Green's theorem provides a connection between path integrals over a well-connected region in the plane and the area of the region bounded in the plane. Given a closed path $P$ bounding a region $R$ with area $A$ , and a vector-valued function $\vec{F}=(f(x,y),g(x,y))$ over the plane, $$\oint_P\vec{F}\cdot d\vec{x} = \int\!\!\!\int_{\!\!R} [g_1(x,y) - f_2(x,y)] dA$$ where $a_n$ is the derivative of $a$ with respect to the $n$ th variable.

Corollary:

The closed path integral over a gradient of a function with continuous partial derivatives is always zero. Thus, gradients are conservative vector fields. The smooth function is called the potential of the vector field.

Proof:

The corollary states that

$$\oint_P\vec{\nabla}_h\cdot d\vec{x} = 0$$

We can easily prove this using Green's theorem.

$$\oint_P\vec{\nabla}_h\cdot d\vec{x} = \int\!\!\!\int_{\!\!R} [g_1(x,y) - f_2(x,y)] dA$$

But since this is a gradient...

$$\int\!\!\!\int_{\!\!R} [g_1(x,y) - f_2(x,y)] dA = \int\!\!\!\int_{\!\!R} [h_{21}(x,y) - h_{12}(x,y)] dA$$

Since $h_{12}=h_{21}$ for any function with continuous partials, the corollary is proven.