Difference between revisions of "Green's Theorem"
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| − | '''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. |
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| + | The "circulation" formulation of Green's Theorem expresses the line integral of two functions over a closed curve in terms of the double integral of the partial derivatives of those same functions: | ||
::<math>\oint_{C} (P\, \mathrm{d}x + Q\, \mathrm{d}y) = \iint_{R} \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\, \mathrm{d}x{d}y.</math> | ::<math>\oint_{C} (P\, \mathrm{d}x + Q\, \mathrm{d}y) = \iint_{R} \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\, \mathrm{d}x{d}y.</math> | ||
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. | 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. | ||
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| + | A different formulation of Green's Theorem | ||
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]]. | 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]]. | ||
Revision as of 00:36, January 4, 2010
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 two functions 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.
A different formulation of Green's Theorem
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 
.
