# Closed curve line integral of gradient using Green's Theorem

1. Apr 22, 2010

### twotwelve

Apostol page 386, problem 5
1. The problem statement, all variables and given/known data
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 $$C$$.

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

3. The attempt at a solution
I need some general direction on this one...

Last edited: Apr 22, 2010
2. Apr 22, 2010