Sja Ch 7 meta notes

Meta Notes for Chapter 7: k-forms on Manifolds 8.18.15 7.1 First definition (of k-form on M) The idea here is to think of a manifold as being covered by a set of embeddings: ψi:Ui→ Vi fi = ψi* f // local representative fi is a pullback of f in the i-world We then have the case of overlap between two regions which leads to f(x) = fi(t) = fj(u) t in Ui u in Uj This leads to a series of statements like fj = (ψi-1 o ψj)*fi // how fi of two regions which overlap are related When this is generalized from the 0-form fi to a k-form αi we get αi = ψi* α αj = (ψi-1 o ψj)*αi which is p 82 (7.1) In all these equations the * indicates a pullback operation. Finally we note that if αi = ΣI fIdtI αj = ΣJ gJdtJ then we can write gJ = ΣI fI(φ(t)) det(Dφ)I,J = ΣI φ*fI(t) det(Dφ)I,J where φ = ψi-1 o ψj No examples are given, this section just gives an approach of breaking things down into separate embeddings i (recall that a manifold M is patched together from such a set of embeddings ψi). By pondering overlap, one obtains strange pullback type relationships between the fi or the αi of two regions. Nothing is too mysterious here, the reader just doesn't see the idea applied to anything and has to trust that it is useful. 7.2 Second definition (of k-form on M) This section is immensely more complicated and perhaps is a presentation of the basics of the Grassman theory of differential forms. Two abstract mathematical tools are used. I was unfamiliar with both of them, so that slowed me down. 1. The concept of a dual space V* of a vector space being the space of linear functionals on the vector space. Two examples are given. Discrete. If V = Rn (or some subset of Rn) with basis vectors vi, then V* = Rn* is the usual space whose vectors are "row vectors". For examples, suppose v = Σicivi Then we can regard ci(v) as a linear functional which maps Rn to R and which gives the coefficients of the expansion of v. ci(v) = viTv in matrix notation or vi v in dot notation. If you suppress the argument of ci(v) you then just have ci = viT . Note that ci is a linear functional, while viT is a row vector. ci = viT ci = a linear functional viT = a row vector This is at first very confusing! How can you have ci a scalar equal to the row vector viT? Well, in the equation Σicivi it is true that ci is a scalar and you could make a vector c out of these scalars, and also a row vector cT. But we have overloaded notation here. The other meaning of ci (and the main meaning) is that ci is a functional (not a scalar number) which acts on vectors in Rn. In this context, ci all by itself is a row vector in the dual space of V. ci = viT linear functional represented as a row vector ci(v) = viTv = vi v linear functional acts on a vector v Sja likes to refer to this ci(v) as λi(v) perhaps to match historical notation, or perhaps to make the distinction between ci as a number and λi as a functional. The λi(v) functionals are called coordinate functions I guess since there is one for each coordinate i. We can repeat things above in this notation v = Σiλi(v)vi // expansion λi = viT // linear functional number i = a row vector Notice that λi all by itself is a row vector , it is not λ which is a row vector. It is easy to show that these functionals λi form a basis in the dual space V*. This basis {λi} is called the dual basis of basis {vi}. The dimensionality of the dual space V* is the same as that of the original space V. It is easy also to show that λi(vj) = δi,j page 84 A vi = basis vectors of V Sjamaar confusion of notation: (1) The notation vi is used sometimes to indicate a basis vector, sometimes an arbitrary vector in Rn. (2) Notation vi indicates a vector, whereas (v)i is the scalar component of a vector v. These two objects are easily confused. In physics one writes the latter as vi without bolding so I guess there is distinction. Suppose you take the particular basis vi = ei for Rn which are the "unit vectors". Then equations above become v = Σiλi(v)ei // expansion λi = eiT // linear functional number i = row vector eiT Continuous. To get a continuum example we could write V = space of functions {f} and f = Σicifi f, fi = real functions on some interval ci(f) = ∫dx f(x) si(x) si = some weight function, element of V Here ci is then a linear functional of the vector f which is a function. I was happy to get to quote the Riesz Representation Theorem which says every linear functional can be associated with some function in the space V, and in the above ci(f) is associated with a weight function si(x) in V, and in the discrete example, the linear functional ci(v) is associated with the vector c of Rn. Since there are an infinite number of si(x) weight functions, the dimensionality of the dual space is ∞, the same as that of the functions space V. I did some reading elsewhere to get this dual space stuff understood. In that scenario, a vector in V is called a vector, whereas a vector in the dual space V* is called a covector. This has nothing to do with covariant or contravariant vectors as far as I can tell. It is just the idea v = Σicivi that a linear combination of vectors partnered with covectors gives you an arbitrary vector in V. It reminds me of how matrix elements and their "cofactors" get added to make a determinant. 2. The second abstract math idea is that of multilinear functions. When this maps to reals, you have a multilinear functional, and that is the obvious generalization of a regular linear functional from one vector argument to k vector arguments. A multilinear function has the usual two properties of a linear function but separately in each of its arguments. Very simple idea. λ(v1,v2, αv3, v4) = α λ(v1,v2, v3, v4) // scalar rule λ(v1,v2, u + w, v4) = λ(v1,v2, u, v4) + λ(v1,v2, w, v4) // addition rule Sja likes to combine these into a single rule like this λ(v1,v2, αu + βw, v4) = αλ(v1,v2, u, v4) + βλ(v1,v2, w, v4) page 85A Of particular interest are those multilinear functions which happen to be "alternating" in all arguments. An example is f(v1,v2...vk) = det(v1,v2...vk). The space of such alternating multilinear functions with k arguments is called AVk . It is shown that this space is in fact a vector space due to the obvious scalar and addition rules. The space AVk is very much like a dual space, but it is not clear what the corresponding underlying space would be. If you think of the starting space as being the direct product space Vk, then maybe the dual space would be the space of multilinear functionals with k arguments, but the alternating property would not be there. So much then for these two math ideas: dual spaces and alternating multilinear functions. Sja then goes on to define a function whose name is not f or g but "λ1λ2....λk" . Here is that definition: "λ1λ2....λk" = det[λi(vj)] where the matrix is λ1(v1) λ1(v2) λ1(v3) ..... λ1(vk) λ2(v1) λ2(v2) λ2(v3) ..... λ2(vk) ... λk(v1) λk(v2) λk(v3) ..... λk(vk) Here these λi(v) are the coordinate functions described just above in the dual space notes. In this definition, we happen to have 1,2,3... in order, but you could think of them in any order, and in fact you could say "λiλi....λi" = det[λi(vj)] where the matrix is λi(v1) λi(v2) λi(v3) ..... λi(vk) λi(v1) λi(v2) λi(v3) ..... λi(vk) ... λi(v1) λi(v2) λi(v3) ..... λi(vk) Notice that the det[...] notation is not very clear until you actually write out the matrix. We think of k ≤ n where n is from Rn. Notice that "λiλi....λi" is an element of the space AVk since it is in fact an alternating multilinear function of order k. As a shorthand, you could write the above as "λI" = "λiλi....λi" = det[λi(vj)] I = multi-index i1i2....ik Or, you could state that I was an ordered multi-index, same notation for both. Sja refers this as an increasing multi-index which is I guess clearer. We know from earlier that λi(vj) = δi,j . Sja shows then on page 86 this interesting generalization: λI(eJ) = δI,J where I and J are increasing multi-indices of order k This is a very dense notation! If you write it out, it really says "λiλi....λi"(ej,ej....ej) = δi,j δi,j ..... δi,j Here I and J are both increasing multi-indices of order k. I don't think this claim is true if you make them arbitrary multi-indices, but I don't know and that is not our interest here. We then arrive at this claim: Theorem 7.14: The set of λI multilinear alternating functionals for all increasing multi-indices I forms a complete basis for the space AVk of general alternating multilinear functionals. The proof is easy once you have shown that λI(eJ) = δI,J . Now I have intentionally deferred the next four very important claims of Sjamaar which serve to conclude the Chapter: Claim 1. (dxiconnection) If we take {ei} as our basis of interest in V = Rn, we showed above that λi = eiT = linear functional = row vector in the dual space Now we suddenly associate this functional or row vector with the symbol "dxi" which somehow is associated with our differential forms differential object dxi. So we now suddenly write dxi = λi = eiT = linear functional = row vector in the dual space and so we now have three notations for the same thing. Then we can write for example "dxI" = "dxidxi....dxi" = det[dxi(vj)] I = multi-index i1i2....ik and this function name in quotes is reminiscent of the products we had in our differential forms in Ch 2. So many equations above can be restated using this new dxi symbol! For example, dxI(eJ) = δI,J I.J both increasing multi-indices So this claim is what starts to make a connection between the math of this chapter and differential forms. Claim 2. The wedge product notation Recall that we defined the function name "λiλi....λi" = det[λi(vj)] which we can now write as "dxidxi....dxi" = det[dxi(vj)] Instead of writing out this long name which looks like a product of factors but really is not, we can write things in this way (λi ˄ λi.... ˄ λi)(v1,v2, ...vk) = det[λi(vj)] (dxi ˄ dxi.... ˄ dxi)(v1,v2, ...vk) = det[dxi(vj)] This then is the definition of the "wedge product" of k linear functionals to form a k-multilinear alternating functional. Various properties of the wedge product are then immediate: (a) if two factors in the wedge product are the same, the wedge product vanishes (because in this case the matrix has two rows which are the same) (b) the wedge product of a dxi with itself must vanish (for same reason) Examples for k = 2 For the case k = 2 we find that (λ1 ˄ λ2)(v1, v2) = det [ ] = det(a, b) where a = and b = which are the two column vectors. I know from tensor doc that this quantity det(a,b) is in fact the area of a 2-piped spanned by a and b. Now if we assume the usual ei basis we could write a = = = = v1 Then we have (λ1 ˄ λ2)(v1, v2) = det(v1,v2) Note: So far, the wedge product is between linear functionals in V*. One could regard two row vectors aT and bT as linear functionals. Then you could say (aT ˄ bT)(v1, v2) = det [ ] which does not seem very exciting. As s special case (aT ˄ bT)(e1, e2) = det [ ] = det [ ] = det [ ] = det(a,b) = number = area of 2-piped. Comments. So far, a 2-wedge product is only defined between two linear functionals. I can regard these as being the two row vectors aT and bT. The value of a 2-wedge product is unspecified unless you display the 2-wedge product arguments! Claim 3. Connection to k-forms : Since the dxI form a basis in AVk, the most general element of AVk can be written α = ΣI fI dxI sum over increasing multi-indices I or α(v1,v2, ...vk) = ΣI fI dxI(v1,v2, ...vk) Due to the earlier noted fact that dxI(eJ) = δI,J we can show that fI = α(eI) Then we can rewrite the above as α = ΣI α(eI) dxI sum over increasing multi-indices I or α(v1,v2, ...vk) = ΣI α(eI) dxI(v1,v2, ...vk) In the first notation, Sjamaar "tacks on" x in order to make a connection to Chapter 2: αx = ΣI fI(x) dxI sum over increasing multi-indices I or αx(v1,v2, ...vk) = ΣI fI(x) dxI(v1,v2, ...vk) fI(x) = αx(eI) So finally αx = ΣI αx(eI) dxI αx(v1,v2, ...vk) = ΣI αx(eI) dxI(v1,v2, ...vk) Claim 4. In the parallel Grassmann World view of differential forms, we have these pullback facts: φ*[αy(eI)] = αφ(x)(eI) // action on the coefficient [φ*dyI(v1,v2.....vk)] = dyI( (Dφ)v1, (Dφ)v2, .,,,, (Dφ)vk) // action on the differential We then find that φ*αy(v1,v2.....vk) = αφ(x)((Dφ)v1,(Dφ)v2.....(Dφ)vk) // page 88A He and I show that if you make this definition of gJ in the Grassmann World, gJ = φ*αy(eJ) // which Sja writes as gJ = (φ*α )x(eJ) then you find that gJ = ΣI fI(φ(x)) det(Dφ)I,J which is the same as the Chapter 2 gJ. Forms on Manifolds. Finally we come to the main topic of this chapter! Back in Chap 2 we learned about forms defined on RN , a whole space, or perhaps an open subset of the space. In the current Chap 7 the goal is to show how you deal with forms which are defined not on RN, but on some surface within RN . Rather than try to do this for an arbitrary surface in RN, we do this for a surface which has the properties of being a manifold within RN. Perhaps that manifold is an n-dimensional surface. We know there is a set of embeddings ψi which together characterize this manifold and its "cushion" at every point x which allows differentiation in all directions. We are able on this manifold to define differential forms αx where the k in k-form is ≤ n. The Grassmann theory produces k-forms which are alternating multilinear functions of k vectors. In the final step, when x is on a manifold, we take these vectors to lie in the tangent space TxM which we know has a local Rn of n dimensions. This tangent space is of course different at every x, and so we cannot have just a single Rn to deal with, but a continuum of local Rn spaces TxM. The arguments of αx must be vectors in this local Rn tangent space. Example 7.17. In this case M = a curve, αx(v) has only one vector since k = n = 1 of interest here. An example of a 1-form is αx(v) = ± || v(x) ||. The tangent space TxM has only one vector v. As we saw in Buck, this form is related to arc length. Remember: you always evaluate a k-form on M at a set of vectors which lie in the tangent space. This whole idea sounds like general relativity and I think I am going to be happy I learned about manifolds now and not at some future time. The final section on page 89 shows that this Grassmann theory of forms on manifolds produces exactly the same relationship between local representative αi that we conjectured in Section 7.1. Here they are derived from the theory, not conjectured. So ends the chapter. There is much still to digest!! I am now ready to start into Chapter 8 which is already printed.