Green’s theorem

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 $\overrightarrow{F}=(f(x,y),g(x,y))$ over the plane,

$${\oint }_P\overrightarrow{F}\cdot 𝑑\overrightarrow{x}=\int {\int }_R[g_1(x,y)-f_2(x,y)]𝑑A$$

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

$${\oint }_P{\overrightarrow{\nabla }}_h\cdot 𝑑\overrightarrow{x}=0$$

We can easily prove this using Green’s theorem.

$${\oint }_P{\overrightarrow{\nabla }}_h\cdot 𝑑\overrightarrow{x}=\int {\int }_R[g_1(x,y)-f_2(x,y)]𝑑A$$

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.