Green's Theorem

The Essentials

The circulation form of Green's theorem is:

\[ \oint_C\vec{F}\cdot d\vec{r}=\iint_D(Q_x-P_y)dA \]

This turns a line integral of a closed line into a double integral.

The flux form of Green's theorem can be used to calculate flux:

\[ \int_C\vec{F}\cdot\vec{N}ds=\iint_D (P_x+Q_y)dA \]

A source free vector field is one where the flux around a closed line is 0. Flux from source free vector fields are independent of path. In addition, for a source free vector field \( P_x+Q_y=0 \) and there exists a stream function \( g(x,y) \) such that \( \vec{F}(x,y)=\nabla g(x,y) \).

For a region with holes in it, the region has to be broken down into a sum of smaller regions in order to use Green's theorem.

Example

Use Green's theorem to calculate the work done by the function \( \vec{F}=\langle e^x+y, y^2-2y+1\rangle \) over the circle centered at the origin with a radius of 2 starting and ending at the point (0, 2).

Work is a line integral, and since are line is a closed curve, we can use the circulation form of Green's theorem to find the work:

\[ \iint_D(Q_x-P_y)dA \]
\[ \int_0^{2\pi}\int_0^2(0-1)rdrd\theta \]
\[ \int_0^{2\pi}-2d\theta=-4\pi \]

Practice

1) Evaluate the integral:

\[ \int_C (x^2+2xy+3x-2)dx+xydy \]

Where C is the region between the graphs of \( y=x \) and \( y=x^2 \) between the points (0, 0) and (1, 1) oriented in the counterclockwise direction.

2) Use Green's theorem to find the flux of \( \vec{F}=\langle xy,x+y\rangle \) across the boundary \( x^2+y^2=9 \) counterclockwise.

Solutions:

1) \( -\frac{1}{10} \)

2) \( 9\pi \)

More Resources

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