ]is nonzero by the proof of Theorem /one.taboldstyle/zero.taboldstyle./two.taboldstyle/zero.taboldstyle ), but only for the unit circle. /one.taboldstyle/three.taboldstyle/four.taboldstyle /one.taboldstyle/zero.taboldstyle. APPLICATIONS TO TOPOLOGY /one.taboldstyle/zero.taboldstyle./two.taboldstyle/seven.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc. H1 DR(S1)is1-dimensional. A basis of H1 DR(S1)is[µ], the class of the element of arc length of S1. P/r.sc/o.sc/o.sc/f.sc. The1-formµis closed but not exact and therefore defines a nonzero class in H1 DR(S1). We need to prove that for every 1-formαonS1(which is necessarily closed because dim(S1)/equalx1) there exists a constant ksuch that [α]/equalx k[µ]. Assuming we could do this, let us first guess what kshould be. The equality [α]/equalxk[µ]means [α−kµ]/equalx[0], i.e.α−kµis exact, i.e.α/equalxkµ+dgfor some smooth function gonS1. Integrating over the circle and using Stokes gives /integraldisplay S1α/equalxk/integraldisplay S1µ+/integraldisplay S1dg/equalx2πk. Let us therefore define k/equalx(2π)−1/integraltext S1α, as we must. Next we determine what g should be by solving dg/equalxα−kµforg. We do this indirectly by first solving the equation dh/equalxc∗(α−kµ), where c(t)/equalx(cost,sint)is the usual parametrization of the circle. We have c∗(α)/equalxf dt, where f:R→Ris a2π-periodic function. The constant kis given in terms of fby k/equalx1 2π/integraldisplay S1α/equalx1 2π/integraldisplay2π 0c∗(α)/equalx1 2π/integraldisplay2π 0f(t)dt. (/one.taboldstyle/zero.taboldstyle./one.taboldstyle) A solution of the equation dh/equalxc∗(α−kµ)/equalx(f−k)dtis the function h(t)/equalx/integraltextt 0(f(s)−k)ds/equalx/integraltextt 0f(s)ds−kt. This function has the property h(t+ 2π)/equalx/integraldisplayt+2π 0f(s)ds−k(t+ 2π)/equalx/integraldisplayt 0f(s)ds−kt+/integraldisplayt+2π tf(s)ds−2πk /equalxh(t)+/integraldisplay2π 0f(s)ds−2πk/equalxh(t), where the last step follows from the periodicity of fand from ( /one.taboldstyle/zero.taboldstyle./one.taboldstyle). This shows that his2π-periodic, and therefore of the form h(t)/equalxg(c(t)), i.e. h/equalxc∗(g), for some smooth function gonS1. The equation dh/equalxc∗(α−kµ)then becomes c∗(dg)/equalxc∗(α−kµ), which implies dg/equalxα−kµbecause cis a parametrization of the circle. QED Exercises /one.taboldstyle/zero.taboldstyle./one.taboldstyle. Write a formula for the map φoccurring in the proof of Brouwer’s fixed point theorem and prove that it is smooth. /one.taboldstyle/zero.taboldstyle./two.taboldstyle. Letx0be any point in Rn. By analogy with the radial contraction onto the origin, write a formula for radial contraction onto the point x0. Deduce that any open or closed ball centred at x0is contractible. /one.taboldstyle/zero.taboldstyle./three.taboldstyle. A subset MofRnisstar-shaped relative to a point x0∈Mif for all x∈Mthe straight line segment joining x0toxis entirely contained in M. Show that if Mis star-shaped relative to x0, then it is contractible onto x0. Give an example of a contractible set that is not star-shaped. /one.taboldstyle/zero.taboldstyle./four.taboldstyle. A subset MofRnisconvex if for all xandyinMthe straight line segment joining xtoyis entirely contained in M. Prove the following assertions. (i)Mis convex if and only if it is star-shaped relative to each of i ts points. Give an example of a star-shaped set that is not convex. EXERCISES /one.taboldstyle/three.taboldstyle/five.taboldstyle (ii) The closed ball B(ε,x)of radiusεcentred at xis convex. (iii) The open ball B◦(ε,x)of radiusεcentred at xis convex. /one.taboldstyle/zero.taboldstyle./five.taboldstyle. Recall that GL(n,R)denotes the general linear group and O(n)the orthogonal group. (See Theorem /six.taboldstyle./one.taboldstyle/eight.taboldstyle and Exercise /six.taboldstyle./one.taboldstyle/three.taboldstyle.) Define a map φ:GL(n,R)→O(n)by φ(A)/equalxQ, where Qis the first factor in the QR-decomposition of A. (See the proof of Theorem /eight.taboldstyle./four.taboldstyle.) Prove the following assertions. (i)φis a retraction. (ii)φis homotopic to the identity mapping of M. /one.taboldstyle/zero.taboldstyle./six.taboldstyle. Compute the turning number (see Exercise /four.taboldstyle./one.taboldstyle/four.taboldstyle) of the loops φ0andφ1of Example /one.taboldstyle/zero.taboldstyle./one.taboldstyle/two.taboldstyle . Despite the fact that the loops are homotopic they do not hav e the same turning number. Why does this not contradict Theorem /one.taboldstyle/zero.taboldstyle./one.taboldstyle/zero.taboldstyle ? /one.taboldstyle/zero.taboldstyle./seven.taboldstyle. Letαbe the k-form f dx I/equalxf dx i1dxi2···dxikonRnand letφ:Rn×[0,1]→Rn be the radial contraction φ(x,t)/equalxtx. Verify that κφ∗(α)/equalxk/summationdisplay m/equalx1(−1)m+1/parenleftbigg/integraldisplay1 0f(tx)tk−1dt/parenrightbigg ximdxi1dxi2···/hatwiderdxim···dxik, and check directly that dκφ∗(α)+κdφ∗(α)/equalxαfork≥1. /one.taboldstyle/zero.taboldstyle./eight.taboldstyle. Letα/equalxf dx dy +g dz dx +h dy dz be a2-form on R3and letφ(x,y,z,t)/equalxt(x,y,z) be the radial contraction of R3onto the origin. Verify that κφ∗(α)/equalx/parenleftbigg/integraldisplay1 0f(tx,t y,tz)t dt/parenrightbigg (x dy−y dx)+/parenleftbigg/integraldisplay1 0g(tx,t y,tz)t dt/parenrightbigg (z dx−x dz) +/parenleftbigg/integraldisplay1 0h(tx,t y,tz)t dt/parenrightbigg (y dz−z dy). /one.taboldstyle/zero.taboldstyle./nine.taboldstyle. Letα/equalx/summationtext IfIdxIbe a closed k-form whose coefficients fIare smooth functions defined on Rn\{0}that are all homogeneous of the same degree p/nequal−k. Let β/equalx1 p+k/summationdisplay Ik/summationdisplay l/equalx1(−1)l+1xilfIdxi1dxi2···/hatwiderdxil···dxik. Show that dβ/equalxα. (Use dα/equalx0and apply the identity proved in Exercise B./six.taboldstyleto each fI; see also Exercise /two.taboldstyle./nine.taboldstyle.) /one.taboldstyle/zero.taboldstyle./one.taboldstyle/zero.taboldstyle. Letα/equalx(2xyz−x2y)dy dz +(xz2−y2z)dz dx +(2xyz−xy2)dx dy . (i) Check that αis closed. (ii) Find a 1-formβsuch that dβ/equalxα. /one.taboldstyle/zero.taboldstyle./one.taboldstyle/one.taboldstyle. LetMand Nbe manifolds and φ0,φ1:M→Nhomotopic maps. Show that/integraltext cφ∗ 0(α)/equalx/integraltext cφ∗ 1(α)for all closed k-chains cinMand all closed k-formsαonN. /one.taboldstyle/zero.taboldstyle./one.taboldstyle/two.taboldstyle. Prove that any two maps φ0andφ1from MtoNare homotopic if MorNis contractible. (First show that every map M→Nis homotopic to a constant map φ(x)/equalxy0.) /one.taboldstyle/zero.taboldstyle./one.taboldstyle/three.taboldstyle. Letx0/equalx(2,0)and let Mbe the twice-punctured plane R2\{0,x0}. Let c1,c2, c3: [0,2π]→Mbe the loops defined by c1(t)/equalx(cost,sint),c2(t)/equalx(2 + cos t,sint)and c3(t)/equalx(1 + 2 cos t,2 sin t). Show that c1,c2and c3are not homotopic. (Construct a 1-form αonMsuch that the integrals/integraltext c1α,/integraltext c2αand/integraltext c3αare distinct.) /one.taboldstyle/zero.taboldstyle./one.taboldstyle/four.taboldstyle. LetMbe a manifold. Let [α]be a class in Hk DR(M)and let [β]be a class in Hl DR(M). Define the product of[α]and[β]to be the class [α]·[β]/equalx[αβ]∈Hk+l DR(M). Show that[α]·[β]is well-defined, i.e. independent of the choice of the repres entatives of the classes [α]and[β]. APPENDIX A Sets and functions A./one.taboldstyle. Glossary We start with a list of set-theoretical notations that are fr equently used in the text. Let Xand Ybe sets. x∈X:xis an element of X. {a,b,c}: the set containing the elements a,band c. X⊆Y:Xis a subset of Y, i.e. every element of Xis an element of Y. X∩Y: the intersection of Xand Y. This is defined as the set of all xsuch that x∈Xandx∈Y. X∪Y: the union of Xand Y. This is defined as the set of all xsuch that x∈Xorx∈Y. X\Y: the complement of YinX. This is defined as the set of xinXsuch that xis not in Y. (x,y): the ordered pair consisting of two elements xand y. X×Y: the Cartesian product of Xand Y. This is by definition the set of all ordered pairs (x,y)with x∈Xand y∈Y. Examples: R×Ris the Euclidean plane, usually written R2;S1×[0,1]is a cylinder wall of height 1; and S1×S1is a torus. R2 S1×[0, 1] S1×S1 (x1,x2,..., xk): the k-tuple, i.e. ordered list, consisting of the kelements x1, x2,...,xk. X1×X2×···× Xk: the k-fold Cartesian product of sets X1,X2,...,Xk. This is by definition the set of all k-tuples (x1,x2,..., xk)with xi∈Xi. Exam- ples: Rn1×Rn2×···× Rnk/equalxRn1+n2+···+nk; and S1×S1×···× S1(ktimes) is ak-torus. /one.taboldstyle/three.taboldstyle/seven.taboldstyle /one.taboldstyle/three.taboldstyle/eight.taboldstyle A. SETS AND FUNCTIONS {x∈X|P(x)}: the set of all x∈Xwhich have the property P(x). Examples: {x∈R|1≤x<3}is the interval [1,3), {x|x∈Xand x∈Y}is the intersection X∩Y, {x|x∈Xorx∈Y}is the union X∪Y, {x∈X|x/nelementY}is the complement X\Y. f:X→Y:fis a function (also called a map) from XtoY. This means that fassigns to each x∈Xa unique element f(x)∈Y. The set Xis called thedomain orsource off, and Yis called the codomain ortarget off. f(A): the image of a Aunder the map f. IfAis a subset of X, then its image under fis by definition the set f(A)/equalx{y∈Y|y/equalxf(x)for some x∈A}. f−1(B): the preimage ofBunder the map f. IfBis a subset of Y, this is by definition the set f−1(B)/equalx{x∈X|f(x)∈B}. (This is a somewhat confusing notation. It is notmeant to imply that fis required to have an inverse.) f−1(c): an abbreviation for f−1({c}), i.e. the set{x∈X|f(x)/equalxc}. This is often called the fibreorlevel set offatc. f|A: the restriction offtoA. IfAis a subset of X,f|Ais the function defined by (f|A)(x)/equalxf(x) ifx∈A, not defined if x/nelementA. In other words, f|Ais equal to fonA, but “forgets” the values of fat points outside A. g◦f: the composition offand g. Iff:X→Yand g:Y→Zare functions, then g◦f:X→Zis defined by (g◦f(x)/equalxg(f(x)). We often say that the function g◦fis obtained by “substituting y/equalxf(x)into g(y)”. A function f:X→Yisinjective orone-to-one ifx1/nequalx2implies f(x1)/nequalf(x2). (Equivalently, fis injective if f(x1)/equalxf(x2)implies x1/equalxx2.) It is called surjective orontoiff(X)/equalxY, i.e. if y∈Ythen y/equalxf(x)for some x∈X. It is called bijective orinvertible if it is both injective and surjective. The function fis bijective if and only if it has an inverse , that is a map g:Y→Xsatisfying both g(f(x))/equalxxfor allx∈Xand f(g(y))/equalxyfor all y∈Y. Both conditions must be satisfied; that’s why an inverse is sometimes called a two-sided inverse for emphasis. The inverse of a bijective map fis unique and is denoted by f−1. Let f:X→Ybe an injective map. We can consider fas a bijection from X onto the image f(X)and then form the inverse f−1:f(X)→X. We can extend f−1to a map g:Y→Xas follows: for y∈f(X)putg(y)/equalxf−1(y). For each y∈Y which is not in f(X)we choose at random an element x∈Xand define g(y)/equalxx. (For instance, we could choose an arbitrary x0∈Xand send all y’s not in f(X)to the same x0.) The map gsatisfies g(f(x))/equalxxfor all x∈X, but not f(g(y))/equalxy for all y∈Y(unless fis bijective). We call galeft inverse off. IfXis a finite set and f:X→Ra real-valued function, the sum of all the numbers f(x), where xranges through X, is denoted by/summationtext x∈Xf(x). The set Xis A./two.taboldstyle. GENERAL TOPOLOGY OF EUCLIDEAN SPACE /one.taboldstyle/three.taboldstyle/nine.taboldstyle called the index set for the sum. This notation is often abbreviated or abused in various ways. For instance, if Xis the collection{1,2,..., n}, one uses the familiar notation/summationtextn i/equalx1f(i). In these notes we will often deal with indices which are pair s ork-tuples of integers, also known as multi-indices . As a simple example, let nbe a fixed nonnegative integer, let Xbe the set of all pairs of integers (i,j)satisfying 0≤i≤j≤n, and let f(i,j)/equalxi+j. For n/equalx3we can display Xand fin a tableau as follows. ij 0123 234 45 6 The sum/summationtext x∈Xf(x)of all these numbers is written as /summationdisplay 0≤i≤j≤n(i+j). You will be asked to evaluate it explicitly in Exercise A./two.taboldstyle. A./two.taboldstyle. General topology of Euclidean space Letxbe a point in Euclidean space Rn. The open ball of radiusεabout a point xis the collection of all points ywhose distance to xis less thanε, B◦(ε,x)/equalx{y∈Rn|/bardbly−x/bardbl<ε}. xε A subset OofRnisopen if for every x∈Othere exists an ε >0such that B◦(ε,x) is contained in O. Intuitively this means that at every point in Othere is a little bit of room inside Oto move around in any direction you like. An open neighbourhood ofxis any open set containing x. A subset CofRnisclosed if its complement Rn\Cis open. This definition is equivalent to the following: Cis closed if and only if for every sequence of points x1,x2,...,xn,... that converges to a point xinRn, the limit xis contained in C. Loosely speaking, closed means “closed under taking limits ”. An example of a closed set is the closed ball of radiusεabout a point x, which is defined as the /one.taboldstyle/four.taboldstyle/zero.taboldstyle A. SETS AND FUNCTIONS collection of all points ywhose distance to xis less than or equal to ε, B(ε,x)/equalx{y∈Rn|/bardbly−x/bardbl≤ε}. xε Closed is not the opposite of open! There exist lots of subset s ofRnthat are neither open nor closed, for example the interval [0,1)inR. (On the other hand, there are not so many subsets that are both open and closed, na mely just the empty set and Rnitself.) A subset AofRnisbounded if there exists some R>0such that/bardblx/bardbl≤ Rfor allxinA. (That is, Ais contained in the ball B(R,0)for some value of R.) A compact subset of Rnis one that is both closed and bounded. The importance of the notion of compactness, as far as these notes are concerned, i s that the integral of a continuous function over a compact subset of Rnis always a well-defined, finite number. Exercises A./one.taboldstyle. Parts ( ii) and ( iii) of this problem require the use of an atlas or the Web. Let Xbe the surface of the earth, let Ybe the real line and let f:X→Ybe the function which assigns to each x∈Xits geographical latitude measured in degrees. (i) Determine the sets f(X),f−1(0),f−1(90), and f−1(−90). (ii) Let Abe the contiguous United States. Find f(A). Round the numbers to whole degrees. (iii) Let B/equalxf(A), where Ais as in part ( ii). Find (a) a country other than Athat is contained in f−1(B); (b) a country that intersects f−1(B)but is not contained inf−1(B); and (c) a country in the northern hemisphere that does not in tersect f−1(B). A./two.taboldstyle. LetS(n)/equalx/summationtext 0≤i≤j≤n(i+j). Prove the following assertions. (i)S(0)/equalx0and S(n+ 1)/equalxS(n)+3 2(n+ 1)(n+ 2). (ii)S(n)/equalx1 2n(n+ 1)(n+ 2). (Use induction on n.) A./three.taboldstyle. Prove that the open ball B◦(ε,x)is open. (This is not a tautology! State your reasons as precisely as you can, using the definition of openn ess stated in the text. You will need the triangle inequality /bardbly−x/bardbl≤/bardbl y−z/bardbl+/bardblz−x/bardbl.) A./four.taboldstyle. Prove that the closed ball is B(ε,x)is closed. (Same comments as for Exercise A./three.taboldstyle.) A./five.taboldstyle. Show that the two definitions of closedness given in the text a re equivalent. EXERCISES /one.taboldstyle/four.taboldstyle/one.taboldstyle A./six.taboldstyle. Complete the following table. Here Sn−1denotes the unit sphere about the origin inRn, that is the set of vectors of length 1. closed? bounded? compact? [−3,5] yes yes yes [−3,5) [−3,∞) (−3,∞) B(ε,x) B◦(ε,x) Sn−1 xy-plane in R3 unit cube [0,1]n APPENDIX B Calculus review This appendix is a brief review of some single- and multi-var iable calculus needed in the study of manifolds. References for this materi al are [ Edw/nine.taboldstyle/four.taboldstyle ], [HH/zero.taboldstyle/nine.taboldstyle ] and [ MT/one.taboldstyle/one.taboldstyle ]. B./one.taboldstyle. The fundamental theorem of calculus Suppose that Fis a differentiable function of a single variable xand that the derivative f/equalxF′is continuous. Let [a,b]be an interval contained in the domain ofF. The fundamental theorem of calculus says that /integraldisplayb af(t)dt/equalxF(b)−F(a). (B./one.taboldstyle) There are two useful alternative ways of writing this theore m. Replacing bwith x and differentiating with respect to xwe find d dx/integraldisplayx af(t)dt/equalxf(x). (B./two.taboldstyle) Writing ginstead of Fand g′instead of fand adding g(a)to both sides in formula (B./one.taboldstyle) we get g(x)/equalxg(a)+/integraldisplayx ag′(t)dt. (B./three.taboldstyle) Formulæ ( B./one.taboldstyle)–(B./three.taboldstyle) are equivalent, but they emphasize different aspects of the fundamental theorem of calculus. Formula ( B./one.taboldstyle) is a formula for a definite integral: it tells you how to find the (oriented) surface area between th e graph of the function fand the x-axis. Formula ( B./two.taboldstyle) says that the integral of a continuous function is a differentiable function of the upper limit; and the derivat ive is the integrand. Formula ( B./three.taboldstyle) is an “integral formula”, which expresses the function gin terms of the value g(a)and the derivative g′. (See Exercise B./one.taboldstylefor an application.) B./two.taboldstyle. Derivatives Letφ1,φ2,...,φmbe functions of nvariables x1,x2,...,xn. As usual we write x/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAx1 x2 ... xn/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA, φ (x)/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAφ1(x) φ2(x) ... φm(x)/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA, and viewφ(x)as a single map from RntoRm. (In calculus the word “map” is often used for vector-valued functions, while the word “function ” is generally reserved /one.taboldstyle/four.taboldstyle/three.taboldstyle /one.taboldstyle/four.taboldstyle/four.taboldstyle B. CALCULUS REVIEW for real-valued functions.) The most basic way to differenti ate a mapφfrom Rnto Rmis to form a limit of the kind dφ(x+tv) dt/barex/barex/barex/barext/equalx0/equalxlim t→0φ(x+tv)−φ(x) t. (B./four.taboldstyle) If this limit exists, it is called the directional derivative of φatxalong v. The expression x+tvas a function of tparametrizes a straight line in Rnpassing through the point xin the direction of the vector v. The directional derivative ( B./four.taboldstyle) measures the rate of change of φalong this straight line. Since φ(x+tv)−φ(x)is a vector in Rmfor allt, the directional derivative is likewise a vector in Rm. If the mapφis not defined everywhere but only on a subset UofRn, it may be difficult to make sense of the limit ( B./four.taboldstyle). Let xbe in the domain Uofφ. The problem is that x+tvmay not be in Ufor all t/nequal0, so that we cannot evaluate φ(x+tv). The solution is to assume that Uis an open subset of Rn. Given any direction vector v∈Rn, we can then find a number ε >0such that the points x+tvare contained in Ufor−ε < t< ε. Therefore φ(x+tv)is well-defined for −ε<t<εand we can legitimately ask whether the limit ( B./four.taboldstyle) exists. Thepartial derivatives ofφatxare by definition the directional derivatives ∂φ ∂xj(x)/equalxdφ(x+tej) dt/barex/barex/barex/barext/equalx0, (B./five.taboldstyle) where e1/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA1 0 0 ... 0 0/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA, e2/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA0 1 0 ... 0 0/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA, ..., en/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA0 0 0 ... 0 1/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA are the standard basis vectors of Rn, i.e. the columns of the identity n×n-matrix. We can write the partial derivative in components as follows : ∂φ ∂xj(x)/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA∂φ1 ∂xj(x) ∂φ2 ∂xj(x) ... ∂φm ∂xj(x)/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA. Thetotal derivative orJacobi matrix ofφatxis obtained by lining these columns up in an m×n-matrix Dφ(x)/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA∂φ1 ∂x1(x)∂φ1 ∂x2(x)...∂φ1 ∂xn(x) ∂φ2 ∂x1(x)∂φ2 ∂x2(x)...∂φ2 ∂xn(x) ......... ∂φm ∂x1(x)∂φm ∂x2(x)...∂φm ∂xn(x)/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA. We say that the map φiscontinuously differentiable orC1if the partial derivatives ∂φi ∂xj(x) B./two.taboldstyle. DERIVATIVES /one.taboldstyle/four.taboldstyle/five.taboldstyle are well-defined for all xin the domain Uofφand depend continuously on xfor alli/equalx1,2,...,nand j/equalx1,2,...,m. If the second partial derivatives ∂2φi ∂xj∂xk(x) exist for all x∈Uand are continuous for all i/equalx1,2,...,nand j,k/equalx1,2,...,m, thenφis called twice continuously differentiable orC2. Likewise, if all r-fold partial derivatives ∂rφi ∂xj1∂xj2···∂xjr(x) exist and are continuous, then φisrtimes continuously differentiable orCr. IfφisCr for all r≥1, then we say that φisinfinitely many times differentiable , orC∞, orsmooth . This means that φcan be differentiated arbitrarily many times with respect to any of the variables. Smooth functions include such familiar on e-variable functions as polynomials, exponentials, logarithms and trig functio ns. (Of course for log andtanone must make the proviso that they are smooth only on their do main of definition.) The following useful fact says that any directional derivat ive can be expressed as a linear combination of partial derivatives. The proof is Exercise B./four.taboldstyle. B./one.taboldstyle. L/e.sc/m.sc/m.sc/a.sc. LetUbe an open subset of Rnand letφ:U→Rmbe a C1map. Let x∈Uandv∈Rn. Then the directional derivative of φatxalong vexists and is equal to dφ(x+tv) dt/barex/barex/barex/barext/equalx0/equalxDφ(x)v, the vector in Rmobtained by multiplying the matrix Dφ(x)by the vector v. Velocity vectors. Suppose n/equalx1. Thenφis a vector-valued function of one variable x, called a pathorparametrized curve inRm. The matrix Dφ(x)consists of a single column vector, called the velocity vector , and is usually denoted simply by φ′(x). Gradients. Suppose m/equalx1. Thenφis a scalar-valued function of nvariables and Dφ(x)is a single row vector. The transpose matrix of Dφ(x)is therefore a column vector, called the gradient ofφ: Dφ(x)T/equalxgrad (φ)(x). The directional derivative of φalong vcan then be written as an inner product, Dφ(x)v/equalxgrad (φ)(x)Tv/equalxgrad (φ)(x)·v. There is an important characterization of the gradient, whi ch is based on the familiar identity a·b/equalx/bardbla/bardbl/bardblb/bardblcosθ. Here 0≤θ≤πis the angle subtended by aandb. Let us fix a point xin the domain of φand let us consider all possible directional derivatives of φatxalong unitvectors v(i.e. vectors of length 1). Then Dφ(x)v/equalxgrad (φ)(x)·v/equalx/bardblgrad (φ)(x)/bardblcosθ, whereθis the angle between grad (φ)(x)andv. So Dφ(x)vtakes on its maximal value if cosθ/equalx1, i.e.θ/equalx0. This means that vpoints in the same direction as grad (φ)(x). Thus the direction of the vector grad (φ)(x)is the direction of steepest ascent , i.e. in which φincreases fastest, and the magnitude ofgrad (φ)(x)is equal /one.taboldstyle/four.taboldstyle/six.taboldstyle B. CALCULUS REVIEW to the directional derivative Dφ(x)v, where vis the unit vector pointing along grad (φ)(x). B./three.taboldstyle. The chain rule Recall that if A,Band Care sets and φ:A→Bandψ:B→Care maps, we can applyψafterφto obtain the composite map (ψ◦φ)(x)/equalxψ(φ(x)). B./two.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc (/c.sc/h.sc/a.sc/i.sc/n.sc /r.sc/u.sc/l.sc/e.sc). LetU⊆RnandV⊆Rmbe open and let φ:U→V andψ:V→RlbeCr. Thenψ◦φisCrand D(ψ◦φ)(x)/equalxDψ(φ(x))Dφ(x) for all x∈U. Here Dψ(φ(x))Dφ(x)denotes the composition or the product of the l×m- matrix Dψ(φ(x))and the m×n-matrix Dφ(x). B./three.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. In the one-variable case n/equalxm/equalxl/equalx1the derivatives Dφand Dψare1×1-matrices (φ′(x))and (ψ′(y)), and matrix multiplication is ordinary multiplication, so we get the usual chain rule (ψ◦φ)′(x)/equalxψ′(φ(x))φ′(x). B./four.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. LetUbe an open subset of Rnand letφ:U→RmbeC1. Let I be an open interval and let c:I→Ube a path in U. Thenφ◦cis a path in Rm. Suppose that at time t0∈Ithe path cpasses through the point c(t0)/equalxxat velocity c′(t0)/equalxv. How to compute the velocity vector of the composite path φ◦cat time t0? The chain rule gives (φ◦c)′(t)/equalxD(φ◦c)(t)/equalxDφ(c(t))Dc(t)/equalxDφ(c(t))c′(t) for all t∈I. Setting t/equalxt0gives (φ◦c)′(t0)/equalxDφ(x)v. Let us write out a formula for the (i,j)-th entry of the Jacobi matrix of ψ◦φ. The i-th component of the map ψ◦φisψi◦φ, so the (i,j)-th entry of D(ψ◦φ)(x) is the partial derivative ∂(ψi◦φ) ∂xj(x). According to Theorem B./two.taboldstylethis entry can be computed by multiplying the i-th row ofDψ(φ(x)), which is /parenleftBig∂ψi ∂y1(φ(x))∂ψi ∂y2(φ(x))···∂ψ ∂ym(φ(x))/parenrightBig , by the j-th column of Dφ(x), which is /parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA∂φ1 ∂xj(x) ∂φ2 ∂xj(x) ... ∂φm ∂xj(x)/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA. This gives the formula ∂(ψi◦φ) ∂xj(x)/equalxm/summationdisplay k/equalx1∂ψi ∂yk(φ(x))∂φk ∂xj(x). B./four.taboldstyle. THE IMPLICIT FUNCTION THEOREM /one.taboldstyle/four.taboldstyle/seven.taboldstyle This is perhaps the form in which the chain rule is most often u sed. Sometimes we are sloppy and abbreviate this identity to ∂(ψi◦φ) ∂xj/equalxm/summationdisplay k/equalx1∂ψi ∂yk∂φk ∂xj. Even sloppier, but nevertheless quite common, notations ar e ∂ψi ∂xj/equalxm/summationdisplay k/equalx1∂ψi ∂yk∂φk ∂xj, or even ∂ψi ∂xj/equalxm/summationdisplay k/equalx1∂ψi ∂yk∂yk ∂xj. In these notes we frequently prefer the so-called “pullback ” notation. Instead ofψ◦φwe often write φ∗(ψ), so thatφ∗(ψ)(x)stands forψ(φ(x)). Similarly, φ∗(∂ψi/∂yk)(x)stands for∂ψi/∂yk(φ(x)). In this notation we have ∂φ∗(ψi) ∂xj/equalxm/summationdisplay k/equalx1φ∗/parenleftbigg∂ψi ∂yk/parenrightbigg∂φk ∂xj. (B./six.taboldstyle) B./four.taboldstyle. The implicit function theorem Letφ:W→Rmbe a continuously differentiable function defined on an open subset WofRn+m. Let us think of a vector in Rn+mas an ordered pair of vectors (u,v)with u∈Rnandv∈Rm. Consider the equation φ(u,v)/equalx0. Under what circumstances is it possible to solve for vas a function of u? The answer is given by the implicit function theorem. B./five.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc. To motivate the general result let us consider the case m/equalxn/equalx1. Thenφis a function of two real variables (u,v), and the equation φ(u,v)/equalx0 represents a curve in the plane, such as the lemniscate (u2+v2)2−2(u2−v2)/equalx0. (u0,v0) Suppose we manage to find a special solution (u0,v0)of the equation. The gradient grad (φ)/equalx(∂φ/∂ u,∂φ/∂ v)is perpendicular to the curve at every point, so if ∂φ/∂ v/nequal0at(u0,v0), then the curve has a nonvertical tangent line at (u0,v0). Then for uclose to u0and vclose to v0(i.e. for (u,v)in a box centred at (u0,v0), such as the little grey box in the picture above) the curve loo ks like the graph of a function, so we can solve the equation φ(u,v)/equalx0forvas a function v/equalxf(u)of u. How to find the derivative of f? By differentiating the relation φ(u,f(u))/equalx0. We rewrite this as φ(ψ(u))/equalx0, whereψis defined by ψ(u)/equalx(u,f(u))foru /one.taboldstyle/four.taboldstyle/eight.taboldstyle B. CALCULUS REVIEW in an open interval Icontaining u0. According to the chain rule, Theorem B./two.taboldstyle, D(φ◦ψ)(u)/equalxDφ(ψ(u)Dψ(u). We have Dφ/parenleftBigg u v/parenrightBigg /equalx/parenleftBig∂φ ∂u(u,v)∂φ ∂v(u,v)/parenrightBig , Dψ(u)/equalx/parenleftBigg 1 f′(u)/parenrightBigg , and therefore D(φ◦ψ)(u)/equalx∂φ ∂u(u,f(u))+∂φ ∂v(u,f(u))f′(u). The relation φ(u,f(u))/equalx0(which holds for all u∈I) tells us that D(φ◦ψ)(u)/equalx0 foru∈I. Solving for f′gives the implicit differentiation formula f′(u)/equalx−∂φ ∂u(u,f(u))/slashbigg∂φ ∂v(u,f(u)). For general mand nwe form the Jacobi matrices of φwith respect to the u- andv-variables separately, Duφ/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA∂φ1 ∂u1...∂φ1 ∂un...... ∂φm ∂u1...∂φm ∂un/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA, Dvφ/equalx/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA∂φ1 ∂v1...∂φ1 ∂vm...... ∂φm ∂v1...∂φm ∂vm/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA. Observe that the matrix Dvφis square. We are in business if we have a point (u0,v0)at whichφis0and Dvφis invertible. B./six.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc (/i.sc/m.sc/p.sc/l.sc/i.sc/c.sc/i.sc/t.sc /f.sc/u.sc/n.sc/c.sc/t.sc/i.sc/o.sc/n.sc /t.sc/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc). Letφ:W→RmbeCr, where W is open in Rn+m. Suppose that (u0,v0)∈Wis a point such that φ(u0,v0)/equalx0and Dvφ(u0,v0)is invertible. Then there are open neighbourhoods U⊆Rnofu0and V⊆Rmofv0such that for each u∈Uthere exists a unique v/equalxf(u)∈Vsatisfying φ(u,f(u))/equalx0. The function f:U→VisCrwith derivative given by implicit differentiation: D f(u)/equalx−Dvφ(u,v)−1Duφ(u,v)/barex/barexv/equalxf(u) for all u∈U. As a special case we take φto be of the form φ(u,v)/equalxg(v)−u, where g:W→Rnis a given function with Wopen in Rn. Solvingφ(u,v)/equalx0here amounts to inverting the function g. Moreover, Dvφ/equalxD g, so the implicit function theorem yields the following result. B./seven.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc (/i.sc/n.sc/v.sc/e.sc/r.sc/s.sc/e.sc /f.sc/u.sc/n.sc/c.sc/t.sc/i.sc/o.sc/n.sc /t.sc/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc). Letg:W→Rnbe continuously differentiable, where Wis open in Rn. Suppose that v0∈Wis a point such that D g(v0) is invertible. Then there is an open neighbourhood U⊆Rnofv0such that g(U)is an open neighbourhood of g(v0)and the map g:U→g(U)is invertible. The inverse g−1:V→Uis continuously differentiable with derivative given by D g−1(u)/equalxD g(v)−1/barex/barexv/equalxg−1(u) for all v∈V. Again let us spell out the one-variable case n/equalx1. Invertibility of D g(v0) simply means that g′(v0)/nequal0. This implies that near v0the function gis strictly monotone increasing (if g′(v0)>0) or decreasing (if g′(v0)<0). Therefore if Iis EXERCISES /one.taboldstyle/four.taboldstyle/nine.taboldstyle a sufficiently small open interval around u0, then g(I)is an open interval around g(u0)and the restricted function g:I→g(I)is invertible. The inverse function has derivative (g−1)′(u)/equalx1 g′(v), with v/equalxg−1(u). B./eight.taboldstyle. E/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc (/s.sc/q.sc/u.sc/a.sc/r.sc/e.sc /r.sc/o.sc/o.sc/t.sc/s.sc). Letg(v)/equalxv2. Then g′(v0)/nequal0whenever v0/nequal0. Forv0>0we can take I/equalx(0,∞). Then g(I)/equalx(0,∞),g−1(u)/equalx√u, and (g−1)′(u)/equalx 1/(2√u). For v0<0we can take I/equalx(−∞,0). Then g(I)/equalx(0,∞),g−1(u)/equalx−√u, and (g−1)′(u)/equalx−1/(2√u). In a neighbourhood of 0it is not possible to invert g. B./five.taboldstyle. The substitution formula for integrals LetVbe an open subset of Rnand let f:V→Rbe a function. Suppose we want to change the variables in the integral/integraltext Vf(y)dy. (This is shorthand for an n-fold integral over y1,y2,...,yn.) This means we substitute y/equalxp(x), where p:U→Vis a map from an open U⊆RntoV. Under a suitable hypothesis we can change the integral over yto an integral over x. B./nine.taboldstyle. T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc (/c.sc/h.sc/a.sc/n.sc/g.sc/e.sc /o.sc/f.sc /v.sc/a.sc/r.sc/i.sc/a.sc/b.sc/l.sc/e.sc/s.sc /f.sc/o.sc/r.sc/m.sc/u.sc/l.sc/a.sc). LetUandVbe open subsets of Rn and let p:U→Vbe a map. Suppose that pis bijective and that pand its inverse are continuously differentiable. Then for any integrable funct ionfwe have /integraldisplay Vf(y)dy/equalx/integraldisplay Uf(p(x))|det(Dp(x))|dx. Again this should look familiar from one-variable calculus : ifp:(a,b)→(c,d) isC1and has a C1inverse, then /integraldisplayd cf(y)dy/equalx/integraltextb af(p(x))p′(x)dx ifpis increasing ,/integraltexta bf(p(x))p′(x)dx ifpis decreasing . This can be written succinctly as/integraltextd cf(y)dy/equalx/integraltextb af(p(x))|p′(x)|dx, which looks more similar to the multidimensional case. Exercises B./one.taboldstyle. Letg: [a,b]→Rbe a Cn+1-function, where n≥0. Suppose a≤x≤band put h/equalxx−a. (i) By changing variables in the fundamental theorem of calc ulus ( B./three.taboldstyle) show that g(x)/equalxg(a)+h/integraldisplay1 0g′(a+th)dt. (ii) Show that g(x)/equalxg(a)+h(t−1)g′(a+th)/barex/barex1 0+h2/integraldisplay1 0(1−t)g′′(a+th)dt /equalxg(a)+hg′(a)+h2/integraldisplay1 0(1−t)g′′(a+th)dt. (Integrate the formula in part ( i) by parts and don’t forget to use the chain rule.) /one.taboldstyle/five.taboldstyle/zero.taboldstyle B. CALCULUS REVIEW (iii) By induction on ndeduce from part ( ii) that g(x)/equalxn/summationdisplay k/equalx0g(k)(a) k!hk+hn+1 n!/integraldisplay1 0(1−t)ng(n+1)(a+th)dt. This is Taylor’s formula with integral remainder term . B./two.taboldstyle. Letφ:R→Rbe the function defined by φ(x)/equalxx|x|. Show that φisC1but not C2. For each r≥1give an example of a function which is Crbut not Cr+1. B./three.taboldstyle. Define a function f:R→Rbyf(0)/equalx0and f(x)/equalxe−1/x2forx/nequal0. (i) Show that fis differentiable at 0and that f′(0)/equalx0. (Use the definition of the derivative, f′(0)/equalxlim x→0f(x)−f(0) x.) (ii) Show that fis smooth and that f(n)(0)/equalx0for all n. (By induction on n, prove the following assertion: f(n)(x)/equalx0 ifx/equalx0, gn(1/x)e−1/x2ifx/nequal0, where, for each n≥0,gn(1/x)is a certain polynomial in 1/x, which you need not determine explicitly.) (iii) Plot the function fover the interval−5≤x≤5. Using software or a graphing calculator is fine, but pay special attention to the behaviou r near x/equalx0. B./four.taboldstyle. (i) Let xandvbe vectors in Rn. Define a path c:R→Rnbyc(t)/equalxx+tv. Find c′(t). (ii) Prove Lemma B./one.taboldstyle. (Letφ:U→Rmbe a C1map, where Uis open in Rn. Combine the result of part ( i) with the formula of Example B./four.taboldstyle.) B./five.taboldstyle. According to Newton’s law of gravitation, a particle of mass m1placed at the origin inR3exerts a force on a particle of mass m2placed at x∈R3\{0}equal to F/equalx−Gm1m2 /bardblx/bardbl3x, where Gis a constant of nature. Show that Fis the gradient of f(x)/equalxGm1m2//bardblx/bardbl. B./six.taboldstyle. A function f:Rn\{0}→ Rishomogeneous of degree piff(tx)/equalxtpf(x)for all x∈Rn\{0}and t>0. Here pis a real constant. (i) Show that the functions f1(x,y)/equalx(x2−xy)/(x2+y2),f2(x,y)/equalx/radicalBig x3+y3, f3(x,y,z)/equalx(x2z6+ 3x4y2z2)−√ 2are homogeneous. What are their degrees? (ii) Let fbe a homogeneous function of degree p. Assume that fis defined at 0and continuous everywhere. Also assume f(x)/nequal0for at least one x∈Rn. Show that p≥0. Show that fis constant if p/equalx0. (iii) Show that if fis homogeneous of degree pand smooth, then n/summationdisplay i/equalx1xi∂f ∂xi(x)/equalxp f(x). (Differentiate the relation f(tx)/equalxtpf(x)with respect to t.) B./seven.taboldstyle. A function f:Rn\{0}→ Risquasihomogeneous of degree pif there exist real constants a1,a2,...,anwith the property that f(ta1x1,ta2x2,..., tanxn)/equalxtpf(x) EXERCISES /one.taboldstyle/five.taboldstyle/one.taboldstyle for all x∈Rn\{0}and t>0. Suppose that fis quasihomogeneous of degree pand smooth. Show that/summationtextn i/equalx1aixi∂f ∂xi(x)/equalxp f(x). B./eight.taboldstyle. Define a map ψfrom Rn−1toRnby ψ(t)/equalx1 /bardblt/bardbl2+ 1/parenleftbig2t+(/bardblt/bardbl2−1)en/parenrightbig. (Here we regard a point t/equalx(t1,t2,..., tn−1)inRn−1as a point in Rnby identifying it with (t1,t2,..., tn−1,0).) (i) Show that ψ(t)lies on the unit sphere Sn−1about the origin. (ii) Show that ψ(t)is the intersection point of the sphere and the line through t he points enandt. (iii) Compute Dψ(t). (iv) Let Xbe the sphere punctured at the “north pole”, X/equalxSn−1\{en}.Stereographic projection from the north pole is the map φ:X→Rn−1given byφ(x)/equalx(1− xn)−1(x1,x2,..., xn−1)T. Show that φis a two-sided inverse of ψ. (v) Draw diagrams illustrating the maps φandψforn/equalx2and n/equalx3. (vi) Now let ybe any point on the sphere and let Pthe hyperplane perpendicular to y. (Ahyperplane inRnis a linear subspace of dimension n−1.) The stereographic projection from yof any point xin the sphere distinct from yis defined as the unique intersection point of the line joining ytoxand the hyperplane P. This defines a map φ:Sn−1\{y}→ P. The point yis called the centre of the projection. Write a formula for the stereographic projection φfrom the south pole −enand for its inverse ψ:Rn−1→Sn−1. B./nine.taboldstyle. Letφ:Rn→Rmbe a C1map. Prove the following assertions. (i)φis constant if and only if Dφ(x)/equalx0for all x∈Rn. (ii)φis linear if and only if Dφ(x)v/equalxφ(v)for all xandvinRn. B./one.taboldstyle/zero.taboldstyle. A mapφ:Rn→Rmis called evenifφ(−x)/equalxφ(x)for all xinRn. Find Dφ(0)if φis even and C1. B./one.taboldstyle/one.taboldstyle. Letf:R2→Rbe a smooth function which satisfies f(x,y)/equalx−f(y,x)for all x and y. Show that ∂f ∂x(a,b)/equalx−∂f ∂y(b,a) for all aand b. B./one.taboldstyle/two.taboldstyle. Letf:R2→Rand g:R→Rbe smooth functions. Show that d dx/integraldisplayg(x) 0f(x,y)dy/equalxf(x,g(x))g′(x)+/integraldisplayg(x) 0∂f(x,y) ∂xdy. B./one.taboldstyle/three.taboldstyle. Thermodynamicists like to use rules such as ∂y ∂x∂x ∂y/equalx1. Explain the rule and show that it is correct. (Assume that the variables are subject to a relation F(x,y)/equalx0defining functions x/equalxf(y),y/equalxg(x), and apply the multivariable chain rule. See also Example B./five.taboldstyle.) Similarly, explain why ∂y ∂x∂z ∂y∂x ∂z/equalx−1. Naively cancelling numerators against denominators gives the wrong answer! /one.taboldstyle/five.taboldstyle/two.taboldstyle B. CALCULUS REVIEW B./one.taboldstyle/four.taboldstyle. Leta0,a1,a2,...,anbe vectors in Rn. A linear combination/summationtextn i/equalx0ciaiisconvex if all coefficients are nonnegative and their sum is 1:ci≥0and/summationtextn i/equalx0ci/equalx1. The simplex∆ spanned by the ai’s is the collection of all their convex linear combinations , ∆/equalx/braceleftbiggn/summationdisplay i/equalx0ciai/barex/barex/barex/barexc1≥0,..., cn≥0,n/summationdisplay i/equalx0ci/equalx1/bracerightbigg . Thestandard simplex inRnis the simplex spanned by the vectors 0,e1,e2,...,en. (i) For n/equalx1,2,3draw pictures of the standard n-simplex as well as a nonstandard n-simplex. (ii) The volume of a region RinRnis defined as vol(R)/equalx/integraldisplay Rdx1dx2···dxn. Show that the volume of the standard n-simplex is 1/n!. (iii) Show that vol(∆)/equalx1 n!|det(A)|, where Ais the n×n-matrix with columns a1−a0,a2−a0,...,an−a0. (Map∆to the standard simplex by an appropriate substitution and app ly the substitution formula for integrals.) The following two calculus problems are not review problems , but the results are needed in Chapter /nine.taboldstyle. B./one.taboldstyle/five.taboldstyle. Forx>0define Γ(x)/equalx/integraldisplay∞ 0e−ttx−1dt and prove the following assertions. (i)Γ(x+ 1)/equalxxΓ(x)for all x>0. (ii)Γ(n)/equalx(n−1)!for positive integers n. (iii)/integraldisplay∞ 0e−u2uadu/equalx1 2Γ/parenleftbigga+ 1 2/parenrightbigg . B./one.taboldstyle/six.taboldstyle. CalculateΓ(n+1 2)(whereΓis the function defined in Exercise B./one.taboldstyle/five.taboldstyle) by establishing the following identities. For brevity write γ/equalxΓ(1 2). (i)γ/equalx/integraldisplay∞ −∞e−s2ds. (ii)γ2/equalx/integraldisplay∞ −∞/integraldisplay∞ −∞e−x2−y2dx dy . (iii)γ2/equalx/integraldisplay2π 0/integraldisplay∞ 0re−r2dr dθ. (iv)γ/equalx√π. (v)Γ/parenleftbigg n+1 2/parenrightbigg /equalx1·3·5···(2n−1) 2n√ πforn≥1. APPENDIX C The Greek alphabet upper case lower case name A α alpha B β beta Γ γ gamma ∆ δ delta Eǫ,ε epsilon Z ζ zeta H η eta Θθ,ϑ theta I ι iota K κ kappa Λ λ lambda M µ mu N ν nu Ξ ξ xi O o omicron Ππ,̟ pi P ρ rho Σσ,ς sigma T τ tau Υ υ upsilon Φφ,ϕ phi X χ chi Ψ ψ psi Ω ω omega /one.taboldstyle/five.taboldstyle/three.taboldstyle Bibliography [Bac/zero.taboldstyle/six.taboldstyle] D. Bachman, A geometric approach to differential forms , second ed., Birkhäuser, Boston, MA, /two.taboldstyle/zero.taboldstyle/one.taboldstyle/two.taboldstyle. [BS/nine.taboldstyle/one.taboldstyle] P. Bamberg and S. Sternberg, A course in mathematics for students of physics , Cambridge Univer- sity Press, Cambridge, /one.taboldstyle/nine.taboldstyle/nine.taboldstyle/one.taboldstyle. [BT/eight.taboldstyle/two.taboldstyle] R. Bott and L. Tu, Differential forms in algebraic topology , Springer-Verlag, New York, /one.taboldstyle/nine.taboldstyle/eight.taboldstyle/two.taboldstyle. Perhaps the best exposition at the graduate level of the appl ications of differential forms to topology. [Bre/nine.taboldstyle/one.taboldstyle] D. Bressoud, Second year calculus , Undergraduate Texts in Mathematics, Springer-Verlag, Ne w York, /one.taboldstyle/nine.taboldstyle/nine.taboldstyle/one.taboldstyle. Multivariable calculus from the point of view of Newton’s la w and special relativity with coverage of differential forms. [Car/nine.taboldstyle/four.taboldstyle] M. do Carmo, Differential forms and applications , Springer-Verlag, Berlin, /one.taboldstyle/nine.taboldstyle/nine.taboldstyle/four.taboldstyle, translated from the /one.taboldstyle/nine.taboldstyle/seven.taboldstyle/one.taboldstyle Portuguese original. Slightly more advanced than these notes, with some coverage of Riemannian geometry, in- cluding the Gauss-Bonnet theorem. [Dar/nine.taboldstyle/four.taboldstyle] R. Darling, Differential forms and connections , Cambridge University Press, Cambridge, /one.taboldstyle/nine.taboldstyle/nine.taboldstyle/four.taboldstyle. Advanced undergraduate text on manifolds and differential f orms, including expositions of Maxwell theory and its modern generalization, gauge theory . [Edw/nine.taboldstyle/four.taboldstyle] H. Edwards, Advanced calculus: A differential forms approach , Birkhäuser Boston, Boston, Mas- sachusetts, /one.taboldstyle/nine.taboldstyle/nine.taboldstyle/four.taboldstyle, corrected reprint of the /one.taboldstyle/nine.taboldstyle/six.taboldstyle/nine.taboldstyle original. One of the earliest undergraduate textbooks covering differ ential forms. Still recommended as an alternative or supplementary source. [Fla/eight.taboldstyle/nine.taboldstyle] H. Flanders, Differential forms with applications to the physical scienc es, second ed., Dover Publi- cations, New York, /one.taboldstyle/nine.taboldstyle/eight.taboldstyle/nine.taboldstyle. Written for /one.taboldstyle/nine.taboldstyle/six.taboldstyle/zero.taboldstyles engineering graduate students. Concise b ut lucid (and cheap!). [Gra/nine.taboldstyle/eight.taboldstyle] A. Gray, Modern differential geometry of curves and surfaces with Mat hematica , second ed., CRC Press, Boca Raton, FL, /one.taboldstyle/nine.taboldstyle/nine.taboldstyle/eight.taboldstyle. Leisurely and thorough exposition at intermediate undergr aduate level with plenty of com- puter graphics. [GP/seven.taboldstyle/four.taboldstyle] V. Guillemin and A. Pollack, Differential topology , Prentice-Hall, Englewood Cliffs, NJ, /one.taboldstyle/nine.taboldstyle/seven.taboldstyle/four.taboldstyle. Written for beginning graduate students, but very intuitiv e and with lots of interesting appli- cations to topology. [HH/zero.taboldstyle/nine.taboldstyle] J. Hubbard and B. Hubbard, Vector calculus, linear algebra, and differential forms: A u nified approach , fourth ed., Matrix Editions, Ithaca, NY, /two.taboldstyle/zero.taboldstyle/zero.taboldstyle/nine.taboldstyle. [Lay/one.taboldstyle/two.taboldstyle] D. Lay, Linear algebra and its applications , fourth ed., Pearson, /two.taboldstyle/zero.taboldstyle/one.taboldstyle/two.taboldstyle. Good reference for the basic linear algebra required in thes e notes. [MT/one.taboldstyle/one.taboldstyle] J. Marsden and A. Tromba, Vector calculus , sixth ed., W. H. Freeman, New York, /two.taboldstyle/zero.taboldstyle/one.taboldstyle/one.taboldstyle. Standard multivariable calculus reference. [Mil/zero.taboldstyle/three.taboldstyle] J. Milnor, Towards the Poincaré conjecture and the classification of 3-manifolds , Notices of the American Mathematical Society /five.taboldstyle/zero.taboldstyle(/two.taboldstyle/zero.taboldstyle/zero.taboldstyle/three.taboldstyle), no. /one.taboldstyle/zero.taboldstyle, /one.taboldstyle/two.taboldstyle/two.taboldstyle/six.taboldstyle–/one.taboldstyle/two.taboldstyle/three.taboldstyle/three.taboldstyle. A discussion of one of the great problems of differential topo logy. Available online at http://www.ams.org/notices/200310/fea-mi/l.Varnor.pdf . [ONe/zero.taboldstyle/six.taboldstyle] B. O’Neill, Elementary differential geometry , revised second ed., Elsevier/Academic Press, Am- sterdam, /two.taboldstyle/zero.taboldstyle/zero.taboldstyle/six.taboldstyle. [Opr/zero.taboldstyle/three.taboldstyle] J. Oprea, Differential geometry and its applications , second ed., Prentice Hall, /two.taboldstyle/zero.taboldstyle/zero.taboldstyle/three.taboldstyle. Curves, surfaces, geodesics and calculus of variations wit h plenty of MAPLE programming. Accessible to intermediate undergraduates. /one.taboldstyle/five.taboldstyle/five.taboldstyle /one.taboldstyle/five.taboldstyle/six.taboldstyle BIBLIOGRAPHY [Sam/zero.taboldstyle/one.taboldstyle] H. Samelson, Differential forms, the early days; or the stories of Deahna’ s theorem and of Volterra’s theorem , American Mathematical Monthly /one.taboldstyle/zero.taboldstyle/eight.taboldstyle(/two.taboldstyle/zero.taboldstyle/zero.taboldstyle/one.taboldstyle), no. /six.taboldstyle, /five.taboldstyle/two.taboldstyle/two.taboldstyle–/five.taboldstyle/three.taboldstyle/zero.taboldstyle. A brief history of the subject, available online at http://www.jstor.org/stab/l.Vare/2695706 . [Sin/zero.taboldstyle/one.taboldstyle] S. Singer, Symmetry in mechanics. A gentle, modern introduction , Birkhäuser, Boston, MA, /two.taboldstyle/zero.taboldstyle/zero.taboldstyle/one.taboldstyle. [Spi/seven.taboldstyle/one.taboldstyle] M. Spivak, Calculus on manifolds. A modern approach to classical theor ems of advanced calculus , Westview Press, /one.taboldstyle/nine.taboldstyle/seven.taboldstyle/one.taboldstyle. Efficient and rigorous treatment of many of the topics in these notes. [Spi/nine.taboldstyle/nine.taboldstyle] ,A comprehensive introduction to differential geometry , third ed., Publish or Perish, Hous- ton, TX, /one.taboldstyle/nine.taboldstyle/nine.taboldstyle/nine.taboldstyle. Differential geometry textbook at advanced undergraduate l evel in five massive but fun to read volumes. [Tu/one.taboldstyle/one.taboldstyle] L. Tu, An introduction to manifolds , second ed., Springer-Verlag, New York, /two.taboldstyle/zero.taboldstyle/one.taboldstyle/one.taboldstyle. Notation Index ∗, Hodge star operator, /two.taboldstyle/four.taboldstyle relativistic, /three.taboldstyle/zero.taboldstyle [0,1]k, unit cube in Rk, /six.taboldstyle/four.taboldstyle #, connected sum, /seven.taboldstyle ·, Euclidean inner product (dot product), /eight.taboldstyle ◦, composition of maps, /three.taboldstyle/nine.taboldstyle, /one.taboldstyle/three.taboldstyle/eight.taboldstyle, /one.taboldstyle/four.taboldstyle/six.taboldstyle/integraltext , integral of a form over a chain, /four.taboldstyle/nine.taboldstyle, /six.taboldstyle/three.taboldstyle over a manifold, /one.taboldstyle/one.taboldstyle/seven.taboldstyle /bardbl /bardbl, Euclidean norm (length), /eight.taboldstyle ⊗, tensor multiplication, /nine.taboldstyle/eight.taboldstyle ∂ ∂xi, partial derivative, /one.taboldstyle/four.taboldstyle/four.taboldstyle ⊥, orthogonal complement, /eight.taboldstyle/two.taboldstyle ∧, exterior multiplication, /one.taboldstyle/seven.taboldstyle, /nine.taboldstyle/four.taboldstyle AT, transpose of a matrix A, /three.taboldstyle/seven.taboldstyle Ak(V), set of alternating k-multilinear func- tions on V, /nine.taboldstyle/four.taboldstyle Aσ, permutation matrix, /four.taboldstyle/five.taboldstyle AI,J,(I,J)-submatrix of A, /four.taboldstyle/four.taboldstyle ∗α, Hodge star of α, /two.taboldstyle/four.taboldstyle relativistic, /three.taboldstyle/zero.taboldstyle/integraltext Mα, integral of αover a manifold M, /one.taboldstyle/one.taboldstyle/seven.taboldstyle/integraltext cα, integral of αover a chain c, /four.taboldstyle/nine.taboldstyle, /six.taboldstyle/three.taboldstyle Alt(µ), alternating form associated to µ, /nine.taboldstyle/eight.taboldstyle B(ε,x), closed ball in Rn, /one.taboldstyle/four.taboldstyle/zero.taboldstyle B◦(ε,x), open ball in Rn, /one.taboldstyle/three.taboldstyle/nine.taboldstyle [B], orientation defined by oriented frame B, /one.taboldstyle/zero.taboldstyle/three.taboldstyle C∞, smooth, /one.taboldstyle/four.taboldstyle/five.taboldstyle Cr,rtimes continuously differentiable, /one.taboldstyle/four.taboldstyle/five.taboldstyle curl, curl of a vector field, /two.taboldstyle/seven.taboldstyle Dφ, Jacobi matrix of φ, /one.taboldstyle/four.taboldstyle/four.taboldstyle ∂, boundary of a chain, /six.taboldstyle/six.taboldstyle of a manifold, /one.taboldstyle/one.taboldstyle/one.taboldstyle d, exterior derivative, /two.taboldstyle/zero.taboldstyle ∆, Laplacian of a function, /two.taboldstyle/nine.taboldstyle δI,J, Kronecker delta, /nine.taboldstyle/four.taboldstyle δi,j, Kronecker delta, /five.taboldstyle/three.taboldstyle det(A), determinant of a matrix A, /three.taboldstyle/one.taboldstylediv, divergence of a vector field, /two.taboldstyle/seven.taboldstyle ∗dx, infinitesimal hypersurface, /two.taboldstyle/six.taboldstyle dx, infinitesimal displacement, /two.taboldstyle/six.taboldstyle dxI, short for dxi1dxi2···dxik, /one.taboldstyle/seven.taboldstyle dxi, covector (“infinitesimal increment”), /one.taboldstyle/seven.taboldstyle, /nine.taboldstyle/three.taboldstyle /hatwiderdxi, omit dxi, /one.taboldstyle/eight.taboldstyle ei,i-th standard basis vector of Rn, /one.taboldstyle/four.taboldstyle/four.taboldstyle f(A), image of Aunder f, /one.taboldstyle/three.taboldstyle/eight.taboldstyle f−1, inverse of f, /one.taboldstyle/three.taboldstyle/eight.taboldstyle f−1(B), preimage of Bunder f, /one.taboldstyle/three.taboldstyle/eight.taboldstyle f−1(c), preimage of{c}under f, /one.taboldstyle/three.taboldstyle/eight.taboldstyle f|A, restriction of ftoA, /one.taboldstyle/three.taboldstyle/eight.taboldstyle g◦f, composition of fand g, /three.taboldstyle/nine.taboldstyle, /one.taboldstyle/three.taboldstyle/eight.taboldstyle, /one.taboldstyle/four.taboldstyle/six.taboldstyle Γ, Gamma function, /one.taboldstyle/two.taboldstyle/one.taboldstyle, /one.taboldstyle/five.taboldstyle/two.taboldstyle GL(n), general linear group, /eight.taboldstyle/seven.taboldstyle grad, gradient of a function, /two.taboldstyle/six.taboldstyle graph , graph of a function, /seven.taboldstyle/four.taboldstyle Hn, upper half-space in Rn, /one.taboldstyle/one.taboldstyle/one.taboldstyle I, multi-index (i1,i2,..., ik)(usually increas- ing), /one.taboldstyle/seven.taboldstyle int(M), interior of a manifold with boundary, /one.taboldstyle/one.taboldstyle/one.taboldstyle ker(A), kernel (nullspace) of a matrix A, /eight.taboldstyle/one.taboldstyle l(σ), length of a permutation σ, /three.taboldstyle/four.taboldstyle µM, volume form of a manifold M, /one.taboldstyle/zero.taboldstyle/five.taboldstyle /parenleftbign k/parenrightbig, binomial coefficient, /one.taboldstyle/nine.taboldstyle, /two.taboldstyle/four.taboldstyle n, unit normal vector field, /one.taboldstyle/zero.taboldstyle/four.taboldstyle nullity (A), dimension of the kernel of A, /eight.taboldstyle/one.taboldstyle O(n), orthogonal group, /eight.taboldstyle/four.taboldstyle Ωk(M), vector space of k-forms on M, /one.taboldstyle/nine.taboldstyle, /nine.taboldstyle/zero.taboldstyle φ∗, pullback of a form, /three.taboldstyle/nine.taboldstyle, /nine.taboldstyle/six.taboldstyle of a function, /three.taboldstyle/nine.taboldstyle, /one.taboldstyle/four.taboldstyle/seven.taboldstyle Rn, Euclidean n-space, /one.taboldstyle /one.taboldstyle/five.taboldstyle/seven.taboldstyle /one.taboldstyle/five.taboldstyle/eight.taboldstyle NOTATION INDEX rank (A), dimension of the column space of A, /eight.taboldstyle/one.taboldstyle Sn, unit sphere about the origin in Rn+1, /eight.taboldstyle, /seven.taboldstyle/two.taboldstyle Sn, permutation group, /three.taboldstyle/four.taboldstyle sign(σ), sign of a permutation σ, /three.taboldstyle/five.taboldstyle SL(n), special linear group, /eight.taboldstyle/seven.taboldstyle SO(n), special orthogonal group, /eight.taboldstyle/five.taboldstyle TxM, tangent space to Matx, /eight.taboldstyle, /seven.taboldstyle/four.taboldstyle, /eight.taboldstyle/one.taboldstyle, /eight.taboldstyle/two.taboldstyle τ(c), turning number of regular closed path c, /six.taboldstyle/zero.taboldstyle V∗, dual of a vector space V, /nine.taboldstyle/one.taboldstyle Vk,k-fold Cartesian product of a vector space V, /nine.taboldstyle/three.taboldstyle voln,n-dimensional Euclidean volume, /nine.taboldstyle/nine.taboldstyle w(M,x), winding number of a hypersurface M about a point x, /one.taboldstyle/three.taboldstyle/one.taboldstyle w(c,x), winding number of a closed path c about a point x, /five.taboldstyle/eight.taboldstyle /bardblx/bardbl, Euclidean norm (length) of a vector x, /eight.taboldstyle x·y, Euclidean inner product (dot product) of vectors xandy, /eight.taboldstyle Index Page numbers in boldface refer to definitions; page numbers in italic refer to theorems, examples, or applications. Italic boldface is used when both types of items occur on the same page. affine space, /eight.taboldstyle,/eight.taboldstyle/zero.taboldstyle alternating algebra, /nine.taboldstyle/zero.taboldstyle multilinear function, /three.taboldstyle/seven.taboldstyle,/nine.taboldstyle/three.taboldstyle,/nine.taboldstyle/five.taboldstyle–/nine.taboldstyle/eight.taboldstyle property, /one.taboldstyle/eight.taboldstyle,/two.taboldstyle/zero.taboldstyle,/two.taboldstyle/one.taboldstyle Ampère, André Marie (/one.taboldstyle/seven.taboldstyle/seven.taboldstyle/five.taboldstyle–/one.taboldstyle/eight.taboldstyle/three.taboldstyle/six.taboldstyle), /three.taboldstyle/zero.taboldstyle Ampère’s Law, /three.taboldstyle/zero.taboldstyle angle form, /two.taboldstyle/three.taboldstyle,/two.taboldstyle/seven.taboldstyle,/three.taboldstyle/eight.taboldstyle,/four.taboldstyle/nine.taboldstyle,/five.taboldstyle/two.taboldstyle,/five.taboldstyle/four.taboldstyle,/six.taboldstyle/zero.taboldstyle,/seven.taboldstyle/one.taboldstyle,/one.taboldstyle/two.taboldstyle/eight.taboldstyle,/one.taboldstyle/three.taboldstyle/zero.taboldstyle function along a path, /five.taboldstyle/six.taboldstyle on an open subset, /five.taboldstyle/four.taboldstyle anticommutativity, /one.taboldstyle/eight.taboldstyle antisymmetric matrix, /eight.taboldstyle/four.taboldstyle,/eight.taboldstyle/seven.taboldstyle multilinear function, seealternating multilinear function arc length, /nine.taboldstyle/seven.taboldstyle,/one.taboldstyle/zero.taboldstyle/six.taboldstyle,/one.taboldstyle/one.taboldstyle/eight.taboldstyle,/one.taboldstyle/two.taboldstyle/zero.taboldstyle Archimedes of Syracuse (/two.taboldstyle/eight.taboldstyle/seven.taboldstyle–/two.taboldstyle/one.taboldstyle/two.taboldstyle BC), /three.taboldstyle,/one.taboldstyle/two.taboldstyle/zero.taboldstyle Archimedes’ Law, /one.taboldstyle/two.taboldstyle/zero.taboldstyle atlas, /seven.taboldstyle/three.taboldstyle,/seven.taboldstyle/four.taboldstyle,/nine.taboldstyle/zero.taboldstyle average of a function, /one.taboldstyle/one.taboldstyle/eight.taboldstyle ball, seeclosed ball, open ball barycentre, /one.taboldstyle/one.taboldstyle/eight.taboldstyle bilinear, /nine.taboldstyle/three.taboldstyle,/nine.taboldstyle/four.taboldstyle,/nine.taboldstyle/eight.taboldstyle block, /one.taboldstyle/zero.taboldstyle/two.taboldstyle rectangular, seerectangular block Bonnet, Pierre (/one.taboldstyle/eight.taboldstyle/one.taboldstyle/nine.taboldstyle–/one.taboldstyle/eight.taboldstyle/nine.taboldstyle/two.taboldstyle), /one.taboldstyle/five.taboldstyle/three.taboldstyle boundary of a chain, /six.taboldstyle/six.taboldstyle,/six.taboldstyle/eight.taboldstyle–/seven.taboldstyle/two.taboldstyle of a manifold, /two.taboldstyle–/seven.taboldstyle,/one.taboldstyle/one.taboldstyle/one.taboldstyle,/one.taboldstyle/one.taboldstyle/two.taboldstyle–/one.taboldstyle/two.taboldstyle/one.taboldstyle,/one.taboldstyle/two.taboldstyle/three.taboldstyle,/one.taboldstyle/two.taboldstyle/eight.taboldstyle bounded, /four.taboldstyle/nine.taboldstyle,/one.taboldstyle/one.taboldstyle/six.taboldstyle,/one.taboldstyle/four.taboldstyle/zero.taboldstyle Brouwer, Luitzen Egbertus Jan (/one.taboldstyle/eight.taboldstyle/eight.taboldstyle/one.taboldstyle–/one.taboldstyle/nine.taboldstyle/six.taboldstyle/six.taboldstyle), /one.taboldstyle/two.taboldstyle/three.taboldstyle,/one.taboldstyle/three.taboldstyle/four.taboldstyle Brouwer’s fixed point theorem, /one.taboldstyle/two.taboldstyle/three.taboldstyle,/one.taboldstyle/three.taboldstyle/four.taboldstyle Cartan, Élie (/one.taboldstyle/eight.taboldstyle/six.taboldstyle/nine.taboldstyle–/one.taboldstyle/nine.taboldstyle/five.taboldstyle/one.taboldstyle), /one.taboldstyle/seven.taboldstyle Cartesian product, /seven.taboldstyle,/one.taboldstyle/one.taboldstyle,/nine.taboldstyle/three.taboldstyle,/one.taboldstyle/two.taboldstyle/six.taboldstyle,/one.taboldstyle/three.taboldstyle/seven.taboldstyle Cartesius, Renatus, seeDescartes, René centroid, /one.taboldstyle/one.taboldstyle/eight.taboldstylechain, /six.taboldstyle/five.taboldstyle,/six.taboldstyle/six.taboldstyle–/seven.taboldstyle/two.taboldstyle chart, /seven.taboldstyle/three.taboldstyle,/seven.taboldstyle/four.taboldstyle,/seven.taboldstyle/seven.taboldstyle,/seven.taboldstyle/eight.taboldstyle circle, /nine.taboldstyle,/four.taboldstyle/nine.taboldstyle,/five.taboldstyle/zero.taboldstyle,/five.taboldstyle/two.taboldstyle,/five.taboldstyle/eight.taboldstyle,/five.taboldstyle/nine.taboldstyle,/six.taboldstyle/nine.taboldstyle,/seven.taboldstyle/one.taboldstyle,/seven.taboldstyle/eight.taboldstyle,/one.taboldstyle/two.taboldstyle/five.taboldstyle,/one.taboldstyle/two.taboldstyle/eight.taboldstyle closed ball, /nine.taboldstyle,/one.taboldstyle/one.taboldstyle/three.taboldstyle,/one.taboldstyle/one.taboldstyle/four.taboldstyle,/one.taboldstyle/two.taboldstyle/one.taboldstyle,/one.taboldstyle/two.taboldstyle/five.taboldstyle,/one.taboldstyle/three.taboldstyle/five.taboldstyle,/one.taboldstyle/four.taboldstyle/zero.taboldstyle chain, /six.taboldstyle/eight.taboldstyle,/seven.taboldstyle/one.taboldstyle,/one.taboldstyle/three.taboldstyle/five.taboldstyle curve, /one.taboldstyle form, /two.taboldstyle/two.taboldstyle,/two.taboldstyle/eight.taboldstyle–/three.taboldstyle/zero.taboldstyle,/four.taboldstyle/two.taboldstyle,/five.taboldstyle/two.taboldstyle,/five.taboldstyle/eight.taboldstyle,/five.taboldstyle/nine.taboldstyle,/one.taboldstyle/two.taboldstyle/seven.taboldstyle,/one.taboldstyle/two.taboldstyle/nine.taboldstyle–/one.taboldstyle/three.taboldstyle/five.taboldstyle path, /five.taboldstyle/two.taboldstyle,/five.taboldstyle/seven.taboldstyle,/six.taboldstyle/eight.taboldstyle set,/four.taboldstyle,/three.taboldstyle/eight.taboldstyle,/four.taboldstyle/nine.taboldstyle,/nine.taboldstyle/one.taboldstyle,/one.taboldstyle/one.taboldstyle/one.taboldstyle,/one.taboldstyle/one.taboldstyle/six.taboldstyle,/one.taboldstyle/three.taboldstyle/nine.taboldstyle,/one.taboldstyle/four.taboldstyle/zero.taboldstyle,/one.taboldstyle/four.taboldstyle/one.taboldstyle codimension, /seven.taboldstyle/four.taboldstyle,/eight.taboldstyle/one.taboldstyle,/eight.taboldstyle/two.taboldstyle,/one.taboldstyle/one.taboldstyle/three.taboldstyle cohomology, /one.taboldstyle/two.taboldstyle/nine.taboldstyle,/one.taboldstyle/three.taboldstyle/two.taboldstyle,/one.taboldstyle/five.taboldstyle/three.taboldstyle column operation, /three.taboldstyle/two.taboldstyle,/four.taboldstyle/five.taboldstyle vector, /one.taboldstyle,/three.taboldstyle/one.taboldstyle,/four.taboldstyle/seven.taboldstyle,/seven.taboldstyle/five.taboldstyle,/nine.taboldstyle/eight.taboldstyle,/one.taboldstyle/zero.taboldstyle/zero.taboldstyle,/one.taboldstyle/four.taboldstyle/five.taboldstyle compact, /six.taboldstyle/three.taboldstyle,/one.taboldstyle/one.taboldstyle/six.taboldstyle–/one.taboldstyle/two.taboldstyle/zero.taboldstyle,/one.taboldstyle/two.taboldstyle/eight.taboldstyle,/one.taboldstyle/three.taboldstyle/one.taboldstyle,/one.taboldstyle/four.taboldstyle/zero.taboldstyle complementary, /two.taboldstyle/four.taboldstyle composition of maps, /one.taboldstyle/three.taboldstyle/eight.taboldstyle configuration space, /one.taboldstyle/zero.taboldstyle,/one.taboldstyle/five.taboldstyle connected, /four.taboldstyle/two.taboldstyle,/five.taboldstyle/two.taboldstyle–/five.taboldstyle/four.taboldstyle,/one.taboldstyle/one.taboldstyle/eight.taboldstyle connected component, /one.taboldstyle/three.taboldstyle,/one.taboldstyle/two.taboldstyle/nine.taboldstyle connected sum, /seven.taboldstyle conservative, /five.taboldstyle/one.taboldstyle,/one.taboldstyle/one.taboldstyle/eight.taboldstyle,/one.taboldstyle/one.taboldstyle/nine.taboldstyle constant form, /one.taboldstyle/nine.taboldstyle,/two.taboldstyle/nine.taboldstyle,/nine.taboldstyle/two.taboldstyle continuously differentiable, /one.taboldstyle/four.taboldstyle/five.taboldstyle contractible, /one.taboldstyle/two.taboldstyle/five.taboldstyle,/one.taboldstyle/two.taboldstyle/nine.taboldstyle–/one.taboldstyle/three.taboldstyle/five.taboldstyle contraction, /one.taboldstyle/two.taboldstyle/five.taboldstyle,/one.taboldstyle/three.taboldstyle/zero.taboldstyle,/one.taboldstyle/three.taboldstyle/one.taboldstyle–/one.taboldstyle/three.taboldstyle/five.taboldstyle contravariance, /four.taboldstyle/zero.taboldstyle convex, /one.taboldstyle/three.taboldstyle/four.taboldstyle linear combination, /one.taboldstyle/five.taboldstyle/two.taboldstyle coordinate map, seechart covariant vector, seecovector covector, /nine.taboldstyle/one.taboldstyle,/nine.taboldstyle/one.taboldstyle,/nine.taboldstyle/four.taboldstyle,/nine.taboldstyle/eight.taboldstyle field, /nine.taboldstyle/two.taboldstyle Coxeter, Harold Scott MacDonald (/one.taboldstyle/nine.taboldstyle/zero.taboldstyle/seven.taboldstyle–/two.taboldstyle/zero.taboldstyle/zero.taboldstyle/three.taboldstyle), /four.taboldstyle/five.taboldstyle Coxeter relations, /four.taboldstyle/five.taboldstyle critical point, /eight.taboldstyle/eight.taboldstyle cross-cap, /seven.taboldstyle cube in an open set, /six.taboldstyle/five.taboldstyle,/six.taboldstyle/six.taboldstyle–/six.taboldstyle/nine.taboldstyle,/seven.taboldstyle/one.taboldstyle–/seven.taboldstyle/two.taboldstyle curl, /two.taboldstyle/seven.taboldstyle,/three.taboldstyle/zero.taboldstyle,/seven.taboldstyle/two.taboldstyle,/one.taboldstyle/one.taboldstyle/nine.taboldstyle /one.taboldstyle/five.taboldstyle/nine.taboldstyle /one.taboldstyle/six.taboldstyle/zero.taboldstyle INDEX curvature, /one.taboldstyle/seven.taboldstyle curve, /one.taboldstyle,/seven.taboldstyle/five.taboldstyle cycle, /six.taboldstyle/eight.taboldstyle,/seven.taboldstyle/one.taboldstyle cylinder formula, /one.taboldstyle/two.taboldstyle/six.taboldstyle with base M,/one.taboldstyle/two.taboldstyle/six.taboldstyle d’Alembert, Jean Le Rond (/one.taboldstyle/seven.taboldstyle/one.taboldstyle/seven.taboldstyle–/one.taboldstyle/seven.taboldstyle/eight.taboldstyle/three.taboldstyle), /three.taboldstyle/zero.taboldstyle d’Alembertian, /three.taboldstyle/zero.taboldstyle de Rham, Georges (/one.taboldstyle/nine.taboldstyle/zero.taboldstyle/three.taboldstyle–/one.taboldstyle/nine.taboldstyle/nine.taboldstyle/zero.taboldstyle), /one.taboldstyle/two.taboldstyle/nine.taboldstyle,/one.taboldstyle/three.taboldstyle/two.taboldstyle,/one.taboldstyle/five.taboldstyle/three.taboldstyle degenerate chain, /six.taboldstyle/eight.taboldstyle,/seven.taboldstyle/one.taboldstyle degree of a form, /one.taboldstyle/seven.taboldstyle of a homogeneous function, /one.taboldstyle/five.taboldstyle/zero.taboldstyle of a multi-index, /one.taboldstyle/seven.taboldstyle of a quasihomogeneous function, /one.taboldstyle/five.taboldstyle/one.taboldstyle degrees of freedom, /one.taboldstyle/zero.taboldstyle,/one.taboldstyle/five.taboldstyle Descartes, René (/one.taboldstyle/five.taboldstyle/nine.taboldstyle/six.taboldstyle–/one.taboldstyle/six.taboldstyle/five.taboldstyle/zero.taboldstyle), /one.taboldstyle/one.taboldstyle,/nine.taboldstyle/three.taboldstyle,/one.taboldstyle/three.taboldstyle/seven.taboldstyle determinant, /three.taboldstyle/three.taboldstyle,/three.taboldstyle/three.taboldstyle–/three.taboldstyle/seven.taboldstyle,/four.taboldstyle/two.taboldstyle–/four.taboldstyle/four.taboldstyle,/four.taboldstyle/six.taboldstyle,/eight.taboldstyle/seven.taboldstyle,/nine.taboldstyle/three.taboldstyle–/nine.taboldstyle/five.taboldstyle, /nine.taboldstyle/nine.taboldstyle,/one.taboldstyle/zero.taboldstyle/zero.taboldstyle,/one.taboldstyle/zero.taboldstyle/three.taboldstyle differential equation, /one.taboldstyle/six.taboldstyle form, seeform dimension, /one.taboldstyle–/one.taboldstyle/two.taboldstyle,/seven.taboldstyle/four.taboldstyle Dirichlet, Lejeune (/one.taboldstyle/eight.taboldstyle/zero.taboldstyle/five.taboldstyle–/one.taboldstyle/eight.taboldstyle/five.taboldstyle/nine.taboldstyle), /one.taboldstyle/two.taboldstyle/zero.taboldstyle Dirichlet integral, /one.taboldstyle/two.taboldstyle/zero.taboldstyle disconnected, /one.taboldstyle/three.taboldstyle divergence, /two.taboldstyle/seven.taboldstyle,/three.taboldstyle/zero.taboldstyle,/seven.taboldstyle/two.taboldstyle,/one.taboldstyle/one.taboldstyle/nine.taboldstyle,/one.taboldstyle/two.taboldstyle/zero.taboldstyle domain, /one.taboldstyle/one.taboldstyle/four.taboldstyle,/one.taboldstyle/one.taboldstyle/nine.taboldstyle,/one.taboldstyle/two.taboldstyle/zero.taboldstyle of a function or a form, /one.taboldstyle/seven.taboldstyle,/two.taboldstyle/two.taboldstyle,/three.taboldstyle/nine.taboldstyle,/five.taboldstyle/zero.taboldstyle,/one.taboldstyle/three.taboldstyle/eight.taboldstyle, /one.taboldstyle/four.taboldstyle/four.taboldstyle dot product, seeinner product dual basis, /nine.taboldstyle/two.taboldstyle,/nine.taboldstyle/four.taboldstyle–/nine.taboldstyle/six.taboldstyle,/nine.taboldstyle/seven.taboldstyle vector, seecovector space, /nine.taboldstyle/one.taboldstyle electromagnetic wave, /three.taboldstyle/zero.taboldstyle electromagnetism, /three.taboldstyle/zero.taboldstyle element of arc length, /nine.taboldstyle/seven.taboldstyle,/one.taboldstyle/zero.taboldstyle/six.taboldstyle,/one.taboldstyle/two.taboldstyle/zero.taboldstyle of surface area, /one.taboldstyle/zero.taboldstyle/six.taboldstyle embedding, /seven.taboldstyle/three.taboldstyle,/seven.taboldstyle/four.taboldstyle–/seven.taboldstyle/six.taboldstyle,/eight.taboldstyle/six.taboldstyle–/eight.taboldstyle/seven.taboldstyle,/eight.taboldstyle/nine.taboldstyle,/nine.taboldstyle/seven.taboldstyle,/one.taboldstyle/zero.taboldstyle/five.taboldstyle–/one.taboldstyle/zero.taboldstyle/eight.taboldstyle, /one.taboldstyle/one.taboldstyle/zero.taboldstyle,/one.taboldstyle/one.taboldstyle/one.taboldstyle,/one.taboldstyle/one.taboldstyle/two.taboldstyle,/one.taboldstyle/one.taboldstyle/eight.taboldstyle,/one.taboldstyle/two.taboldstyle/zero.taboldstyle equivalence relation, /one.taboldstyle/three.taboldstyle/two.taboldstyle Euclid of Alexandria ( ca./three.taboldstyle/two.taboldstyle/five.taboldstyle–/two.taboldstyle/six.taboldstyle/five.taboldstyle BC), /one.taboldstyle,/seven.taboldstyle/three.taboldstyle, /seven.taboldstyle/four.taboldstyle,/nine.taboldstyle/nine.taboldstyle,/one.taboldstyle/two.taboldstyle/one.taboldstyle,/one.taboldstyle/two.taboldstyle/four.taboldstyle,/one.taboldstyle/two.taboldstyle/five.taboldstyle,/one.taboldstyle/three.taboldstyle/seven.taboldstyle,/one.taboldstyle/three.taboldstyle/nine.taboldstyle Euclidean motion, /nine.taboldstyle/nine.taboldstyle plane, /one.taboldstyle/three.taboldstyle/seven.taboldstyle space, /one.taboldstyle,/seven.taboldstyle/three.taboldstyle,/seven.taboldstyle/four.taboldstyle,/one.taboldstyle/two.taboldstyle/one.taboldstyle,/one.taboldstyle/two.taboldstyle/four.taboldstyle,/one.taboldstyle/two.taboldstyle/five.taboldstyle,/one.taboldstyle/three.taboldstyle/nine.taboldstyle volume, /nine.taboldstyle/nine.taboldstyle ε´υρηκα ,/one.taboldstyle/two.taboldstyle/zero.taboldstyle even map, /one.taboldstyle/five.taboldstyle/one.taboldstyle permutation, /three.taboldstyle/five.taboldstyle,/four.taboldstyle/five.taboldstyle exact form, /two.taboldstyle/two.taboldstyle,/two.taboldstyle/eight.taboldstyle,/five.taboldstyle/one.taboldstyle–/five.taboldstyle/eight.taboldstyle,/seven.taboldstyle/one.taboldstyle,/one.taboldstyle/two.taboldstyle/seven.taboldstyle,/one.taboldstyle/two.taboldstyle/nine.taboldstyle–/one.taboldstyle/three.taboldstyle/one.taboldstyle exterior algebra, /nine.taboldstyle/zero.taboldstylederivative, /two.taboldstyle/zero.taboldstyle,/two.taboldstyle/two.taboldstyle,/two.taboldstyle/six.taboldstyle,/two.taboldstyle/seven.taboldstyle,/six.taboldstyle/seven.taboldstyle,/seven.taboldstyle/zero.taboldstyle on a manifold, /nine.taboldstyle/zero.taboldstyle differential calculus, /one.taboldstyle/seven.taboldstyle product, seeproduct of forms Faraday, Michael (/one.taboldstyle/seven.taboldstyle/nine.taboldstyle/one.taboldstyle–/one.taboldstyle/eight.taboldstyle/six.taboldstyle/seven.taboldstyle), /three.taboldstyle/zero.taboldstyle Faraday’s Law, /three.taboldstyle/zero.taboldstyle fibre, seelevel set fixed point, /one.taboldstyle/two.taboldstyle/three.taboldstyle flux, /two.taboldstyle/six.taboldstyle,/one.taboldstyle/zero.taboldstyle/eight.taboldstyle,/one.taboldstyle/one.taboldstyle/nine.taboldstyle form as a vector-eating animal, /nine.taboldstyle/one.taboldstyle,/nine.taboldstyle/seven.taboldstyle closed, seeclosed form exact, seeexact form on a manifold, /nine.taboldstyle/zero.taboldstyle,/nine.taboldstyle/seven.taboldstyle on Euclidean space, /one.taboldstyle/seven.taboldstyle,/one.taboldstyle/eight.taboldstyle–/three.taboldstyle/zero.taboldstyle,/nine.taboldstyle/two.taboldstyle,/nine.taboldstyle/six.taboldstyle volume, seevolume form frame, /one.taboldstyle/zero.taboldstyle/three.taboldstyle oriented, seeoriented frame orthonormal, seeorthonormal frame free space, /three.taboldstyle/zero.taboldstyle Freedman, Michael (/one.taboldstyle/nine.taboldstyle/five.taboldstyle/one.taboldstyle–), /one.taboldstyle/three.taboldstyle/two.taboldstyle function, /one.taboldstyle/three.taboldstyle/eight.taboldstyle,/one.taboldstyle/four.taboldstyle/four.taboldstyle fundamental theorem of calculus, /two.taboldstyle/eight.taboldstyle,/four.taboldstyle/nine.taboldstyle,/five.taboldstyle/one.taboldstyle, /seven.taboldstyle/one.taboldstyle,/one.taboldstyle/four.taboldstyle/three.taboldstyle,/one.taboldstyle/four.taboldstyle/nine.taboldstyle inRn,/five.taboldstyle/one.taboldstyle,/seven.taboldstyle/zero.taboldstyle,/one.taboldstyle/one.taboldstyle/eight.taboldstyle Gamma function, /one.taboldstyle/two.taboldstyle/one.taboldstyle,/one.taboldstyle/five.taboldstyle/two.taboldstyle Gauss, Carl Friedrich (/one.taboldstyle/seven.taboldstyle/seven.taboldstyle/seven.taboldstyle–/one.taboldstyle/eight.taboldstyle/five.taboldstyle/five.taboldstyle), v,/three.taboldstyle/zero.taboldstyle,/seven.taboldstyle/zero.taboldstyle, /seven.taboldstyle/two.taboldstyle,/one.taboldstyle/zero.taboldstyle/five.taboldstyle,/one.taboldstyle/one.taboldstyle/nine.taboldstyle,/one.taboldstyle/five.taboldstyle/three.taboldstyle Gauss map, /one.taboldstyle/zero.taboldstyle/five.taboldstyle Gauss’ Law, /three.taboldstyle/zero.taboldstyle general linear group, /eight.taboldstyle/seven.taboldstyle,/one.taboldstyle/three.taboldstyle/five.taboldstyle generalized angle form, /one.taboldstyle/three.taboldstyle/zero.taboldstyle gluing diagram, /five.taboldstyle,/one.taboldstyle/four.taboldstyle,/one.taboldstyle/five.taboldstyle graded commutativity, /one.taboldstyle/eight.taboldstyle,/one.taboldstyle/nine.taboldstyle gradient, /two.taboldstyle/six.taboldstyle,/two.taboldstyle/nine.taboldstyle,/five.taboldstyle/one.taboldstyle,/seven.taboldstyle/two.taboldstyle,/eight.taboldstyle/one.taboldstyle–/eight.taboldstyle/eight.taboldstyle,/one.taboldstyle/zero.taboldstyle/five.taboldstyle,/one.taboldstyle/zero.taboldstyle/nine.taboldstyle,/one.taboldstyle/one.taboldstyle/zero.taboldstyle, /one.taboldstyle/one.taboldstyle/eight.taboldstyle,/one.taboldstyle/two.taboldstyle/zero.taboldstyle,/one.taboldstyle/four.taboldstyle/five.taboldstyle Gram, Jørgen (/one.taboldstyle/eight.taboldstyle/five.taboldstyle/zero.taboldstyle–/one.taboldstyle/nine.taboldstyle/one.taboldstyle/six.taboldstyle), /one.taboldstyle/zero.taboldstyle/zero.taboldstyle,/one.taboldstyle/zero.taboldstyle/one.taboldstyle Gram matrix, /one.taboldstyle/zero.taboldstyle/zero.taboldstyle Gram-Schmidt process, /one.taboldstyle/zero.taboldstyle/one.taboldstyle graph, /seven.taboldstyle/four.taboldstyle,/seven.taboldstyle/five.taboldstyle,/eight.taboldstyle/one.taboldstyle,/one.taboldstyle/zero.taboldstyle/six.taboldstyle,/one.taboldstyle/one.taboldstyle/zero.taboldstyle,/one.taboldstyle/one.taboldstyle/one.taboldstyle,/one.taboldstyle/four.taboldstyle/three.taboldstyle Grassmann, Hermann (/one.taboldstyle/eight.taboldstyle/zero.taboldstyle/nine.taboldstyle–/one.taboldstyle/eight.taboldstyle/seven.taboldstyle/seven.taboldstyle), /nine.taboldstyle/zero.taboldstyle gravitation, /five.taboldstyle/eight.taboldstyle,/one.taboldstyle/five.taboldstyle/zero.taboldstyle Greek alphabet, /one.taboldstyle/five.taboldstyle/five.taboldstyle Green, George (/one.taboldstyle/seven.taboldstyle/nine.taboldstyle/three.taboldstyle–/one.taboldstyle/eight.taboldstyle/four.taboldstyle/one.taboldstyle), v,/seven.taboldstyle/zero.taboldstyle,/seven.taboldstyle/two.taboldstyle,/one.taboldstyle/two.taboldstyle/one.taboldstyle Green’s theorem, /seven.taboldstyle/two.taboldstyle,/one.taboldstyle/two.taboldstyle/one.taboldstyle half-space, /one.taboldstyle/one.taboldstyle/one.taboldstyle Hodge, William (/one.taboldstyle/nine.taboldstyle/zero.taboldstyle/three.taboldstyle–/one.taboldstyle/nine.taboldstyle/seven.taboldstyle/five.taboldstyle), /two.taboldstyle/four.taboldstyle,/two.taboldstyle/six.taboldstyle,/two.taboldstyle/nine.taboldstyle,/three.taboldstyle/zero.taboldstyle,/four.taboldstyle/zero.taboldstyle Hodge star operator, /two.taboldstyle/four.taboldstyle,/two.taboldstyle/six.taboldstyle,/two.taboldstyle/nine.taboldstyle,/four.taboldstyle/zero.taboldstyle,see also relativity homogeneous function, /two.taboldstyle/eight.taboldstyle,/eight.taboldstyle/seven.taboldstyle,/one.taboldstyle/three.taboldstyle/five.taboldstyle,/one.taboldstyle/five.taboldstyle/zero.taboldstyle homotopy, /one.taboldstyle/two.taboldstyle/four.taboldstyle formula, /one.taboldstyle/two.taboldstyle/seven.taboldstyle of loops, /one.taboldstyle/two.taboldstyle/five.taboldstyle,/one.taboldstyle/two.taboldstyle/nine.taboldstyle,/one.taboldstyle/three.taboldstyle/one.taboldstyle of paths, /one.taboldstyle/two.taboldstyle/five.taboldstyle hyperplane, /seven.taboldstyle/five.taboldstyle,/one.taboldstyle/zero.taboldstyle/eight.taboldstyle,/one.taboldstyle/five.taboldstyle/one.taboldstyle hypersurface, /two.taboldstyle/six.taboldstyle,/seven.taboldstyle/five.taboldstyle,/one.taboldstyle/zero.taboldstyle/four.taboldstyle–/one.taboldstyle/one.taboldstyle/zero.taboldstyle,/one.taboldstyle/one.taboldstyle/four.taboldstyle,/one.taboldstyle/one.taboldstyle/eight.taboldstyle,/one.taboldstyle/three.taboldstyle/one.taboldstyle INDEX /one.taboldstyle/six.taboldstyle/one.taboldstyle image of a set under a map, /three.taboldstyle/seven.taboldstyle,/six.taboldstyle/three.taboldstyle,/six.taboldstyle/five.taboldstyle,/seven.taboldstyle/two.taboldstyle,/seven.taboldstyle/three.taboldstyle, /seven.taboldstyle/five.taboldstyle,/eight.taboldstyle/one.taboldstyle,/eight.taboldstyle/six.taboldstyle,/eight.taboldstyle/nine.taboldstyle,/one.taboldstyle/zero.taboldstyle/six.taboldstyle,/one.taboldstyle/one.taboldstyle/one.taboldstyle,/one.taboldstyle/two.taboldstyle/zero.taboldstyle,/one.taboldstyle/three.taboldstyle/eight.taboldstyle image of a set under a map ,/four.taboldstyle/nine.taboldstyle increasing multi-index, /one.taboldstyle/nine.taboldstyle,/two.taboldstyle/four.taboldstyle,/two.taboldstyle/nine.taboldstyle,/four.taboldstyle/four.taboldstyle,/nine.taboldstyle/four.taboldstyle,/nine.taboldstyle/five.taboldstyle, see also complementary index of a vector field, /five.taboldstyle/nine.taboldstyle inner product, /eight.taboldstyle,/nine.taboldstyle/three.taboldstyle,/one.taboldstyle/zero.taboldstyle/zero.taboldstyle,/one.taboldstyle/zero.taboldstyle/seven.taboldstyle,/one.taboldstyle/four.taboldstyle/five.taboldstyle of forms, /two.taboldstyle/nine.taboldstyle integrability condition, /two.taboldstyle/three.taboldstyle integral of a1-form over a path, /four.taboldstyle/nine.taboldstyle,/five.taboldstyle/zero.taboldstyle–/five.taboldstyle/two.taboldstyle of a form over a chain, /six.taboldstyle/three.taboldstyle,/six.taboldstyle/four.taboldstyle–/six.taboldstyle/five.taboldstyle,/seven.taboldstyle/zero.taboldstyle–/seven.taboldstyle/two.taboldstyle,/one.taboldstyle/one.taboldstyle/five.taboldstyle over a manifold, /one.taboldstyle/one.taboldstyle/seven.taboldstyle,/one.taboldstyle/one.taboldstyle/seven.taboldstyle–/one.taboldstyle/one.taboldstyle/nine.taboldstyle,/one.taboldstyle/two.taboldstyle/seven.taboldstyle,/one.taboldstyle/two.taboldstyle/nine.taboldstyle, /one.taboldstyle/three.taboldstyle/one.taboldstyle,/one.taboldstyle/three.taboldstyle/five.taboldstyle inverse, /four.taboldstyle/five.taboldstyle,/seven.taboldstyle/three.taboldstyle,/seven.taboldstyle/four.taboldstyle,/seven.taboldstyle/seven.taboldstyle,/eight.taboldstyle/six.taboldstyle,/one.taboldstyle/three.taboldstyle/one.taboldstyle,/one.taboldstyle/three.taboldstyle/eight.taboldstyle,/one.taboldstyle/three.taboldstyle/eight.taboldstyle,/one.taboldstyle/four.taboldstyle/eight.taboldstyle, /one.taboldstyle/five.taboldstyle/one.taboldstyle inversion, /three.taboldstyle/four.taboldstyle inward pointing, /one.taboldstyle/one.taboldstyle/four.taboldstyle k-chain, seechain k-cube, seecube k-form, seeform k-multilinear function, seemultilinear function k-tuple, seetuple Klein, Felix (/one.taboldstyle/eight.taboldstyle/four.taboldstyle/nine.taboldstyle–/one.taboldstyle/nine.taboldstyle/two.taboldstyle/five.taboldstyle), /five.taboldstyle Klein bottle, /five.taboldstyle Kronecker, Leopold (/one.taboldstyle/eight.taboldstyle/two.taboldstyle/three.taboldstyle–/one.taboldstyle/eight.taboldstyle/nine.taboldstyle/one.taboldstyle), /five.taboldstyle/three.taboldstyle,/nine.taboldstyle/one.taboldstyle,/nine.taboldstyle/four.taboldstyle Kronecker delta, /five.taboldstyle/three.taboldstyle,/nine.taboldstyle/one.taboldstyle,/nine.taboldstyle/four.taboldstyle Laplace, Pierre-Simon (/one.taboldstyle/seven.taboldstyle/four.taboldstyle/nine.taboldstyle–/one.taboldstyle/eight.taboldstyle/two.taboldstyle/seven.taboldstyle), /two.taboldstyle/nine.taboldstyle Laplacian, /two.taboldstyle/nine.taboldstyle left inverse, /seven.taboldstyle/nine.taboldstyle,/one.taboldstyle/three.taboldstyle/eight.taboldstyle Leibniz, Gottfried Wilhelm von (/one.taboldstyle/six.taboldstyle/four.taboldstyle/six.taboldstyle–/one.taboldstyle/seven.taboldstyle/one.taboldstyle/six.taboldstyle), /two.taboldstyle/zero.taboldstyle,/two.taboldstyle/one.taboldstyle,/four.taboldstyle/two.taboldstyle Leibniz rule for forms, /two.taboldstyle/one.taboldstyle,/four.taboldstyle/two.taboldstyle for functions, /two.taboldstyle/zero.taboldstyle length of a permutation, /three.taboldstyle/four.taboldstyle,/four.taboldstyle/five.taboldstyle of a vector, /eight.taboldstyle,/eight.taboldstyle/four.taboldstyle,/eight.taboldstyle/eight.taboldstyle,/one.taboldstyle/zero.taboldstyle/four.taboldstyle,/one.taboldstyle/one.taboldstyle/four.taboldstyle,/one.taboldstyle/two.taboldstyle/five.taboldstyle,/one.taboldstyle/four.taboldstyle/one.taboldstyle level curve, /eight.taboldstyle/one.taboldstyle hypersurface, /eight.taboldstyle/one.taboldstyle set,/eight.taboldstyle/zero.taboldstyle,/one.taboldstyle/three.taboldstyle/eight.taboldstyle surface, /eight.taboldstyle/one.taboldstyle lightlike, /two.taboldstyle/nine.taboldstyle line segment, /three.taboldstyle/one.taboldstyle linear functional, seecovector local parametrization of a manifold, /seven.taboldstyle/four.taboldstyle,/seven.taboldstyle/nine.taboldstyle,/one.taboldstyle/zero.taboldstyle/four.taboldstyle representative, /nine.taboldstyle/zero.taboldstyle,/nine.taboldstyle/seven.taboldstyle,/one.taboldstyle/zero.taboldstyle/five.taboldstyle Lotka, Alfred (/one.taboldstyle/eight.taboldstyle/eight.taboldstyle/zero.taboldstyle–/one.taboldstyle/nine.taboldstyle/four.taboldstyle/nine.taboldstyle), /one.taboldstyle/six.taboldstyle,/eight.taboldstyle/seven.taboldstyle Lotka-Volterra model, /one.taboldstyle/six.taboldstyle,/eight.taboldstyle/seven.taboldstyle manifold, /one.taboldstyle–/one.taboldstyle/six.taboldstyle,/seven.taboldstyle/four.taboldstyle,/seven.taboldstyle/five.taboldstyle–/eight.taboldstyle/eight.taboldstyle abstract, /one.taboldstyle/zero.taboldstyle–/one.taboldstyle/four.taboldstyle,/seven.taboldstyle/four.taboldstyle given explicitly, /nine.taboldstyle,/eight.taboldstyle/zero.taboldstylegiven implicitly, /seven.taboldstyle,/eight.taboldstyle/zero.taboldstyle with boundary, /two.taboldstyle–/seven.taboldstyle,/one.taboldstyle/one.taboldstyle/one.taboldstyle,/one.taboldstyle/one.taboldstyle/two.taboldstyle–/one.taboldstyle/two.taboldstyle/one.taboldstyle,/one.taboldstyle/two.taboldstyle/three.taboldstyle,/one.taboldstyle/two.taboldstyle/eight.taboldstyle map, /one.taboldstyle/three.taboldstyle/eight.taboldstyle,/one.taboldstyle/four.taboldstyle/four.taboldstyle Maxwell, James Clerk (/one.taboldstyle/eight.taboldstyle/three.taboldstyle/one.taboldstyle–/one.taboldstyle/eight.taboldstyle/seven.taboldstyle/nine.taboldstyle), /three.taboldstyle/zero.taboldstyle mean of a function, /one.taboldstyle/one.taboldstyle/eight.taboldstyle measurable set, /three.taboldstyle/eight.taboldstyle Milnor, John (/one.taboldstyle/nine.taboldstyle/three.taboldstyle/one.taboldstyle–), /one.taboldstyle/three.taboldstyle/two.taboldstyle minimum, /eight.taboldstyle/three.taboldstyle Minkowski, Hermann (/one.taboldstyle/eight.taboldstyle/six.taboldstyle/four.taboldstyle–/one.taboldstyle/nine.taboldstyle/zero.taboldstyle/nine.taboldstyle), /two.taboldstyle/nine.taboldstyle Minkowski inner product, /two.taboldstyle/nine.taboldstyle space, /two.taboldstyle/nine.taboldstyle Möbius, August (/one.taboldstyle/seven.taboldstyle/nine.taboldstyle/zero.taboldstyle–/one.taboldstyle/eight.taboldstyle/six.taboldstyle/eight.taboldstyle), /four.taboldstyle,/one.taboldstyle/one.taboldstyle/four.taboldstyle Möbius band, /four.taboldstyle,/one.taboldstyle/one.taboldstyle/four.taboldstyle multi-index, /one.taboldstyle/seven.taboldstyle,/one.taboldstyle/three.taboldstyle/nine.taboldstyle,see also increasing multi-index multilinear algebra, /one.taboldstyle/seven.taboldstyle function, /nine.taboldstyle/three.taboldstyle,/nine.taboldstyle/eight.taboldstyle n-manifold, seemanifold naturality of pullbacks, /four.taboldstyle/zero.taboldstyle,/five.taboldstyle/zero.taboldstyle,/six.taboldstyle/four.taboldstyle Newton, Isaac (/one.taboldstyle/six.taboldstyle/four.taboldstyle/three.taboldstyle–/one.taboldstyle/seven.taboldstyle/two.taboldstyle/seven.taboldstyle), /five.taboldstyle/eight.taboldstyle,/one.taboldstyle/five.taboldstyle/zero.taboldstyle,/one.taboldstyle/five.taboldstyle/three.taboldstyle norm of a vector, /eight.taboldstyle normal vector field, seeunit normal vector field odd permutation, /three.taboldstyle/five.taboldstyle,/four.taboldstyle/five.taboldstyle open ball, /one.taboldstyle/three.taboldstyle/nine.taboldstyle neighbourhood, /one.taboldstyle/three.taboldstyle/nine.taboldstyle set,/one.taboldstyle/three.taboldstyle/nine.taboldstyle,/one.taboldstyle/four.taboldstyle/zero.taboldstyle,/one.taboldstyle/four.taboldstyle/four.taboldstyle is a manifold, /seven.taboldstyle/five.taboldstyle ordered basis, seeframe ordered pair, /one.taboldstyle/three.taboldstyle/seven.taboldstyle orientation of a boundary, /one.taboldstyle/one.taboldstyle/five.taboldstyle,/one.taboldstyle/one.taboldstyle/seven.taboldstyle of a hypersurface, /one.taboldstyle/zero.taboldstyle/four.taboldstyle,/one.taboldstyle/one.taboldstyle/zero.taboldstyle of a manifold, /nine.taboldstyle/seven.taboldstyle,/one.taboldstyle/zero.taboldstyle/four.taboldstyle,/one.taboldstyle/one.taboldstyle/eight.taboldstyle of a vector space, /nine.taboldstyle/nine.taboldstyle,/one.taboldstyle/zero.taboldstyle/three.taboldstyle,/one.taboldstyle/one.taboldstyle/zero.taboldstyle preserving, /five.taboldstyle/zero.taboldstyle,/six.taboldstyle/three.taboldstyle,/one.taboldstyle/zero.taboldstyle/four.taboldstyle,/one.taboldstyle/zero.taboldstyle/five.taboldstyle,/one.taboldstyle/zero.taboldstyle/seven.taboldstyle,/one.taboldstyle/one.taboldstyle/zero.taboldstyle,/one.taboldstyle/one.taboldstyle/five.taboldstyle reversing, /five.taboldstyle/zero.taboldstyle,/six.taboldstyle/three.taboldstyle,/one.taboldstyle/zero.taboldstyle/four.taboldstyle oriented frame, /one.taboldstyle/zero.taboldstyle/three.taboldstyle orthogonal complement, /eight.taboldstyle/two.taboldstyle,/one.taboldstyle/zero.taboldstyle/four.taboldstyle group, /eight.taboldstyle/four.taboldstyle,/eight.taboldstyle/seven.taboldstyle,/one.taboldstyle/one.taboldstyle/zero.taboldstyle,/one.taboldstyle/three.taboldstyle/five.taboldstyle matrix, /eight.taboldstyle/four.taboldstyle,/nine.taboldstyle/nine.taboldstyle,/one.taboldstyle/zero.taboldstyle/zero.taboldstyle operator, /two.taboldstyle/nine.taboldstyle projection, /one.taboldstyle/zero.taboldstyle/one.taboldstyle orthonormal, /two.taboldstyle/nine.taboldstyle,/one.taboldstyle/zero.taboldstyle/zero.taboldstyle frame, /one.taboldstyle/one.taboldstyle/zero.taboldstyle outward pointing, /one.taboldstyle/one.taboldstyle/four.taboldstyle,/one.taboldstyle/two.taboldstyle/zero.taboldstyle pair of pants, /one.taboldstyle/one.taboldstyle/four.taboldstyle paraboloid, /four.taboldstyle parallelepiped, /three.taboldstyle/one.taboldstyle,/three.taboldstyle/one.taboldstyle,/four.taboldstyle/four.taboldstyle,/nine.taboldstyle/nine.taboldstyle,/one.taboldstyle/zero.taboldstyle/five.taboldstyle parallelogram, /one.taboldstyle/four.taboldstyle,/one.taboldstyle/five.taboldstyle,/three.taboldstyle/one.taboldstyle,/one.taboldstyle/zero.taboldstyle/nine.taboldstyle parametrization, /nine.taboldstyle,/six.taboldstyle/three.taboldstyle,/seven.taboldstyle/three.taboldstyle parametrized curve, seepath /one.taboldstyle/six.taboldstyle/two.taboldstyle INDEX partial differential equation, /two.taboldstyle/three.taboldstyle operator, /two.taboldstyle/zero.taboldstyle path, /one.taboldstyle/zero.taboldstyle,/one.taboldstyle/four.taboldstyle,/four.taboldstyle/nine.taboldstyle,/five.taboldstyle/one.taboldstyle,/five.taboldstyle/eight.taboldstyle–/five.taboldstyle/nine.taboldstyle,/seven.taboldstyle/nine.taboldstyle,/seven.taboldstyle/nine.taboldstyle,/eight.taboldstyle/six.taboldstyle–/eight.taboldstyle/seven.taboldstyle,/one.taboldstyle/two.taboldstyle/five.taboldstyle, /one.taboldstyle/four.taboldstyle/five.taboldstyle pentagon, /one.taboldstyle/six.taboldstyle Perelman, Grigori (/one.taboldstyle/nine.taboldstyle/six.taboldstyle/six.taboldstyle–), /one.taboldstyle/three.taboldstyle/two.taboldstyle permutation, /three.taboldstyle/four.taboldstyle,/three.taboldstyle/five.taboldstyle,/three.taboldstyle/six.taboldstyle,/four.taboldstyle/four.taboldstyle,/four.taboldstyle/five.taboldstyle,/nine.taboldstyle/three.taboldstyle,/one.taboldstyle/one.taboldstyle/zero.taboldstyle group, /three.taboldstyle/four.taboldstyle,/four.taboldstyle/five.taboldstyle matrix, /four.taboldstyle/five.taboldstyle pinch point, /seven.taboldstyle plane curve, /one.taboldstyle,/one.taboldstyle/four.taboldstyle,/seven.taboldstyle/five.taboldstyle Poincaré, Jules Henri (/one.taboldstyle/eight.taboldstyle/five.taboldstyle/four.taboldstyle–/one.taboldstyle/nine.taboldstyle/one.taboldstyle/two.taboldstyle), /one.taboldstyle/two.taboldstyle/nine.taboldstyle,/one.taboldstyle/three.taboldstyle/two.taboldstyle Poincaré conjecture, /one.taboldstyle/three.taboldstyle/one.taboldstyle lemma, /one.taboldstyle/two.taboldstyle/nine.taboldstyle potential, /five.taboldstyle/one.taboldstyle,/five.taboldstyle/eight.taboldstyle,/one.taboldstyle/one.taboldstyle/eight.taboldstyle predator, /one.taboldstyle/six.taboldstyle preimage of a set or a point under a map, /eight.taboldstyle/zero.taboldstyle, /eight.taboldstyle/two.taboldstyle,/eight.taboldstyle/seven.taboldstyle,/one.taboldstyle/zero.taboldstyle/five.taboldstyle,/one.taboldstyle/one.taboldstyle/three.taboldstyle,/one.taboldstyle/three.taboldstyle/eight.taboldstyle prey, /one.taboldstyle/six.taboldstyle product of forms, /one.taboldstyle/nine.taboldstyle,/two.taboldstyle/seven.taboldstyle–/two.taboldstyle/eight.taboldstyle on a manifold, /nine.taboldstyle/zero.taboldstyle of permutations, /three.taboldstyle/five.taboldstyle,/four.taboldstyle/five.taboldstyle of sets, seeCartesian product product rule, seeLeibniz rule projective plane, /five.taboldstyle,/one.taboldstyle/three.taboldstyle/three.taboldstyle space, /eight.taboldstyle/six.taboldstyle,/one.taboldstyle/one.taboldstyle/zero.taboldstyle pullback of a form on a manifold, /nine.taboldstyle/seven.taboldstyle,/one.taboldstyle/zero.taboldstyle/six.taboldstyle,/one.taboldstyle/two.taboldstyle/four.taboldstyle,/one.taboldstyle/two.taboldstyle/six.taboldstyle,/one.taboldstyle/two.taboldstyle/seven.taboldstyle on Euclidean space, /three.taboldstyle/nine.taboldstyle,/four.taboldstyle/two.taboldstyle–/four.taboldstyle/four.taboldstyle,/four.taboldstyle/nine.taboldstyle,/five.taboldstyle/zero.taboldstyle,/six.taboldstyle/three.taboldstyle, /six.taboldstyle/four.taboldstyle,/nine.taboldstyle/zero.taboldstyle,/nine.taboldstyle/six.taboldstyle of a function, /one.taboldstyle/four.taboldstyle/seven.taboldstyle punctured Euclidean space, /eight.taboldstyle/eight.taboldstyle,/one.taboldstyle/two.taboldstyle/four.taboldstyle,/one.taboldstyle/two.taboldstyle/nine.taboldstyle–/one.taboldstyle/three.taboldstyle/one.taboldstyle plane, /four.taboldstyle/nine.taboldstyle,/five.taboldstyle/four.taboldstyle,/seven.taboldstyle/one.taboldstyle,/eight.taboldstyle/six.taboldstyle,/one.taboldstyle/two.taboldstyle/six.taboldstyle,/one.taboldstyle/two.taboldstyle/nine.taboldstyle,/one.taboldstyle/three.taboldstyle/zero.taboldstyle QR-decomposition, /one.taboldstyle/zero.taboldstyle/one.taboldstyle,/one.taboldstyle/three.taboldstyle/five.taboldstyle quadrilateral, /one.taboldstyle/two.taboldstyle quasihomogeneous function, /one.taboldstyle/five.taboldstyle/one.taboldstyle rectangular block, /one.taboldstyle/eight.taboldstyle,/six.taboldstyle/three.taboldstyle,/seven.taboldstyle/two.taboldstyle regular path, /six.taboldstyle/zero.taboldstyle value, /eight.taboldstyle/zero.taboldstyle,/eight.taboldstyle/two.taboldstyle–/eight.taboldstyle/five.taboldstyle,/eight.taboldstyle/seven.taboldstyle–/eight.taboldstyle/eight.taboldstyle,/one.taboldstyle/zero.taboldstyle/five.taboldstyle,/one.taboldstyle/zero.taboldstyle/nine.taboldstyle,/one.taboldstyle/one.taboldstyle/three.taboldstyle relativity, /two.taboldstyle/nine.taboldstyle,/three.taboldstyle/zero.taboldstyle,/one.taboldstyle/five.taboldstyle/three.taboldstyle reparametrization of a path, /four.taboldstyle/nine.taboldstyle,/five.taboldstyle/zero.taboldstyle,/five.taboldstyle/two.taboldstyle,/six.taboldstyle/four.taboldstyle of a rectangular block, /six.taboldstyle/three.taboldstyle,/six.taboldstyle/four.taboldstyle restriction of a map, /one.taboldstyle/two.taboldstyle/one.taboldstyle,/one.taboldstyle/three.taboldstyle/eight.taboldstyle retraction, /one.taboldstyle/two.taboldstyle/three.taboldstyle Riemann, Bernhard (/one.taboldstyle/eight.taboldstyle/two.taboldstyle/six.taboldstyle–/one.taboldstyle/eight.taboldstyle/six.taboldstyle/six.taboldstyle), /one.taboldstyle/five.taboldstyle/three.taboldstyle rigid body, /one.taboldstyle/one.taboldstyle row operation, /four.taboldstyle/five.taboldstyle vector, /one.taboldstyle,/four.taboldstyle/seven.taboldstyle,/eight.taboldstyle/one.taboldstyle,/nine.taboldstyle/eight.taboldstyle,/one.taboldstyle/four.taboldstyle/five.taboldstylesaddle point, /eight.taboldstyle/three.taboldstyle Schmidt, Erhard (/one.taboldstyle/eight.taboldstyle/seven.taboldstyle/six.taboldstyle–/one.taboldstyle/nine.taboldstyle/five.taboldstyle/nine.taboldstyle), /one.taboldstyle/zero.taboldstyle/one.taboldstyle sign of a permutation, /three.taboldstyle/five.taboldstyle,/four.taboldstyle/five.taboldstyle simple permutation, /four.taboldstyle/five.taboldstyle simplex, /one.taboldstyle/five.taboldstyle/two.taboldstyle simply connected, /one.taboldstyle/two.taboldstyle/nine.taboldstyle,/one.taboldstyle/three.taboldstyle/one.taboldstyle singular cube, seecube value, /eight.taboldstyle/zero.taboldstyle,/eight.taboldstyle/two.taboldstyle–/eight.taboldstyle/four.taboldstyle,/eight.taboldstyle/seven.taboldstyle singularity, /two.taboldstyle,/eight.taboldstyle,/one.taboldstyle/four.taboldstyle,/one.taboldstyle/six.taboldstyle,/seven.taboldstyle/three.taboldstyle,/seven.taboldstyle/seven.taboldstyle,/one.taboldstyle/one.taboldstyle/three.taboldstyle Smale, Stephen (/one.taboldstyle/nine.taboldstyle/three.taboldstyle/zero.taboldstyle–), /one.taboldstyle/three.taboldstyle/two.taboldstyle smooth curve, /seven.taboldstyle/five.taboldstyle function or map, /one.taboldstyle/four.taboldstyle/five.taboldstyle,/one.taboldstyle/five.taboldstyle/zero.taboldstyle hypersurface, /seven.taboldstyle/five.taboldstyle manifold, /four.taboldstyle point, /two.taboldstyle surface, /seven.taboldstyle/five.taboldstyle solution curve, /one.taboldstyle/zero.taboldstyle,/one.taboldstyle/six.taboldstyle space curve, /one.taboldstyle space-time, /three.taboldstyle/zero.taboldstyle spacelike, /two.taboldstyle/nine.taboldstyle special linear group, /eight.taboldstyle/seven.taboldstyle special orthogonal group, /eight.taboldstyle/five.taboldstyle sphere, /four.taboldstyle,/eight.taboldstyle,/one.taboldstyle/zero.taboldstyle,/one.taboldstyle/one.taboldstyle,/one.taboldstyle/five.taboldstyle,/seven.taboldstyle/two.taboldstyle,/seven.taboldstyle/three.taboldstyle,/eight.taboldstyle/three.taboldstyle,/eight.taboldstyle/eight.taboldstyle,/one.taboldstyle/zero.taboldstyle/five.taboldstyle,/one.taboldstyle/zero.taboldstyle/nine.taboldstyle, /one.taboldstyle/one.taboldstyle/three.taboldstyle,/one.taboldstyle/two.taboldstyle/one.taboldstyle,/one.taboldstyle/two.taboldstyle/three.taboldstyle,/one.taboldstyle/two.taboldstyle/four.taboldstyle,/one.taboldstyle/two.taboldstyle/nine.taboldstyle–/one.taboldstyle/three.taboldstyle/one.taboldstyle,/one.taboldstyle/four.taboldstyle/one.taboldstyle,/one.taboldstyle/five.taboldstyle/one.taboldstyle spherical coordinates, /four.taboldstyle/six.taboldstyle pendulum, /one.taboldstyle/zero.taboldstyle standard basis of Rn,/two.taboldstyle/five.taboldstyle,/three.taboldstyle/six.taboldstyle,/four.taboldstyle/six.taboldstyle,/eight.taboldstyle/four.taboldstyle,/eight.taboldstyle/nine.taboldstyle,/nine.taboldstyle/two.taboldstyle–/nine.taboldstyle/four.taboldstyle,/nine.taboldstyle/six.taboldstyle,/one.taboldstyle/zero.taboldstyle/three.taboldstyle, /one.taboldstyle/zero.taboldstyle/four.taboldstyle,/one.taboldstyle/zero.taboldstyle/seven.taboldstyle,/one.taboldstyle/zero.taboldstyle/eight.taboldstyle,/one.taboldstyle/four.taboldstyle/four.taboldstyle orientation of Rn,/one.taboldstyle/zero.taboldstyle/three.taboldstyle,/one.taboldstyle/zero.taboldstyle/four.taboldstyle,/one.taboldstyle/one.taboldstyle/five.taboldstyle simplex, /one.taboldstyle/five.taboldstyle/two.taboldstyle star-shaped, /one.taboldstyle/three.taboldstyle/four.taboldstyle state space, seeconfiguration space steepest ascent, /eight.taboldstyle/three.taboldstyle,/one.taboldstyle/four.taboldstyle/six.taboldstyle stereographic projection, /seven.taboldstyle/eight.taboldstyle,/one.taboldstyle/five.taboldstyle/one.taboldstyle Stirling, James (/one.taboldstyle/six.taboldstyle/nine.taboldstyle/two.taboldstyle–/one.taboldstyle/seven.taboldstyle/seven.taboldstyle/zero.taboldstyle), /one.taboldstyle/two.taboldstyle/one.taboldstyle Stirling’s formula, /one.taboldstyle/two.taboldstyle/one.taboldstyle Stokes, George (/one.taboldstyle/eight.taboldstyle/one.taboldstyle/nine.taboldstyle–/one.taboldstyle/nine.taboldstyle/zero.taboldstyle/three.taboldstyle), v,/four.taboldstyle/nine.taboldstyle,/seven.taboldstyle/zero.taboldstyle,/seven.taboldstyle/two.taboldstyle,/one.taboldstyle/one.taboldstyle/seven.taboldstyle, /one.taboldstyle/one.taboldstyle/nine.taboldstyle Stokes’ theorem classical version, /seven.taboldstyle/two.taboldstyle,/one.taboldstyle/one.taboldstyle/nine.taboldstyle for chains, /four.taboldstyle/nine.taboldstyle,/seven.taboldstyle/zero.taboldstyle,/seven.taboldstyle/one.taboldstyle for manifolds, /one.taboldstyle/one.taboldstyle/one.taboldstyle,/one.taboldstyle/one.taboldstyle/seven.taboldstyle,/one.taboldstyle/two.taboldstyle/six.taboldstyle,/one.taboldstyle/two.taboldstyle/eight.taboldstyle,/one.taboldstyle/three.taboldstyle/zero.taboldstyle submanifold, /seven.taboldstyle/four.taboldstyle support of a differential form, /one.taboldstyle/one.taboldstyle/five.taboldstyle surface, /four.taboldstyle,/one.taboldstyle/five.taboldstyle,/seven.taboldstyle/five.taboldstyle,/eight.taboldstyle/nine.taboldstyle,/one.taboldstyle/one.taboldstyle/nine.taboldstyle area, /nine.taboldstyle,/one.taboldstyle/seven.taboldstyle,/one.taboldstyle/zero.taboldstyle/six.taboldstyle,/one.taboldstyle/one.taboldstyle/eight.taboldstyle,/one.taboldstyle/two.taboldstyle/zero.taboldstyle,/one.taboldstyle/four.taboldstyle/three.taboldstyle symmetric bilinear function, /nine.taboldstyle/four.taboldstyle matrix, /eight.taboldstyle/five.taboldstyle,/eight.taboldstyle/eight.taboldstyle tangent bundle, /eight.taboldstyle/seven.taboldstyle hyperplane, /seven.taboldstyle/five.taboldstyle line, /one.taboldstyle,/one.taboldstyle/four.taboldstyle,/seven.taboldstyle/five.taboldstyle,/seven.taboldstyle/six.taboldstyle,/eight.taboldstyle/six.taboldstyle INDEX /one.taboldstyle/six.taboldstyle/three.taboldstyle map, /eight.taboldstyle/six.taboldstyle plane, /one.taboldstyle/five.taboldstyle,/seven.taboldstyle/five.taboldstyle,/eight.taboldstyle/six.taboldstyle space, /one.taboldstyle,/one.taboldstyle/five.taboldstyle,/seven.taboldstyle/four.taboldstyle,/seven.taboldstyle/five.taboldstyle,/eight.taboldstyle/one.taboldstyle–/eight.taboldstyle/four.taboldstyle,/eight.taboldstyle/seven.taboldstyle,/eight.taboldstyle/eight.taboldstyle,/nine.taboldstyle/seven.taboldstyle,/one.taboldstyle/zero.taboldstyle/four.taboldstyle, /one.taboldstyle/zero.taboldstyle/five.taboldstyle,/one.taboldstyle/one.taboldstyle/four.taboldstyle vector, /five.taboldstyle/zero.taboldstyle,/seven.taboldstyle/four.taboldstyle,/eight.taboldstyle/eight.taboldstyle,/nine.taboldstyle/seven.taboldstyle,/one.taboldstyle/one.taboldstyle/four.taboldstyle Taylor, Brook (/one.taboldstyle/six.taboldstyle/eight.taboldstyle/five.taboldstyle–/one.taboldstyle/seven.taboldstyle/three.taboldstyle/one.taboldstyle), /one.taboldstyle/five.taboldstyle/zero.taboldstyle Taylor’s formula, /one.taboldstyle/five.taboldstyle/zero.taboldstyle tensor product, /one.taboldstyle/seven.taboldstyle,/nine.taboldstyle/four.taboldstyle,/nine.taboldstyle/eight.taboldstyle timelike, /two.taboldstyle/nine.taboldstyle topological manifold, /four.taboldstyle torus, /four.taboldstyle,/seven.taboldstyle/seven.taboldstyle n-torus, /seven.taboldstyle trace, /eight.taboldstyle/seven.taboldstyle trajectory, /one.taboldstyle/six.taboldstyle,/eight.taboldstyle/seven.taboldstyle transformation law, /nine.taboldstyle/zero.taboldstyle transpose of a matrix or a vector, /two.taboldstyle/six.taboldstyle,/three.taboldstyle/seven.taboldstyle,/eight.taboldstyle/one.taboldstyle,/nine.taboldstyle/three.taboldstyle, /one.taboldstyle/four.taboldstyle/five.taboldstyle transposition, /four.taboldstyle/five.taboldstyle tuple, /one.taboldstyle,/three.taboldstyle/four.taboldstyle,/seven.taboldstyle/three.taboldstyle,/nine.taboldstyle/three.taboldstyle,/nine.taboldstyle/seven.taboldstyle,/one.taboldstyle/zero.taboldstyle/three.taboldstyle,/one.taboldstyle/zero.taboldstyle/six.taboldstyle,/one.taboldstyle/three.taboldstyle/seven.taboldstyle turning number of regular closed path, /six.taboldstyle/zero.taboldstyle two-sided inverse, seeinverse unit circle, seecircle cube, /three.taboldstyle/three.taboldstyle,/three.taboldstyle/seven.taboldstyle,/four.taboldstyle/five.taboldstyle,/six.taboldstyle/four.taboldstyle,/seven.taboldstyle/two.taboldstyle,/one.taboldstyle/four.taboldstyle/one.taboldstyle interval, /three.taboldstyle/one.taboldstyle,/three.taboldstyle/seven.taboldstyle,/six.taboldstyle/four.taboldstyle,/nine.taboldstyle/nine.taboldstyle,/one.taboldstyle/two.taboldstyle/four.taboldstyle,/one.taboldstyle/two.taboldstyle/six.taboldstyle normal vector field, /one.taboldstyle/zero.taboldstyle/four.taboldstyle,/one.taboldstyle/zero.taboldstyle/seven.taboldstyle,/one.taboldstyle/zero.taboldstyle/nine.taboldstyle,/one.taboldstyle/one.taboldstyle/zero.taboldstyle,/one.taboldstyle/one.taboldstyle/four.taboldstyle, /one.taboldstyle/one.taboldstyle/eight.taboldstyle–/one.taboldstyle/two.taboldstyle/zero.taboldstyle,/one.taboldstyle/three.taboldstyle/zero.taboldstyle sphere, seesphere square, /three.taboldstyle/seven.taboldstyle,/six.taboldstyle/six.taboldstyle,/six.taboldstyle/nine.taboldstyle vector, /five.taboldstyle/four.taboldstyle,/one.taboldstyle/four.taboldstyle/five.taboldstyle vector, see also column, length, position, row, tangent, unit field, /two.taboldstyle/five.taboldstyle,/two.taboldstyle/nine.taboldstyle,/three.taboldstyle/zero.taboldstyle,/five.taboldstyle/zero.taboldstyle,/five.taboldstyle/nine.taboldstyle,/seven.taboldstyle/two.taboldstyle,/nine.taboldstyle/two.taboldstyle,/one.taboldstyle/zero.taboldstyle/seven.taboldstyle,/one.taboldstyle/one.taboldstyle/eight.taboldstyle,/one.taboldstyle/two.taboldstyle/zero.taboldstyle, see also conservative, curl, divergence, gradient, index, potential, unit normal velocity, /one.taboldstyle/zero.taboldstyle,/six.taboldstyle/zero.taboldstyle,/seven.taboldstyle/seven.taboldstyle,/one.taboldstyle/four.taboldstyle/five.taboldstyle Volterra, Vito (/one.taboldstyle/eight.taboldstyle/six.taboldstyle/zero.taboldstyle–/one.taboldstyle/nine.taboldstyle/four.taboldstyle/zero.taboldstyle), /one.taboldstyle/six.taboldstyle,/eight.taboldstyle/seven.taboldstyle volume change, /three.taboldstyle/eight.taboldstyle,/four.taboldstyle/four.taboldstyle element, /one.taboldstyle/seven.taboldstyle,/one.taboldstyle/zero.taboldstyle/six.taboldstyle Euclidean, seeEuclidean volume form, /seven.taboldstyle/two.taboldstyle,/one.taboldstyle/zero.taboldstyle/five.taboldstyle,/one.taboldstyle/zero.taboldstyle/six.taboldstyle,/one.taboldstyle/one.taboldstyle/one.taboldstyle of a hypersurface, /one.taboldstyle/zero.taboldstyle/nine.taboldstyle onRn,/one.taboldstyle/eight.taboldstyle,/two.taboldstyle/four.taboldstyle,/four.taboldstyle/four.taboldstyle on a hypersurface, /one.taboldstyle/one.taboldstyle/zero.taboldstyle,/one.taboldstyle/two.taboldstyle/one.taboldstyle,/one.taboldstyle/three.taboldstyle/zero.taboldstyle of a block, /one.taboldstyle/eight.taboldstyle,/one.taboldstyle/zero.taboldstyle/two.taboldstyle of a manifold, /nine.taboldstyle/seven.taboldstyle,/one.taboldstyle/one.taboldstyle/eight.taboldstyle,/one.taboldstyle/two.taboldstyle/zero.taboldstyle of a parallelepiped, /three.taboldstyle/one.taboldstyle,/nine.taboldstyle/nine.taboldstyle,/one.taboldstyle/zero.taboldstyle/zero.taboldstyle of a simplex, /one.taboldstyle/five.taboldstyle/two.taboldstyle wave operator, /three.taboldstyle/zero.taboldstyle wedge product, /one.taboldstyle/seven.taboldstyle,/nine.taboldstyle/four.taboldstyle,/nine.taboldstyle/eight.taboldstyle winding number of closed path, /five.taboldstyle/eight.taboldstyle,/five.taboldstyle/eight.taboldstyle–/five.taboldstyle/nine.taboldstyle,/one.taboldstyle/two.taboldstyle/eight.taboldstyle,/one.taboldstyle/three.taboldstyle/one.taboldstyle of hypersurface, /one.taboldstyle/three.taboldstyle/one.taboldstyle work, /one.taboldstyle/seven.taboldstyle,/two.taboldstyle/six.taboldstyle,/five.taboldstyle/zero.taboldstyle,/five.taboldstyle/two.taboldstyle,/six.taboldstyle/eight.taboldstyle,/one.taboldstyle/one.taboldstyle/eight.taboldstyle,/one.taboldstyle/one.taboldstyle/nine.taboldstylezero of a vector field, /two.taboldstyle/five.taboldstyle