section 10_10 v 1

Section 10.10 v1 PhL 4.29.16 10.10 Integration of functions over surfaces and curves In Section 10.11 we are going to make this claim concerning the integration of an arbitrary differential k-form over a manifold "surface" x' = F(x) embedded in Rm : α' = Σ'I fI(x') λ'^I = Σ'I fI(x') dx'i ^ dx'i .... ^ dx'i // the k-form in x'-space ∫F α' = Σ'I Σ'J∫[0,1]k fI(F(x)) det(RIJ(x)) dxj ^ dxj ^ ...^ dxj R= (DF) = Σ'I Σ'J (!Syntax Error, I!Syntax Error, I ..... !Syntax Error, I) fI(F(x)) det(RIJ(x)) dxjdxj....dxj . (10.10.1) The general idea is that the k-form α' in dual x'-space is first pulled back to a different k-form in dual x-space, and then the wedge product dxj ^ dxj ^ ...^ dxj of basis functionals appearing in this pulled back k-form is mysteriously replaced by a product of ordinary calculus differentials dxjdxj....dxj . The end result is that ∫F α' is some calculus-computable real number. We assume that the manifold M can be covered by a single mapping x' = F(x), otherwise we create at atlas of mappings as described at the end of Section 10.2. Before delving into this subject, it seems useful to have some discussion of integrals of functions over surfaces and curves in R3 without any mention of differential forms. This discussion takes the form of a set of seemingly simple examples. INTEGRATION OVER SURFACES Example 1: Compute the average temperature on a flat plate S' in the z' = 0 plane in R3 (x'-space). (10.10.2) What is the meaning of "average temperature"? We partition the plate into a large array of N x N tiny squares of equal area dA'i = Δx'Δy', and measure the temperatures Ti simultaneously in all N2 locations. The average temperature is then <T> = (1/A') limN→∞ [ΣiTidA'i] where A' = ab is the area of the plate. So this is an area-weighted average temperature which is cast into a standard-issue 2D Riemann integral, <T> = (1/A') ∫S'dA' T(x') = (1/ab)!Syntax Error, Idx'!Syntax Error, Idy' T(x',y',0) . dA' = dx'dy' (10.10.3) This same kind of integral would be used to compute the average mass density of a flat plate which has areal mass density ρ(x',y') <ρ> = (1/A') ∫S'dA' ρ(x',y') . (10.10.4) One could compute the center of mass location of a plate of mass m with similar integrals, <x'> = (1/m) ∫S'dA' x' ρ(x',y') <y'> = (1/m) ∫S'dA' y' ρ(x',y') m = ∫S' dA' ρ(x',y') . (10.10.5) In this Example we put primes on the variables because they exist in x'-space. There is no need to do any "pulling back" of the area element dA' = dx'dy' to some x-space. The integral is done directly in x'-space. Example 2: Compute the average normal component of a magnetic field B on the same flat plate. (10.10.6) This problem is similar to the temperature problem with T → Bn' and the result is <Bn'> = (1/A') ∫S'dA' Bn'(x') = (1/A') ∫S'dA' B(x') ' = (1/A')∫S'dA' B(r'), dA' = dA' ' (10.10.7) where in this problem it happens that ' = unit normal vector, normal to the surface of the plate at (x',y',0) = ' = constant . Then <Bn'> = <Bz'> = (1/ab)!Syntax Error, Idx'!Syntax Error, Idy' Bz(x',y',0) . (10.10.8) For both these examples, one could consider a round plate instead of a square plate, and then one would use dA' = dx'dy' → (r)drdθ where the Jacobian J = r appears. One could show that such differential area patches dA' cover the plate surface perfectly with no overlaps and no missed regions. Example 1a: Compute the average temperature on a spherical shell of radius R in R3. Example 2a: Compute the average normal component of a magnetic field B on this shell. (10.10.9) Treating this smooth surface as behaving locally like a flat plate, we use the general expressions (10.10.3) and (10.10.7) obtained for Examples 1 and 2 above, <T> = (1/A') ∫S'dA' T(x') <Bn'> = (1/A') ∫S'dA' B(x') ' . (10.10.10) In spherical coordinates, A' = 4πR2, ' = and dA' = R2sinθdθdφ. This area measure can be deduced by looking at a picture of spherical coordinates where dA' = (Rdθ)(Rsinθdφ) is a surface patch. Then, <T> = (1/4π) !Syntax Error, Idφ !Syntax Error, Idθ sinθ T(Rsinθcosφ, Rsinθsinφ, Rcosθ) <Bn'> = <Br> = (1/4π) !Syntax Error, Idφ !Syntax Error, Idθ sinθ Br(Rsinθcosφ, Rsinθsinφ, Rcosθ) . (10.10.11) In the language of our earlier sections, we can think of this surface being defined by an underlying transformation x' = F(θ,φ) : x' = Rsinθcosφ y' = Rsinθsinφ z' = Rcosθ (10.10.12) where we would draw "parameter space" = Rn = R2 on the left (with coordinates θ and φ) and Rm = R3 on the right. So one has (θ,φ)-space on the left, and x'-space on the right. In writing dA' = R2sinθdθdφ, we are "pulling back" an area patch on the sphere in x'-space to a rectangular area dθdφ in (θ,φ)-space, and we pick up an area conversion factor R2sinθ. Similarly, the functions T and Br are "pulled back" so they are written in the form T(F(θ,φ) and Br(F(θ,φ). Although nothing has been said about "differential forms", one suspects that this example can somehow be cast into a 2-form scenario. Comment: We hope the reader will overlook the fact that if B really is a magnetic field, then <Bn'> = 0 when integrated over any closed surface S' (like a spherical shell) due to the divergence theorem and the non-existence of magnetic monopoles, div B = 0. The concerned reader can think of B as some other vector field. Example 1b: Compute the average temperature on an arbitrary smooth surface S' in x'-space. Example 2b: Compute the average normal component of a magnetic field B on such a surface. (10.10.13) Start again with (10.10.10), <T> = (1/A') ∫S' dA' T(x') <Bn'> = (1/A')∫S' dA' B(x') ' = (1/A')∫S dA' B(x') , dA' = dA' ' . (10.10.10) The meaning of these integrals is clear: dA' is a local area element at point x' on the surface, ' is a local unit normal at a point x' on the surface, and A' is the total area of the surface. One just has to figure out what these quantities are for a given surface. Notice that ∫S' dA' B(x') is the classic "surface integral of a vector field" as one might encounter in an electrostatic flux calculation (B = E) or in a fluid flow situation (B = v). At point x' on the surface there is a tangent space Tx'M (Section 10.2) which is spanned by the tangent base vectors u'1 and u'2 which appear in the kinematics package (10.6.a.1). Recall that these vectors are generally not orthogonal. The magnitude of the area of the 2-piped subtended by these vectors is |u'1 x u'2|. But we want a differential 2-piped at point x' with some small extents dξ1 and dξ2 in these two directions, so then dA' = | (dξ1u'1) x (dξ2u'2) | = dξ1dξ2 | u'1 x u'2| . Meanwhile, we know from (10.6.e.2) that u'1 = Ru1 and u'2 = Ru2, these being vector transformations under x' = F(x). Therefore, R(dx1u1) = dx1u'1 R(dx2u2) = dx2u'2 . (10.10.14) Thus, the small rectangle spanned by (dx1u1,dx2u2) in x-space is mapped into a small 2-piped spanned by (dx1u'1,dx'2u2) in x'-space. We can take this 2-piped to be the 2-piped discussed above by setting dξ1 = dx1 and dξ2 = dx2 and then we have dA' = | u'1 x u'2 | dx1dx2. Recall from (10.6.e.3) that u'3 is constructed "as needed" so as to form a complete basis for R3 at point x' on the surface S'. We can take u'3 = u'1 x u'2 and then u'3 can be identified with n', a normal vector at point x' on the surface. We then need to know that magnitude of this vector to know dA'. Since Rm = R3 is a Cartesian space, up and down vector component indices are the same, so (implied sums) | u'3|2 = | n' |2 = | u'1 x u'2 |2 = ( u'1 x u'2) ( u'1 x u'2) = ( u'1 x u'2)i( u'1 x u'2)i = [εiab (u'1)a(u'2)b] [εicd (u'1)c(u'2)d] = εiabεicd(u'1)a(u'2)b(u'1)c(u'2)d = (δacδbd - δadδbc) (u'1)a(u'2)b(u'1)c(u'2)d // see e.g. Tensor (D.10.22) = (u'1)a(u'2)b(u'1)a(u'2)b – (u'1)a(u'2)b(u'1)b(u'2)a = Ra1Ra2 Ra1Rb2 – Ra1Rb2 Rb1Ra2 // kin. package (10.6.a.1) item (e) = Σa (Ra1)2 Σb (Rb2)2 – (ΣaRa1Ra2) (ΣbRb1Rb2) = [Σa (Ra1)2] [Σa (Ra2)2] – [ΣaRa1Ra2]2 ≡ [ K(x) ]2 // since Rij = Rij(x) in general or | u'3| = | n'| = K(x) = . (10.10.15) We could have used the vector identity (A x B) (A x B) = A2B2 – (AB)2 in place of the εε product method, but εε products are good to know about and we give a reasonable source above for the reader interested in their generalizations. Finally then we have an expression for differential area dA', dA' = | u'1 x u'2 | dx1dx2 = | u'3| dx1dx2 = | n' | dx1dx2 = K(x) dx1dx2 . (10.10.16) The vector n' = u'3 has the following components. (n')i = (u'1 x u'2)i = εiab (u'1)a(u'2)b = εiab Ra1Rb2 so (n')1 = ε1ab Ra1Rb2 = R21R32 - R31R22 = det = (n')2 = ε2ab Ra1Rb2 = R31R12 - R11R32 = det = (n')3 = ε3ab Ra1Rb2 = R11R22 - R21R12 = det = . (10.10.17) On the far right we use a common Jacobian-like notation for the 2x2 determinants, where recall from (2.1.2) that Rij ≡ (∂x'i/∂xj). We thus obtain this alternate expression for K2, K2 = | n'|2 = det2 + det2 + det2 = [ ]2 + [ ]2 + [ ]2 . (10.10.18) From (10.10.17) the vector n' and the unit normal vector ' = n' / |n'| = n'/ K may then be written, n' = ( , , ) ' = ( , , ) . (10.10.19) Using (10.10.18) for dA' the solutions to our problems are, <T> = (1/A') ∫S' dA' T(x') = (1/A') ∫S T(F(x)) K(x) dx1dx2 <Bn'> = (1/A')∫S' dA' B(x') ' = (1/A')∫S B(F(x)) ' K(x) dx1dx2 = (1/A')∫S B(F(x)) n' dx1dx2 (10.10.20) where ' and K(x) are as shown above. Notice that the resulting integral is over the region S in x-space which maps into the surface S' in x'-space under x' = F(x). The area A' is given by A' = ∫S' dA' = ∫S K(x) dx1dx2 . (10.10.21) The scalar integral shown in (10.10.20) appears on Buck page 368 (7-3) where T = f, S' = Σ, S = D, and where x1,x2 = u,v. The vector integral appears on p 403 where B = F . The reader will no doubt notice that in writing dA' = K(x) dx1dx2 we are in fact "pulling back" some tilted non-rectangular 2-piped patch dA' on the surface S' in x'-space to a rectangular patch dx1dx2 in x-space and in doing so we pick up a Jacobian-like factor K(x). We are also "pulling back" the integrand functions T(x') and B(x') by writing them as T(F(x)) and B(F(x)). Again we arrive at this "pulling back" concept without ever mentioning "differential forms". The pullback integrals shown above are completely well-defined and it is then just a matter of doing the integrals analytically or numerically. Recall that the square of the area transformation factor K is given by either K2 = [ Σa=13 (Ra1)2] [Σa=13 (Ra2)2] – [Σa=13Ra1Ra2]2 . (10.10.15) or K2 = det2 + det2 + det2 . (10.10.18) In the second form K2 is the sum of the squares of the three 2 x 2 minors of the 3 x 2 "tall" R matrix. See for example Buck page 299 where K = k and Rij = aij. The two expressions above for K2 look totally unrelated and it seems strange that they are equal. It turns out that K2 can be written in yet another way, K2 = det(RTR) (10.10.22) where RT is the "matrix transpose" of R and not the "covariant transpose" discussed in Section 2.11 (f). Recall from Fig (10.6.c.1) that RTR is a square n x n matrix and therefore has a determinant. Lest one have doubts, we have Maple compute K2 in all three ways and show that the three results are the same: Create a general 3 x 3 R matrix: Compute K2 using (10.10.15) and call it K2a: Extract the three 2x2 submatrices from R and call them A,B,C: Compute K2 using (10.10.18), call it K2b: Compute the matrix RTR: Compute K2 using (10.10.22), call it K2c: Show that all three K2 expressions are the same: Later we shall show more generally how these various expressions for K2 arise. INTEGRATION OVER CURVES Example 3: Compute the average temperature on a piece of straight wire C' of length a in R3 in x'-space. (10.10.23) Let ' be a unit vector which is tangent to the wire at some point x' on the wire. Let dx' be an arbitrary differential distance vector whose tail is located at position x' on the wire. Then ds' = dx' ' is a small distance along the wire. In analogy with the flat plate of Example 1, the length-weighted average temperature of a straight wire is <T> = (1/L')∫C'ds' T(x') ds' = dx' ' . (10.10.24) In this particular example, the wire is placed on the x' axis so L' = a, ' = ' , ds' = dx' '= dx' . Then, <T> = (1/a)!Syntax Error, Idx' T(x',0,0) . (10.10.25) Example 4: Compute the average tangential magnetic field B on this same straight wire C'. (10.10.26) This problem is similar to Example 2 (but ' → ') and the solution is <Bt'> = (1/L') ∫C'ds' Bt'(x') = (1/L') ∫C'ds' B(x') ' , ds' = dx' ' = dx' = (1/L') ∫C'dx' B(x') , ds' ' = dx' . (10.10.27) so <Bt'> = (1/a)!Syntax Error, Idx' Bx'(x') . (10.10.28) Notice that ∫C'dx' B(x') is the classic form of a "line integral of a vector field". Example 3a: Compute the average temperature on a ring of wire C' of radius R in the x',y' plane of R3. Example 4a: Compute the average normal component of a magnetic field B on this ring. (10.10.29) Treating this smooth curve as behaving locally like a straight wire, we use the general expressions (10.10.24) and (10.10.27) obtained for Example 3 and 4 above, <T> = (1/L')∫C'ds' T(x') <Bt'> = (1/L') ∫C'dx' B(x') . (10.10.30) The ring is assumed centered at the origin of the x',y' plane so we use cylindrical coordinates with z' = 0, which then is just polar coordinates, so ' = , dx' = Rdθ , ds' = dx' ' = Rdθ, and L' = 2πR. Then, <T> = (1/2π) !Syntax Error, Idθ T(Rcosθ,Rsinθ,0) <Bt'> = (1/2π)!Syntax Error, Idθ Bθ(Rcosθ,Rsinθ,0) (10.10.31) where the last argument of the integrand functions indicates z' = 0 for our placement of the ring in R3. Again, the differential distance element ds' = Rdθ is being "pulled back" from x'-space = R3 to θ-space = R1, and the integrand functions are pulled back according to T(F(θ)) and Bθ(F(θ)) where x' = F(θ) : x' = Rcosθ y' = Rsinθ z' = 0 . (10.10.32) Example 3b: Compute the average temperature on an arbitrary smooth wire C' in R3. Example 4b: Compute the average normal component of a magnetic field B on this wire. (10.10.33) Start again with (10.10.30), <T> = (1/L')∫C' ds' T(x') = (1/L')∫C'dx' ' T(x') <Bt> = (1/L') ∫C'ds' B(x') ' = (1/L') ∫C'dx' B(x'), ds' = dx' ' . (10.10.34) The curve C' exists in x'-space Rm = R3 and we take x-space to be Rn = R1. Then curve C in x-space is just the line segment there from x1 = 0 to a and this maps into curve C' under x' = F(x). In other words, the curve C' in x'-space is being pulled back to a straight line segment C of x-space. To be consistent, we should be calling the x'-space curve F instead of C', and in Example 2b we should call the surface F instead of S', since in both cases the curve and surface are generated by x' = F(x), but we shall sacrifice consistency for clarity. The most pressing issue now is how to compute the unit tangent vector '. Reaching into our kinematics package (10.6.a.1) and nearby discussion, we realize that t' = u'1 , ' = u'1 / | u'1| . (10.10.35) This is because the tangent space Tx'M is spanned by the single tangent base vector u'1, while the other two vectors u'2 and u'3 are selected "as needed" to span the perp space to Tx'M in R3. We invent some differential distance dξ so that dx' = dξ u'1 = dξ t' points along the curve C' at point x'. We know from (10.6.e.2) that u'1 = Ru1, this being a vector transformation under x' = F(x). Therefore R(dx1u1) = dx1u'1 . (10.10.36) Thus, the small differential vector dx = dx1u1 in x-space (tangent to C) is mapped into a small differential vector dx' = dx1u'1 in x'-space, tangent to C' at point x' on C'. Thus we select dξ = dx1 and conclude that dx' = dx1 u'1 so ds' = |dx'| = | dx1 u'1| = | u'1| dx1 = | t' | dx1 . (10.10.37) The distance ds' in x'-space is thus being pulled back to distance dx1 in x-space with scaling factor | t' | . The components of the vector t' = u'1 are, from (10.6.a.1) item (e), (t')i = Ri1(x) t' = (R11, R21, R31 ) = ( (∂x'1/∂x1), (∂x'2/∂x1), (∂x'3/∂x1) ) (10.10.38) and then | t'| 2 = (R11)2 + (R21)2 + (R31)2 = (∂x'1/∂x1)2 + (∂x'2/∂x1)2 + (∂x'3/∂x1)2 ≡ K2(x) // a new and different K from that of Example 2b or | u'1| = | t' | = K(x) = (10.10.39) and then ds' = | t'| dx1 = K(x) dx1. // x = x1 (10.10.40) Just as in (10.10.15), (10.10.18) and (10.10.22), the factor K2(x) can be written three ways, K2(x) = (R11)2 + (R21)2 + (R31)2 = Σa=13(Ra1)2 K2(x) = det2(R11) + det2(R21) + det2(R31) // minors of R are all 1 x 1 K2(x) = det[RTR] = RTR = (R11, R21, R31 ) = (R11)2 + (R21)2 + (R31)2 or K2(x) = det[RTR] = (RTR)11 = Σa=13 (RT)1aRa1 = Σa=13 Ra1Ra1 = Σa=13(Ra1)2 . (10.10.41) Here we don't need a Maple program to verify that all three forms give the same result. Note that the "tall" R matrix is the 3x1 matrix shown on the second last line above. The solutions to our two exercise problems are then (we write <Bt'> in many equivalent ways), <T> = (1/L') ∫C'ds' T(x') = (1/L')!Syntax Error, Idx1 K(x) T(F(x)) <Bt'> = (1/L') ∫C'ds' B(x') ' = (1/L') ∫C'dx' B(x') = (1/L') ∫C dx B(F(x)) = (1/L')!Syntax Error, Idx1 B(F(x)) u1 = (1/L') !Syntax Error, Idx1 B(F(x)) t' = (1/L') !Syntax Error, Idx1 Bi(F(x)) (t')i = (1/L') !Syntax Error, Idx1 Bi(F(x)) Ri1(x) (10.10.42) where dx = dx1u1 // below (10.10.36) dx' = dx1 u'1 = dx1 t' // (10.10.37) and (10.10.35) ds' = | t' | dx1 = K(x) dx1 // (10.10.40) L' = ∫C'ds' = !Syntax Error, Idx1 K(x) = arc length of the curve C' in x'-space . (10.10.43) Notice that ∫C' dx' B(x') is the classic "line integral of a vector field". If the variable x1 = x were time t, then the above K2(t) = (∂x'1/∂t)2 + (∂x'2/∂t)2 + (∂x'3/∂t)2 could be interpreted as the square of the velocity of a particle moving along the curve C', K2(t) = v'12 + v'22 + v'32 = (v')2 K(t) = | v' | = | ∂tx' | = | ∂tF(t) | . (10.10.44) The scalar integral in (10.10.42) appears on Buck page 367 (7-1) where T = f, C' = γ , x = t, [0,a] → [a,b] and K = |∂tγ| so !Syntax Error, Idx1 K(x) T(F(x)) → !Syntax Error, Idt |∂tγ| f(γ(t)) . // Buck 367 (7-1) The vector integral in (10.10.42) appears for R2 on Buck page 376 (7-7) where B = (A,B), (x'1, x'2) = (φ,ψ), t' = ( (∂x'1/∂t), (∂x'2/∂t) ) = (∂tφ,∂tψ), and [0,a] → [a,b] so <Bt'> = !Syntax Error, Idx1 B(F(x)) t' → !Syntax Error, Idt [ A(γ(t)) (∂tφ) + B(γ(t)) ∂tψ ] . // Buck 376 (7-7) Comments on the above examples As will be seen formally in the next section, the surface and curve integrations discussed above fall into the realm of 2-form and 1-form integrations. In the above examples, there was no mention of "functionals" or "dual spaces" or "wedge products" or "cosmetic notation" or even of "differential forms". No mention was made of "surface orientation". The calculations were performed on an ad hoc basis as any journeyman might approach these problems. There was, however, some discussion of "pulling back" integrand functions and differential areas and differential lengths from Rm to Rn, but there was no mention of pulling back functionals between the corresponding dual spaces. The method of differential forms provides a systematic method for doing integrations over "surfaces" (manifolds) of any dimension embedded in a space of any same or larger dimension, where the spaces can have arbitrary metric tensors, and where orientation is tracked.