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317-w09-202-hw-08 (1)

317-w09-202-hw-08 (1) - Math 317 Section 202 Homework no...

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Math 317, Section 202, Homework no. 8 (due Friday March 19, 2010) [8 problems, 40 points + 5 bonus points] Problem 1 (4 points) . Suppose F ( x, y, z ) = g ( r ) r with r = h x, y, z i , r = | r | and a continu- ously differentiable function g ( r ), r > 0. Find all such functions g ( r ) for which ∇ · F = 0. Problem 2 (4 points) . Suppose f ( x, y, z ) = h ( r ) with r = | r | , r = h x, y, z i and some twice continuously differentiable function h ( r ), r > 0. Find all such functions h ( r ) for which f satisfies Laplace’s equation. Problem 3 (6 points, 2 points each) . Suppose F ( x, y, z ) is a vector field, and f ( x, y, z ) and g ( x, y, z ) are real valued functions such that all second order partial derivatives exist. Show that (a) ∇ × ( f F ) = f ( ∇ × F ) + ( f ) × F , (b) ∇ · ( f F ) = f ( ∇ · F ) + ( f ) · F , (c) ∇ · ( f g - g f ) = f 2 g - g 2 f. Problem 4 (6 points) . Stewart, Chapter 17.6, Exercises 13–18 (1 point each). No explanation required; only the answer counts. Problem 5 (10 points, 2 points each) . Parameterize the following surfaces. [Hint: Make sure you do not forget to state the domain of your parameterization r (
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