Problem: Let and . Assuming is continuous, prove that is exact.

Notation notes:

is a vector space denoting the set of 1-forms on .

is the length of the vector , so .

So this is how I tried to approach the problem. I need to find a function, say , such that .

So

+ + ... +

Then I want to find , but I don't know how to take the derivative of since I don't know what is - and what I'm taking the derivative with respect to each time is within .

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