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 theabbreviationr=hx, y, ziandr=|r|.(a) (4 points)f(x, y, z) =c rkwherek= 0,1,-1,2,-2, . . .andcis a constant.(b) (4 points)g(x, y, z) =cre-mr, wherecandm >0 are constants.(c) (4 points)h(x, y, z) =clnrr0, wherecandr0>0 are constants.[Hint: Try to be as economical as possible and present the results in a concise way.Youwill need them again.Background information:These gradients appear in the followingapplications.-∇gis a good approximation to the strong nuclear force for scattering processesof not very high energy.-∇ffork=-1 is the form of the electrostatic and the gravitationalforce.-∇ffork= 2 is the force that acts on an object attached to the origin by an elasticspring.]Problem 2(2 points).In your garden, there is a wooden fence whose base is the curvex2+y2= 2,y≥0, and whose height is given byh(x, y) = 1 +x2+xy2. You plan to paint
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2 K, Strong interaction, Fundamental physics concepts, Gradient