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2313f10-final-7-soln - 78 C HAPTER 5 T HREE D IMENSIONAL A...

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78 C HAPTER 5 T HREE D IMENSIONAL A NALOGUES C HANGING THE SURFACE Stoke’s Theorem tells us that the circulation of F across a surface S is determined solely by the values of F along the boundary of the surface, S . Therefore the circulation of F across S is the same as the circulation of F across any surface with the same boundary, so if neither the surface integral nor the line integral form of Stoke’s Theorem simplify into manageable integrals, we have one more trick at our disposal: we can compute the surface integral across a nicer surface with the same boundary. The following example demonstrates this approach. Example 1. Compute the circulation of F ye xy y 3 3 , x 2 y xe xy , sin xy across the upper hemisphere of the 3 d unit sphere, that is, the surface S x, y, z : x 2 y 2 z 2 1 and z 0 . Solution. First notice that computing circulation as a line integral is going to be impos- sible. The boundary of S is the unit circle, so parameterizing this in the standard way, we have r t cos t, sin t, 0 , r t sin t,
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