l_notes27 - Lecture XXVII Stokess Theorem In the previous...

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Lecture XXVII Stokes’s Theorem In the previous lecture, we saw how Green’s theorem deals with integrals on elementary regions in E 2 and their boundaries. We also saw how the divergence thoerem deals with elementary regions in E 3 and their boundaries. We will now look at Stokes’s theorem, which applies to integrals on elementary surfaces in E 3 and their boundaries. First let us extend the concept of rotational derivatives to E 3 . Defnition 1 Let F be a vector feld with a domain D in E 3 , and let P be an interior point oF D . Given a unit vector ˆ u in E 3 let M be the plane that goes through P and is normal to ˆ u . Consider this plane directed, with the direction given by ˆ u . ±or a > 0 , let C a be the circle in M oF radius a and center P . Consider the limit 1 d lim F R a 0 πa 2 C a · IF the limit exists, then it is called the rotational derivative of F in the direction ˆ F ˆ u at P and it is denoted by rot | u,P . F
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l_notes27 - Lecture XXVII Stokess Theorem In the previous...

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