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Stoke�s theorem

Stoke�s theorem - Lizzie Wagner Stoke's Theorem...

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Lizzie Wagner Stoke’s Theorem Stoke’s theorem is a three dimensional elaboration of Green’s theorem. In Green’s theorem, we related a line integral to a double integral over some region R . In Stoke’s theorem, we are going to relate a line integral to a surface integral. However, please note that in this theorem the surface S can actually be any surface as long as its boundary curve is given by C . Stoke’s theorem states that the closed line integral over C of a vector field F dotted with the positive unit tangent vector T to the boundary curve C is equal to the surface integral (calculated with the use of double integration) of the normal component of the curl of that vector field ((gradient cross F ) dotted with n ) over some surface S . The full theorem is shown below. C F . dr = C F . T d s = ∫ ∫ S (curl F) . n d S For me, this is the trickiest theorem to get a good grasp on. To use this theorem, you must remember to parameterize the surface prior to computing anything. Also, you
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