Situation: I have a smooth curve $\gamma:(a,b)\to\mathbb R^n$ and a locally integrable function $f:(a,b)\to\mathbb R$ for which I know that $\int_a^b f(t)\phi(\gamma(t))\text{d}t = 0$ for any smooth, compactly supported $\phi:\mathbb R^n\to \mathbb R$.
I'm trying to figure out: Does this imply that $f = 0$ almost everywhere?
Some ideas...: The situation is very similar to the statement that if $\int_a^b f(t)\tilde \phi(t)\text{d}t = 0$ for any smooth, compactly supported $\tilde\phi:\mathbb R\to\mathbb R$ then this does indeed imply that $f = 0$ almost everywhere. If $\gamma$ were injective, I believe we could reduce our case to this simpler case, because given any $\tilde \phi$ we could choose $\phi$ such that $\phi\circ\gamma = \tilde\phi$. However, in when $\gamma$ is not injective, I have no idea how to prove the statement or even if it holds at all.
Any help is appreciated!