Greens theorem and closed curves

Question

For a vector field of the form $\underline{v}(x,y) = (a(x), b(y)).$ How would you use Green's Theorem to prove that $$\oint_C\underline{v}.d\underline{r} = 0$$ for any simple closed curve $C\subset \mathbb{R}^2$? I know that when $\underline{v}$ is conservative you just use the FTC for vector fields but when $\underline{v}$ isn't conservative?

Answer 1

Assume the domain of ${\bf v}$ is convex and let $x\mapsto A(x)$, $y\mapsto B(y)$ be primitives of the functions $x\mapsto a(x)$ and $y\mapsto b(y)$ respectively. Then ${\bf v}=\nabla F$ for the scalar function $F(x,y):=A(x)+B(y)$. It follows that ${\bf v}$ is conservative, whence $\int_C{\bf v}\cdot d{\bf r}$ is zero for any closed curve $C\subset{\rm dom}({\bf v})$.