divergance_thm - 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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V10. The Divergence Theorem 1. Introduction; statement of the theorem. The divergence theorem is about closed surfaces, so let's start there. By a closed surface S we will mean a surface consisting of one connected piece which doesn't intersect itself, and which completely encloses a single finite region D of space called its interior. The closed surface S is then said to be the boundary of D; we include S in D. A sphere, cube, and torus (an inflated bicycle inner tube) are all examples of closed surfaces. On the other hand, these are not closed surfaces: a plane, a sphere with one point removed, a tin can whose cross-section looks like a figure-8 (it intersects itself), an infinite cylinder. h n A closed surface always has two sides, and it has a natural positive direction - the one for which n points away from the interior, i.e., points toward the outside. We shall always understand that the closed surface has been oriented this way, unless otherwise specified. We now generalize to 3-space the normal form of Green's theorem (Section V4). Definition. Let F(x, y, z) = Mi + N j + Pk be a vector field differentiable in some region D. By the divergence of F we mean the scalar function div F of three variables defined in D by The divergence theorem. Let S be a positively-oriented closed surface with interior D, and let F be a vector field continuously differentiable in a domain contatining D. Then We write dV on the right side, rather than dxdy dz since the triple integral is often calculated in other coordinate systems, particularly spherical coordinates.
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divergance_thm - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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