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# 12.02 - Section 13.9 wrap-up Let's use the Divergence...

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Section 13.9 wrap-up: Let’s use the Divergence Theorem to prove (div v )( P ) = lim r 0 ( ∫∫ S ( r ) v d S) / Vol( B ( r )) where B ( r ) is a ball with center P and radius r , and S ( r ) = B ( r ): By the Divergence Theorem, the flux of v out of B ( r ) is given by ∫∫ ∫ B ( r ) (div v ) dV . Thus (1) the average divergence of v in B ( r ) equals (2) the flux of v out of B ( r ) divided by the volume of B ( r ). If we let r go to 0, (1) converges to the divergence of v at P , and the right hand side converges to lim r 0 (flux of v out of B ( r ))/vol( B ( r )). In other words, divergence can be interpreted as flux per unit volume, just as the components of the curl vector can be interpreted as circulation per unit area. True/False questions for chapter 13: 1. “If F is a vector field, then div F is a vector field.” ..?.. False; div F is a scalar field.

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2. “If F is a vector field, then curl F is a vector field.” ..?.. True. See Definition 13.5.1. 3. “If f has continuous partial derivatives of all orders on R 3 , then div(curl f ) = 0.” ..?..
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12.02 - Section 13.9 wrap-up Let's use the Divergence...

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