lec_week12

lec_week12 - MIT OpenCourseWare http://ocw.mit.edu 18.02...

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Unformatted text preview: 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 . 18.02 Lecture 29. Tue, Nov 20, 2007 Recall statement of divergence theorem: S F dS = D div F dV . Del operator. = /x, /y, /z (symbolic notation!) f = f/x, f/y, f/z = gradient. F = /x, /y, /z P, Q, R = P x + Q y + R z = divergence. Physical interpretation. div F = source rate = ux generated per unit volume. Imagine an incompressible uid ow (i.e. a given mass occupies a fixed volume) with velocity F , then div F dV = F n dS = ux through S is the net amount leaving D per unit time = total D S amount of sources (minus sinks) in D . Proof of divergence theorem. To show S P, Q, R dS = ( P x + Q y + R z ) dV , we can D separate into sum over components and just show S R k dS = D R z dV & same for P and Q . If the region D is vertically...
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This note was uploaded on 05/04/2011 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.

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lec_week12 - MIT OpenCourseWare http://ocw.mit.edu 18.02...

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