Pullback in terms of page 88A
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Short note by Phil dated 2.18.16 that matches Sjamaar's notation (φ*α, Dφ(x), page 37 figure) to Spivak's (f*ω, f, p) for the pullback of a k-form. It builds a concordance between the two equations, then reads the result in Dirac-style notation as the pulled-back form being the transpose of the differential matrix R applied to the form at y, with no determinants involved.
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Pullbacks in terms of 88A PhL 2.18.16
1. Getting Spivak and Sjamaar to agree.
Equation 88a of Sjamaar says this:
(φ*α)x(v1,v2...vk) = αφ(x)( [Dφ(x)]v1, [Dφ(x)]v2,.... [Dφ(x)]vk)
I think this is in the context of the figure on page 37 where U and x-space is on the left, and where V and y-space is on the right, and where y = φ(x) is the transformation.
This looks a lot like Spivak's two equations,
Let's try to translate Spivak into Sjammar notation. Assume that
Sjamaar Spivak
α ω
φ f
x p
φ(x) = y f(p) = y
With just these three connections, Spivak's first equation becomes
φ*(vp) = ((Dφ(x)(v))φ(x) = ((Dφ(x)(v))y
Now y is a "point on the manifold on the right side of page 37", so I understand perhaps why Spivak shows it as a marking label. I think since (Dφ(x)(v) is linear, I can translate further
((Dφ(x)(v))y = (Dφ(x))y v = a matrix times a vector v
With this assumption, Spivak's second equation becomes
φ*α*(x)(v1,v2...vk) = α(φ(x)) ( (Dφ(x))y v1, (Dφ(x))y v2 ..... (Dφ(x))y vk)
Now let's further change Spivak's notation so that the argument of a form is made a subscript and not a paren argument. Then Spivak becomes
φ*α*x(v1,v2...vk) = αφ(x) ( (Dφ(x))y v1, (Dφ(x))y v2 ..... (Dφ(x))y vk)
which we can compare with Sjamaar
(φ*α)x(v1,v2...vk) = αφ(x)( [Dφ(x)]v1, [Dφ(x)]v2,.... [Dφ(x)]vk)
So maybe I should be interpreting the original Spivak LHS as (f*ω)(p) .
I can take Spivak's (Dφ(x))y in light of page 37 as saying you compute the matrix Dφ(x) where derivatives are with respect to xi coordinates, but then you evaluate the result at the x which corresponds to y on the manifold, x = φ-1(x).
So I think I have a concordance between Spivak and Sjamaar on this pullback equation.
2. What does this equation mean?
Here it is I think in its simplest form re page 37,
(φ*α)x(v1,v2...vk) = αy( [Dφ(x)]v1, [Dφ(x)]v2,.... [Dφ(x)]vk)
The tensor function on the left, (φ*α)x , is obtained from the tensor function on the right by shuffling the arguments as shown. There are no determinants here.
Dφ is the matrix of derivatives, it is the R matrix for the transformation, the differential matrix. So then I could write in shorter notation
(φ*α)x(v1,v2...vk) = αy( Rv1, Rv2,.... Rvk)
or
<(φ*α)x | v1,v2...vk > = <αy | Rv1, Rv2,.... Rvk > = <αy | R | v1,v2...vk >
= <RTαy | v1,v2...vk >
in my Dirac notation. Then in the dual space where these k-forms live,
<(φ*α)x | = <RTαy|
We start with a form αy defined at point y on page 37, we apply operator RT to it, and we end up with the form <β| = <(φ*α)x | which is defined at point x on the left on page 37. This is the pulled back form. The form αy has been pulled back from y-space to x-space where it is the form (φ*α)x .