Green's Theorem in the Plane
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Green's theorem in the plane
Green's theorem enables us to transform double integrals over a region into line integrals over the boundary of the region. The value of this is that the line integral over the boundary of the region may be easier to evaluate. It also has theoretical uses, since the transformation of one kind of integral into another can be always effected by use of this theorem. Green's theorem in the plane Let R be a closed bounded region in the xy-plane contained in a domain D. Let the boundary of R be C such that C is made up of finitely many smooth curves. Let [Equation goes here - download the original pdf to see it.] be continuous functions that have continuous partial derivatives [Equation goes here - download the original pdf to see it.] everywhere in D. Then [Equation goes here - download the original pdf to see it.] Comment The theorem does not require that the region R is simply connected, so it may, for example, have "holes" in it. [Diagram goes here - download the original pdf to see it.] We have to integrate along the whole boundary. We adopt the convention that the integration will be in an anticlockwise sense if the boundary contains the region, and clockwise if the boundary encloses a "hole" within the region.[Example goes here - download the original pdf to see it.]
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Contents of
Green's Theorem in the Plane
1 Green's theorem in the plane
2 Applications of Green's Theorem
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