section 10.11

This is the Title PhL 1.11.15 10.11 Integration of differential k-forms over surfaces Using our cosmetic notation for k-form functionals, we write, αx' = Σ'I fI(x') dx'^I // original k-form in Λ'k(Rm) (10.8.15#6) F*(αx') = Σ'I fI(F(x)) Σ'J det(RIJ) dx^J // k-form pulled back into Λk(Rn) . (10.8.19) The pulled back k-form can be rewritten compactly as βx ≡ F*(αx') = Σ'J gJ(x) dx^J  where gJ(x) = Σ'I fI(F(x)) det(RIJ) . (10.11.1) Besides grouping terms into gJ(x) we have assigned the name βx to the pulled-back k-form. This pulled-back k-form is written out in detail in (10.8.4) with a display of the RIJ matrix. The point x' lies on a "surface" in x'-space which is generated by a defining transformation x' = F(x). A region we shall call S in x-space maps into a region S' on the manifold as shown in Fig (10.2.1) with U and V. The letter S suggests the word "Surface". The integral of the original k-form αx' over surface S' is then set equal to the integral of the pulled-back k-form F*(αx') over the pulled-back surface S, ∫S'αx' ≡ ∫S βx . (10.11.2) Below we shall refer to this as our first definition, as if the right side defines the meaning of the left side. That is to say, the integral of a k-form αx' over some complicated surface (manifold) in x'-space is reduced to an integral of a different k-form βx over a relatively simple surface in x-space. This is reminiscent of our examples in Section 10.10 where we had, for example, ∫S' dA' B(x') ' = ∫S B(F(x)) ' K(x) dx1dx2 (10.10.20) where ∫S' is over an arbitrary surface in x'-space while ∫S is a straightforward integral in Cartesian x-space. The big difference however is that in Section 10.10 we were dealing with calculus integrals, whereas here we are dealing with k-forms which are functionals in certain dual spaces. In the case of the calculus integral examples, one can regard the shift from x'-space to x-space as nothing more than a "change of variables" and there is no "first definition" of anything. We then come to our second definition which is this: ∫S βx = ∫S [ Σ'J gJ(x) dx^J ] ≡  ∫S Σ'J gJ(x) dxjdxj ... dxj (10.11.3) and we then end up the a well-defined multivariable calculus integral. One must ask: how is it that the functional dx^J = λ^J = λj ^ λj^ ... ^ λj = dxj ^ dxj ^ ... ^ dxj disappears and is replaced by dxjdxj ... dxj? This disappearance is papered over in some sources because the wedge product hats are suppressed and dxj and dxj are typeset identically. An answer to this question is the following. One writes βx(S) ≡ ∫S βx ≡ ∫S Σ'J gJ(x) dxjdxj ... dxj . (10.11.4) In this point of view, one regards the k-form βx as a functional which acts on regions of Rn to produce a real number so there is a mapping (a "functional" is any mapping to the reals), βx : S Rn → R . (10.11.5) This is a different kind of functional from the functional dx^J = λ^J ϵ Λk(Rn) which is a vector in the dual space shown. Whereas dx^J is a linear functional, βx(S) is not a linear functional, for example, since doubling the region S is not likely to double the resulting real number βx(S). This seems to be the approach taken by Loring Tu, where we quote from his p 263, (10.11.6) Here he is stating our "second definition". Buck also takes this approach, referring in his Definition on page 381 to a k-form ω as a "regional-functional". He writes as a 3-form example, ω = A(x,y,z) dxdydz // meaning A(x,y,z) dx^dy^dz ω(Ω) = ∫∫∫Ω A(x,y,z) dxdydz Ω = a region in the definition domain of ω Arm-waving comment: We know that an exterior derivative increases the rank of a k-form by one unit. It is not unreasonable then to say that a k-fold integration of a k-form reduces the rank of that k-form by k units down to rank 0, which is a scalar function and in the above situation just a number, the value of the integral. In any event, the end result for the integral of a k-form over a manifold region S' in x'-space is this: ∫S' αx' = ∫S'[Σ'I fI(x') dx'i ^ dx'i ^ ... ^ dx'i ] // αx' = Σ'I fI(x') dx'^I ≡ ∫S [ Σ'J gJ(x) dxj ^ dxj ^ ... ^ dxj ] // first definition (pullback) ≡ ∫S [ Σ'J gJ(x) dxjdxj ... dxj] // second definition where gJ(x) = Σ'I fI(F(x)) det(RIJ) and x' = F(x) , R = (DF) . (10.11.7) As we shall see in Section 10.12, this specification reproduces the "journeyman" integration results shown in the examples of Section 10.10. Recall that Σ'I and Σ'J are ordered sums. An Alternate Approach In this thread we take a narrower view of the functional sense of the k-form integral. We treat ∫S' αx' as if it were a discrete sum over the points x' on the surface S'. Since αx' is a certain functional, the integral ∫S' αx' is then also a functional, being a sum of functionals. In some sense the analysis below is a microscale interpretation of the regional-functional approach noted above. The development below is done in Dirac notation, but it could be restated using the pullback function F* . In a first step, we write the pullback of the original k-form "sum" as, [∫S' αx'] R = ∫S βx (10.11.8) where R is the Dirac pullback operator used in Section 10.7. Recall that the pulled-back k-form is, βx = Σ'J [gJ(x)] dx^J = Σ'J [gJ(x)] λ^J // (10.11.1) and definition (10.1.9) that dx^J ≡ λ^J = Σ'J [Σ'I fI(F(x)) det(RIJ)] λ^J // insert gJ(x) from (10.11.7) = Σ'I fI(F(x)) [Σ'J det(RIJ) λ^J] // reorder = Σ'I fI(F(x)) [ΣJ RIJ λ^J] // (10.8.1) to get symmetric J sum and no det = Σ'I fI(F(x)) [ΣM RIM λ^M] . // rename dummy multiindex J→M (10.11.9) Since λ^M = <u^M | from (2.11.c.2), we rewrite (10.11.8) in Dirac notation, <∫S' αx' | R = ∫S <βx| = ∫S Σ'I fI(F(x)) ΣM RIM <u^M | (10.11.10) which is interpreted as a functional in Λk(Rn) (see fiber comment below). In a second step we close this functional with a certain "measure ket" |μ> defined as | μ> ≡ Σ'J | dxJ> = Σ'J | dxj> | dxj>. ... | dxj> = Σ'J | dxj, dxj ... dxj > (10.11.11) where the differential vectors are aligned with the axes of x-space Rn, dxj ≡ dxj uj // no sum on j or | dxj> = dxj | uj> . (10.11.12) Here dxj is a vector in Rn and | dxJ> is a vector in (Rn)k called Vk in Chapter 5. Thus, | μ > = Σ'J dxjdxj ... dxj | uj, uj ... uj > = Σ'J dxJ | uJ > . // multiindex notation dxJ ≡ dxjdxj ... dxj (10.11.13) For example, for k = 1,2,3 in Rn = R3 the vector | μ> would be | μ > = | dx1> + | dx2> + | dx3> // k = 1 | μ > = | dx1, dx2> + | dx1, dx3> + | dx2, dx3> // k = 2 | μ > = | dx1, dx2, dx3> . // k = 3 (10.11.14) We then define "the integral of the k-form αx' in x'-space" as follows, " ∫S' αx'" ≡ < ∫S' αx' | R | μ > = < ∫S βx | μ > = ∫S <βx | μ > (10.11.15) where we end up with the integral of a certain tensor function over S. Next, write ∫S <βx | μ > = ∫S Σ'I fI(F(x)) ΣM RIM <u^M | μ > // (10.11.10) = ∫S Σ'I fI(F(x)) ΣM RIM Σ'J <u^M | dxJ> // (10.11.11) = ∫S Σ'I fI(F(x)) ΣM RIM Σ'J dxJ <u^M | uJ > . // (10.11.13) (10.11.16) In our Chapter 8 normalization for wedge products, we write <u^M | uJ > = (λm ^ λm ^ ... ^ λm) (uj,uj, ...uj) // (2.11.c.2) = (1/k!) det(δMJ) . // (8.3.9.b) (10.11.17) In the Spivak normalization of the wedge product discussed in Section 8.9 (g) the (1/k!) is replaced by 1, and we shall now continue in the Spivak normalization, so <u^M | uJ > = det(δMJ) . // using Spivak wedge product normalization (10.11.18) Inserting this into (10.11.16) gives ∫S < βx | μ > = ∫S Σ'I fI(F(x)) Σ'J dxJ [ΣM RIM det(δMJ)] (10.11.19) where we have shifted the M sum to the right. Now consider, ΣM RIM [det(δMJ )] = ΣM RIM [ΣP (-1)S(P) δMP(J) ] // (A.1.19) = ΣP (-1)S(P) ΣM RIM δMP(J) // reorder = ΣP(-1)S(P) RIP(J) // k matrix multiplications = det(RIJ) . // (A.1.19) (10.11.20) Inserting this result into (10.11.19) gives ∫S < βx | μ > = ∫S Σ'I fI(F(x)) Σ'J dxJ det(RIJ)] = ∫S Σ'I fI(F(x)) Σ'J det(RIJ) dxJ . (10.11.21) The final result then is " ∫S' αx'" ≡ < ∫S' αx' | R | μ > = < ∫S βx | μ > = ∫S < βx | μ > = ∫S Σ'I fI(F(x)) Σ'J det(RIJ) dxjdxj ... dxj = ∫S gJ(x) dxjdxj ... dxj . // see (10.11.7) (10.11.22) This result then is the same as (10.11.7) obtained by making the "two definitions". Our resulting tensor function turns out to be just a constant function which is a real number which is the result of doing the above regular calculus multivariable integration. Our alternate approach lacks rigor since the integration is treated as a sum over points x' on a manifold and this really means that the Dirac space used above is some kind of fiber bundle space (the tangent bundle of Section 10.2). Moreover, the measure ket |μ> = Σ'J dxJ | uJ > seems arbitrary, but it does manage to "sweep up" all contributions to the integration and we do get the correct result. The method does at least provide an alternative explanation of how the functional dx^J wedge product is replaced by the calculus product dxJ.