If $f$ and $g$ are continuously differntiable in an open connected set $S$ in the plane, show that $\oint f\nabla g \cdot d\alpha = -\oint g \nabla f \cdot d\alpha $ for every piecewise smooth Jordan curve $C$ in $S$.
Attempt:$\oint f\nabla g \cdot d\alpha + \oint g \nabla g \cdot d\alpha = f ~d(g(\alpha)) + g~d(f(\alpha)) = \iint_R ( \dfrac {\partial g}{\partial f}- \dfrac {\partial f}{\partial g} ) df~dg $ .
( By Green's Theorem)
How do I proceed now? We need to prove that $\dfrac {\partial g(\alpha)}{\partial f(\alpha)}= \dfrac {\partial f(\alpha)}{\partial g(\alpha)}$ ?
I just have a connecting idea that in an open connected space, a gradient follows the same behavior what we are looking to prove, but then, nothing is mentioned about the presence of a gradient function either.
Thank you very much for the help.