Is there a math theorem by which a contour integral is equal to a double integral? I was reading Maxwell's relations and came across: $$\oint pdV=\oint TdS\Rightarrow \iint dpdV=\iint dTdS.$$
I know this is straightforward to see since they both represent the surface area, but I've never seen a math theorem on textbooks that indicates $$\oint ydx=\iint dxdy.$$ Is this just a trivial corollary?

This is Stokes' theorem,
$$ \int_S \mathrm{d}\omega = \int_{\partial S} \omega$$
for a 1-form $\omega = y\mathrm{d}x$, $S$ a surface and $\partial S$ its boundary.
