For the above expression, I was told that it can be proven using Green's Theorem on the line integral on the RHS, however I can't seem the prove the equality.

Note that $G$, $H$, $f$ are functions of $x_1$ and $x_2$.

So I apply Green's Theorem:

$\displaystyle{\oint_C f\left(Gdx_1 - H dx_2\right) = \oint fG dx_1 + \left(-fH\right)dx_2 = \iint_D -\frac{\partial \left(fH\right)}{\partial x_1} - \frac{\partial \left(fG\right)}{\partial x_2} dA}$

But then what? I can't seem to get the RHS to equal the LHS.

Any help would be appreciated