Green's Theorem

    Recall to evaluate a line integral we have the option of using a parametrization or using the Fundamental Theorem of Line Integrals. However both have their drawbacks, in particular it may be difficult to come up with a parametrization. On one hand the value of a line integral is independent of the parametrization used, even so, sometimes it may be hard to find any parametrization at all.
    If a vector field is conservative then Fundamental Theorem of Line Integrals can be used, but this also has its difficulties. First, not all vector fields are conservative, and even if a vector field is conservative it may be difficult to find the scalar potential function.The advantage, though, of this method is path independence. If you in fact do know that the vector field is conservative you have two choices, find a connivent path from starting point to ending join and either use the scalar potential function or use some parametrization of that path to find the value of the line integral.
    Green's Theorem in the plane allows use evaluate a line integral even though the vector field may or may not be conservative -- without using a parametrization; but rather a double integral. One of the most beautiful theorems in all of mathematics, Green's theorem allows us to evaluate a line integral by iterated integration over the region enclosed by the closed path.
    A Jordan curve is a closed curve that does not intersect itself and a simply connected region has the property that it is connected and the interior of every Jordan curve in also lies in

Proposition (Green's Theorem) Let be a simply connected region with a piecewise smooth boundary curve oriented counterclockwise and let be a continuously differentiable vector field on then

Example (Green's Theorem) Use Green's theorem to evaluate the line integral around the curve defined by the ellipse oriented counterclockwise.  

    Solution.  Let , then which is continuously differentiable over the ellipse as well as Therefore, and satisfy the hypothesis of Green's theorem, and
                












Knowing the area of an ellipse is

Example (Green's Theorem) Use Green's theorem to evaluate the line integral around the curve defined by the unit circle oriented clockwise.  

    Solution. Let , then which is continuously differentiable over the unit circle as well as Therefore, and satisfy the hypothesis of Green's theorem, and
    




  (we use a minus sign to switch the orientation)

Example (Green's Theorem) Use Green's theorem to evaluate the line integral around the curve defined by the square with vertices oriented counterclockwise.

    Solution. Let , then which is continuously differentiable over the square as well as Therefore, and satisfy the hypothesis of Green's theorem, and

Example (Green's Theorem) Find the work done when an object moves in the force field once counterclockwise around the circular path

    Solution.  Let , then which is continuously differentiable over the circle as well as Let be the region bounded by the curve Then, and satisfy the hypothesis of Green's theorem, and













Converting to polar coordinates,

Example (Green's Theorem) Find the work done when an object moves in the force field once counterclockwise around the circular path  

    Solution.  Let , then which is continuously differentiable over the circle as well as Let be the region bounded by the curve Then, and satisfy the hypothesis of Green's theorem, and

Proposition (Green's Theorem for Doubly-Connected Regions) Let be a doubly-connected region with a piecewise smooth outer boundary curve oriented counterclockwise and a piecewise smooth inner boundary curve oriented clockwise and let be a continuously differentiable vector field on then

Example (Green's Theorem for Doubly-Connected Regions) Evaluate where is any Jordan curve whose interior does not contain the point transversed counterclockwise.

    Solution. Let , then which is continuously differentiable over the circle as well as Let be the region bounded by the given curve Then, and satisfy the hypothesis of Green's theorem, and
    

Example (Green's Theorem for Doubly-Connected Regions) Evaluate where is any Jordan curve whose interior contains the point transversed counterclockwise.

    Solution. Let be a circle centered at with radius so small that all of is contained within Let be oriented clockwise and let be the region between and Then by Green's Theorem for Doubly-Connected Regions,
    








Thus,



which is something that can be easily evaluated, using say, the parametrization



Therefore,

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Cite this as:
Green Theorem
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/green-theorem.html