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317-w09-202-hw-05

# 317-w09-202-hw-05 - Math 317 Section 202 Homework no 5(due...

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Math 317, Section 202, Homework no. 5 (due Friday February 12, 2010) [7 problems, 26 points + 4 bonus points] Problem 1 (12 points) . Compute the gradient of the following potentials. We use the abbreviation r = h x, y, z i and r = | r | . (a) (4 points) f ( x, y, z ) = c r k where k = 0 , 1 , - 1 , 2 , - 2 , . . . and c is a constant. (b) (4 points) g ( x, y, z ) = c r e - mr , where c and m > 0 are constants. (c) (4 points) h ( x, y, z ) = c ln r r 0 , where c and r 0 > 0 are constants. [Hint: Try to be as economical as possible and present the results in a concise way. You will need them again. Background information: These gradients appear in the following applications. -∇ g is a good approximation to the strong nuclear force for scattering processes of not very high energy. -∇ f for k = - 1 is the form of the electrostatic and the gravitational force. -∇ f for k = 2 is the force that acts on an object attached to the origin by an elastic spring.] Problem 2 (2 points) . In your garden, there is a wooden fence whose base is the curve x 2 + y 2 = 2, y 0, and whose height is given by h ( x, y ) = 1 + x 2 + xy 2 . You plan to paint
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