Greens Theorem

Greens Theorem in the Plane

It introduces two concepts:

• circulation density around axis perpendicular to the plane
• divergence (or flux density)

A closed curve $C$ is positively oriented, if the region it encloses is always to the left when moving along the curve.

Circulation Density

The circulation density of a vector field $F=Mi+Nj$ at the point $(x,y)$ is the scalar expression

$$\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}$$

This equation is also referred to as $k$-component of the curl of $F$ $\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}=(curlF)\cdot k$. It measures the rate of the fluid’s rotation at a point, being positive if it is counterclockwise.

Forms of Green’s Theorem

Circulation-Curl (Tangential Form)

If $C$ is a smooth, simple closed curve enclosing a region $R$ in the plane, and $F=Mi+Nj$ a vector field, the counterclockwise circulation of $F$ around $C$ equals:

$$\oint F\cdot Tds=\oint Mdx+Ndy=\iint \_R(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y})dxdy$$

Flux-Divergence (Normal Form)

If $C$ is a smooth, simple closed curve enclosing a region $R$ in the plane, and $F=Mi+Nj$ a vector field, the outward flux of $F$ across $C$ equals:

$$\oint F\cdot \vec{n}ds=\oint Mdy-Ndx=\iint \_R(\frac{\partial M}{\partial x}+\frac{\partial N}{\partial y})dxdy$$

Other Applications

Calculating Area

If a simple, closed curve in the plane and its area satisfy the hypotheses of Green’s Theorem, the area $R$ is given by

$$A\_R=\frac{1}{2}\oint \_Cxdy-ydx$$