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

Notation notes:

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

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

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

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

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

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

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