# Closed and exact forms

1. Jul 26, 2005

### rocket

let $$f:U \rightarrow R^n$$ be a differentiable function with a differentiable inverse $$f^{-1}: f(u) \rightarrow R^n$$. if every closed form on U is exact, show that the same is true for f(U).

Hint: if dw=0 and $$f^{\star}w = d\eta,$$ consider $$(f^{-1})^{\star}\eta.$$
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I'm not quite sure what the hint means or how to use it. is it true that $$f^{\star}$$ is basically another way of writing a differential - eg. df? I didn't really get a clear definition of it in my text.

anyway here's my thoughts so far:

consider $$w$$ as a form on U. suppose $$w$$ is closed. then dw = 0. since every closed form on U is exact, then there exists a $$\eta$$ on U such that $$w = d\eta$$.

but how is it that $$f^{\star}w = d\eta$$ (given in the hint)? like, how is this relationship derived? if $$w = d\eta$$ and also $$f^{\star}w = d\eta$$, then we have $$w = f^{\star}w$$? I find that really confusing, and I'm not sure how to continue the problem. Any help is greatly appreciated. thanks in advance!

2. Jul 26, 2005

### AKG

I'm rusty on this stuff, but if w is a form on f(U), then w' = (f-1)*w is a form on U. Show that if w is closed, then so is w'. If w' is closed, then there is some h' such that w' = dh'. Try to use this to show that w is also exact. Perhaps you will find that w = d(f*h') or something like that.