Why is in Green’s Theorem replaced by in Stokes’ Theorem?
refers to ordinary area while refers to the surface area. So, in Green's Theorem is replaced by in Stokes' Theorem because .
in Green’s Theorem replaced by in Stokes’ Theorem.
Green's Theorem states that;"Let be a region in the plane with smooth boundary curve oriented counterclockwise by for .If a vector field is defined by , then, Stokes' Theorem states that;"Let be an oriented, smooth or piecewise-smooth surface bounded by a curve . Suppose that is an oriented unit normal vector of and has a parametrization that traverses in the counterclockwise direction with respect to .If a vector field is defined on , then, .
Both the notations and refer to area.In the case of Green's Theorem, refers to the ordinary area.In the case of Stokes' Theorem, refers to the surface area. In this case, keeps track of any distortion to the area of the surface caused by transforming a patch of the plane to parametrize . Hence, in Green's Theorem can be replaced by in Stokes' Theorem.
The current through a certain passage of the San Juan Islands in Washington State is given by
Consider a disk R of radius 1 mile and centered on this region. Denote the boundary of the disk by localid="1650455155947" .
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