Green's Theorem for Doubly-Connected Regions

In this topic:

    (1) State Green's theorem for multiply-connected regions.
    
    (2) Evaluate where is any Jordan curve whose interior does not contain the point transversed counterclockwise.

    (3) Evaluate where is any Jordan curve whose interior contains the point transversed counterclockwise.
    

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. We have,
    












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:
Greens Theorem For Doubly Connected Regions
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/greens-theorem-for-doubly-connected-regions.html