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lec_week12

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

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 200 7 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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�� ��� 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 = ﬂux generated per unit volume. Imagine an incompressible ﬂuid ﬂow (i.e. a given mass occupies a fixed volume) with velocity F , then ��� �� div F dV = F n ˆ dS = ﬂux 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 ˆ
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