Let $\varphi$ be a scalar function having a continuous gradient throughout a region $R$ of space, let $\ell$ be the boundary curve of any surface $S$ lying entirely within the region $R$, then prove that:
$$\int \varphi \, d\ell=-\int \operatorname{grad}\varphi\wedge dS $$
It look like Stokes's theorems but not exactly the same, I think we can prove it by reducing it to stokes theorem or by a proof similar to the one of stokes'theorem but I don't figure out how to do that, any hints?
This is a homework problem from Electromagnetism course.