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Unformatted text preview: with a nonzero curl would. Just as Gauss' Theorem relates surface integrals and volume integrals using divergence, Stokes' Theorem relates surface integrals and line integrals using curl. Given a closed curve that encompasses a surface, ds = da where the left side is a line integral and the right side is a surface integral. Again, we pay special attention to the special case in which the curl is zero. In this case, the integral of the field around any closed loop is zero. Electric fields have this property. It is incredibly important when using Gauss' and Stokes' Theorem to remember that one must deal with closed surfaces (with Gauss' Theorem) or closed loops (with Stokes' Theorem). Otherwise the equations are not applicable. Having established the concepts of curl and divergence, and related them to vector calculus through our two theorems, we can apply these concepts to magnetic fields....
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- Fall '10