Prove that the following is "exact"...

**Redding1234** · May 1st 2010, 10:19 AM

Problem: Let $f: \Re \to \Re$ and $\omega = f(||x||) (\sum_{i=1}^n{x_{i}dx_{i}}) \in A^1(\Re^n)$ . Assuming $f$ is continuous, prove that $\omega$ is exact.

Notation notes:

$A^1(\Re^n)$ is a vector space denoting the set of 1-forms on $\Re^n$ .

$||x||$ is the length of the vector $x$ , so $||x|| = \sqrt{(x_1)^2+(x_2)^2 + ... + (x_n)^2}$ .

So this is how I tried to approach the problem. I need to find a function, say $G$ , such that $dG = \omega$ .

So $\omega = f(\sqrt{(x_1)^2+(x_2)^2 + ... + (x_n)^2}) (x_1dx_1 + x_2dx_2 + ... + x_ndx_n)$

$\omega = f(\sqrt{(x_1)^2+(x_2)^2 + ... + (x_n)^2})x_1dx_1$ + $f(\sqrt{(x_1)^2+(x_2)^2 + ... + (x_n)^2})x_2dx_2$ + ... + $f(\sqrt{(x_1)^2+(x_2)^2 + ... + (x_n)^2})x_ndx_n$

Then I want to find $\frac{\partial G}{\partial x_1}, \frac{\partial G}{\partial x_2}, ..., \frac{\partial G}{\partial x_n}$ , but I don't know how to take the derivative of $f$ since I don't know what $f$ is - and what I'm taking the derivative with respect to each time is within $f$ .

You can probably tell I'm not familiar with proofs dealing with sum notation. Any tips?

**Redding1234** · May 3rd 2010, 03:57 PM

I've been thinking about this, and maybe it would be okay to just integrate all of the last line of $\omega$ and leave it in integral form. Instead of actually trying evaluate the integral, perhaps it's enough to just state that the integral exists (since $f$ is continuous)? Then I could just let $G$ equal that integral and have that complete the proof.

I don't see it as an ideal solution, but could it be sufficient?

**maddas** · May 3rd 2010, 04:52 PM

Basically, yea. Let $g(x) := f(\sqrt{x})$ so $f(||x||) = g(||x||^2)$ , and let G be a primitive for g. Then ${\partial\over\partial x_i} (\frac12 G(||x||^2)) = g(||x||^2)x_i=f(||x||)x_i$ ...

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