(14 points) Let f: R3 R and T: R3. →R³ be defined by f(x, y, z) = ln(x²+ y²+2²), T(p, 0,4)=(psin cos 0, psin sin, pcos). (a) (4 points) Write out the composition g(p, 0, 4) = (foT)(p,, ) explicitly. Then calculate the gradient Vg directly, i.e. without using the chain rule. (b) (4 points) Calculate the gradient Vf(x, y, z) where (x, y, z) = T(p, 0,4). (c) (6 points) Calculate the derivative matrix DT(p, 0, p). Then use the Chain Rule to calculate Vg(r,0,4).

Compute the Jacobian at the indicated point. G(r.1) = (r sin (1), r- cos (f)), (r.t) = (5,-) (Express numbers in exact form. Use symbolic notation and fractions where needed.) Jac(G) (5,) =

FCx): 2x-5xナ3 find f'co).

Let i be the unit imaginary number. Thus i2 = -1. (1) Observe that the usual algebra rule to distribute (a + b) (c + d) yields )= (cos u) ( cos v) – (sin u) (sin e) cos u + i sin u Cos v + i sin v COS [(- (cos u) (sin v) + (sin u) (cos v) + i COS 1 Rewrite the right-hand side of the above identity using addition formulas : { (cos u) ( cos v) (- sin u) (sin v) co« (u + v). COs u %3D COS (cos u) (sin v) + (sin u) (cos v) sin (u + v) . COS Answer u + v + i sin COS

Write the function f(z) = z + 1/z in the form f(z) = u(r, theta) + iv(r, theta) (Answer is f(z) = (r + 1/r)cos(theta) + i(r - 1/r)sin(theta)