Given an integral of the form , what considerations would lead you to evaluate the integral with Stokes’ Theorem?
If right-hand side of Stokes' Theorem is used, the result obtained is the same with a single area integral.
An integral of the form .
"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, .
If is an oriented, smooth or piecewise-smooth surface bounded by a curve , Stokes' Theorem relates a line integral of a vector field around the boundary curve to a surface integral of the curl of the vector field. When the line integral has piecewise-continuous boundary, for example, if the boundary curve is a rectangle or triangle , it requires the evaluation of several smooth pieces.If you use the right-hand side of Stokes' Theorem, you obtain the same result with a single area integral.
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