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Basic Differential Forms Problem

by masterfool
Tags: basic, differential, forms
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masterfool
#1
May29-11, 10:57 AM
P: 5
I'm reading Flanders' Differential Forms with Applications to the Physical Sciences and I have some issues with problems 2 and 3 in chapter 3, which appear to ask the reader to compute the pullback a mapping from X to Y applied to a form over X, and I'm not sure how to interpret such a thing. Problem 3 reads:
Consider the mapping [tex]\phi : (x,y)\rightarrow (xy,1)[/tex] on E2 into E2. Compute [tex]\phi^{*}(dx)[/tex], [tex]\phi^{*}(dy)[/tex], and [tex]\phi^{*}(ydx)[/tex].
What I'm tempted to do is to consider phi as a mapping from [tex](x,y)[/tex] to [tex](x',y')[/tex], and instead compute [tex]\phi^{*}(dx')[/tex], [tex]\phi^{*}(dy')[/tex], and [tex]\phi^{*}(y'dx')[/tex], which would make everything easy as pie since then I'd just be computing the pullback of a form defined over the image space. But I'm not sure if that dumb little notational issue is all there is to it, or maybe I'm supposed to do something else, like first apply the identity map to [tex](dx,dy)[/tex] and then apply the pullback, which I must admit makes my head spin a little.
I have essentially the same issue with problem 2, which asks for the pullback of a form defined on the domain, and I'm tempted to say that's just the identity transformation (i.e. that the pullback is a projection operator). I also did a funky calculation and found just that, but I'm far from confident that I even understood what the question was.

Thanks for your help.

I apologize for the hideous appearance; I'm still wrestling with LATEX.

I also realized a bit too late that I should have posted this in "homework help" even though I'm not in a class, so sorry for that too.
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henry_m
#2
May30-11, 06:34 AM
P: 160
Hi!

I think you've understood it; the notation in the question is just a bit confusing as you say. Here's a different way of writing the function:
[tex]\phi:(u,v)\mapsto (uv,1)[/tex]
and call the coordinates on the target space x and y. The reason it's confusing is that the map is an endomorphism, i.e. it's a map from one space to itself. I agree that the author's choice of notation is not all that clear...

(P.s if you want some inline tex without starting a new line, you can use "itex" tags rather than "tex".)
masterfool
#3
May30-11, 08:46 AM
P: 5
Thanks! I found your post [itex]doubly[/itex] helpful.


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