Green's Theorem

Green's Theorem has two formulations: one formulation to find the circulation of a two-dimensional function around a closed contour (a loop), and another formulation to find the flux of a two-dimensional function around a closed contour. Applications of Green's Theorem include finding the area enclosed by a two-dimensional curve, as well as many engineering applications.

The "circulation" formulation of Green's Theorem expresses the line integral of a vector function Pi + Qj over a closed curve in terms of the double integral of the partial derivatives of those same functions:

Simply stated, Green's Theorem converts a line integral over a closed curve (a loop) into a double integral that is often easier to solve. This theorem is an extension of calculus to the context of integrals in planar regions.

The "flux" formulation of Green's Theorem equates the outward flux of a vector function over a closed contour to the double integral of its divergence over the enclosed region.

Green's Theorem is a popular topic in advanced calculus courses, but is not as useful in physics and engineering as its three-dimensional counterpart, the Divergence Theorem.

Stokes' Theorem is the generalization of Green's Theorem to non-planar surfaces.

Example

Problem: Calculate the following along the contour from the origin to (2,0) to (2,6):

Solution: Using Green's Theorem, P=x²y and Q=x²y⁴, and the above line integral equals:

Next we have to determine the limits in terms of x and y for the surface R. The surface is a triangle for which y=3x, so the inner integral should be from y=0 to y=3x. The outer integral can then be from x=0 to x=2. This yields:

Solving the integrals equals:

Another Example

Problem: Calculate the following along the perimeter of the ellipse

Solution: Using Green's Theorem, P=y and Q=x, and the above line integral equals:

This is just twice the area of the ellipse, or .