Equality of Line Integrals for One Forms

Question

I saw a post about how if the equality between integrals of two functions f and g holds over any domain V, then it implies an equality of the integrands too, so f=g. Can we extend this idea to C1 one forms on Rn? If we have two C1 1-forms that have the same line integral along every piecewise smooth curve Ψ from [a,b] to Rn, then does this imply the one forms are equal too?

Answer 1

This is an excellent question. Yes. Suppose $\omega,\eta$ are $1$-forms on $\Bbb R^n$ and they are not equal at some point $p$. Then (by continuity) there is a neighborhood of $p$ on which they are not equal. By subtracting, we reduce to the case of a single $1$-form $\phi$ which is nonzero at $p$, and therefore nonzero on some neighborhood. Look at the hyperplane $\Pi = \{v\in\Bbb R^n: \phi(p)(v) = 0\}$. $\Pi$ divides $\Bbb R^n$ into two halves, if you like; choose a $w\in\Bbb R^n$ on the positive side of $\Pi$. Let $\epsilon>0$ be suitably small. If you let $\Psi$ be the line segment from $p$ to $p+\epsilon w$, then I claim that $$\int_\Psi\phi > 0,$$ which settles your question (by taking the contrapositive). Can you finish the argument?