stokes_theorem

stokes_theorem - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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V13. Stokes' Theorem 1. Introduction; statement of the theorem. The normal form of Green's theorem generalizes in 3-space to the divergence theorem. What is the generalization to space of the tangential form of Green's theorem? It says where C is a simple closed curve enclosing the plane region R. Since the left side represents work done going around a closed curve in the plane, its natural generalization to space would be the integral $ F dr representing work done going around a closed curve in 3-space. In trying to generalize the right-hand side of (I), the space curve C can only be the boundary of some piece of surface S - which of course will no longer be a piece of a plane. So it is natural to look for a generalization of the form gjc F . dr = /L(something derived from F)dS The surface integral on the right should have these properties: a) If curl F = 0 in Bspace, then the surface integral should be 0; (for F is then a gradient field, by V12, (4), so the line integral is 0, by Vll, (12)). b) If C is in the xy-plane with S as its interior, and the field F does not depend on z and has only a k-component, the right-hand side should be These JL curl F dS . things suggest that the theorem we are looking for in space is Stokes' theorem For the hypotheses, first of all C should be a closed curve, since it is the boundary of S, and it should be oriented, since we have to calculate a line integral over it. S is an oriented surface, since we have to calculate the flux of curl F through it. This means that S is two-sided, and one of the sides designated as positive; then the unit normal n is the one whose base is on the positive side. (There is no "standard" choice for positive side, since the surface S is not closed.) cubical surface: no boundary It is important that C and Sbe compatibly oriented. By this we mean that the right-hand rule applies: when you walk in the positive direction on C, keeping S to your left, then your head should point in the direction of n. The pictures give some examples.
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2 V. VECTOR INTEGRAL CALCLUS The field F = Mi + N j + Pk should have continuous first partial derivatives, so that we will be able to integrate curl F. For the same reason, the piece of surface S should be piecewise smooth and should be finite- i.e., not go off to infinity in any direction, and have
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stokes_theorem - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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