# 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 $\vec{F}=(f(x,y),g(x,y))$ 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.

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