Green's Theorem

In this topic:

    (1) State Green's theorem for imply connected regions.
    
    (2) Use Green's theorem to evaluate the line integral around the curve defined by the ellipse oriented counterclockwise.
    
    (3) Use Green's theorem to evaluate the line integral around the curve defined by the unit circle oriented clockwise.  
    
    (4) Use Green's theorem to evaluate the line integral around the curve defined by the square with vertices oriented counterclockwise.
    
    (5) Find the work done when an object moves in the force field once counterclockwise around the circular path
    
    (6) Find the work done when an object moves in the force field once counterclockwise around the circular path  
    

    Green's theorem elegantly yields another technique to evaluate a line integral of a vector field and is especially important when the vector field is not conservative. To do so, Green's theorem assumes that we have a simply connected region that is bounded by a positively oriented piecewise smooth Jordan curve. In this case, Green's theorem offers us a way to evaluate a line integral of a continuously differentiable vector field by computing a double integral involving the partials of the component functions of the vector field. It can be thought of as a generalization of the fundamental theorem of calculus for definite integrals because it gives us a way to evaluate a double integral using only the values of partial antiderivatives on the boundary of the domain of integration.
    Green's theorem for doubly connected regions and alternate forms of Green's theorem involving the curl and div of a vector field are also detailed.

Proposition (Green's Theorem) Let be a simply connected region with a piecewise smooth boundary curve oriented counterclockwise, be a continuously differentiable vector field on and be the tangential and normal components of respectively then



and, in vector form,

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

    Solution.  Using polar coordinates we have,
    





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

    Solution. Using polar coordinates we have,
    





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. We have,
    



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

    Solution.  We have,  








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

    Solution.  We have,

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