Closed curve line integral of gradient using Green's Theorem

[twotwelve]

Apostol page 386, problem 5

Homework Statement

Given $$f,g$$ continuously differentiable on open connected $$S$$ in the plane, show
$$\oint_C{f\nabla g\cdot d\alpha}=-\oint_C{g\nabla f\cdot d\alpha}$$
for any piecewise Jordan curve $$C$$.

Homework Equations

1. Green's Theorem
2. $$\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$$ for $$\nabla f,\nabla g$$

The Attempt at a Solution

I need some general direction on this one...

 

[Dick]

Use that grad(fg)=f*grad(g)+g*grad(f), maybe?

 

[twotwelve]

\begin{facepalm*}
that makes it about a 2 second proof then
\end{facepalm*}
-thanks