362-6 - MAT 362 Answers to selected exercises week of Oct 3...

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MAT 362 Answers to selected exercises, week of Oct. 3 Graded problems 1. The divergence theorem applies to flux of a smooth vector field across a surface that encloses a volume . The fallacy in the argument lies in its use of Stokes’ theorem, which cannot be applied here. Stokes’ theorem applies to a capping surface . A sphere is not a capping surface: it encloses a volume (hence is not a “cap”) and has no unique boundary curve. 2. If u and v are smooth functions (which you may assume), then u v is a smooth vector field, so the divergence theorem applies. All that is needed here is to show that ∇ · ( u v ) = u 2 v + ( u ) · ( v ) , which you can do by writing out the individual terms: ∇ · ( u v ) = ∇ · ( uv x , uv y , uv z ) = ( u x v x + uv xx , u y v y + uv yy , u z v z + uv zz ) = ( u ) · ( v ) + u 2 v . The second step follows from the product rule. (To simplify the notation, I’ve used v x = v /∂ x and v xx = 2 v /∂ x 2 , etc.) 3. This statement follows immediately from Exercise 2: just substitute v = u in Exercise 2 and use the hypothesis that 2 u = 0. The dot product x · x = k x k 2 for any vector
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362-6 - MAT 362 Answers to selected exercises week of Oct 3...

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