I AM HAVING DIFFICULTY ANSWERING C,D,E.

Change of Coordinate System [70 points] Consider the polar coordinate system: I = T Cos 8; y = r sin 0. (1) By the chain rule, we can rewrite O/Or and 0/00 in terms of 8/Or and a/dy: a o or a dy a ay ' ae -rsine ar - + r cose . a dy ay' ar ar or dy ar = cos 0 . ar - + sine . a (2) ay Notice that 0/Or and 8/dy are operators. We can solve 8/Or and /dy algebraically with a/Or and 0/20 when r * 0: a a cos e a r sind ae sin e ay 1 a 1 cos e sin e a 1 a sing a rsin e 28 cose Or sin e cos 8 ay (3) cos 8 ar ar cos 8 dy Thus, we can solve for o/dy by cos 8 a r + sin 0 . (4) Substitute equation 4 into equation 2 and we can solve for O/Or by sine a r + cos 0 . ar (5) Now, we will be solving for the second-order partials in the Cartesian coordinate system with polar coordinate systems. Since a2 a a2 a ay (6) Substituting equations 4 and 5 into equations 6, we can express 0"/Or using the polar coordinate: a2 Arz sine a + cose . sine a ae ar r an + cose . sin 8 a sine a + cos 0 . a + cos e . sin 8 ar a ar ae + cos 8 . - ar Using the product rule, we can simplify this expression by a2 072 = cos' 0 . sin 8 a 2sine cose 2 sin & cos e sin 0 r arde 292 . (7) PAGE 2 Similarly, we can express 0"/Or using the polar coordinate and simplify into: ay 2 = sin?9 . cos- 8 0 2 sine cose 02 2 sin & cos # a cos' B Orde a92 . (8) Finally, we can now express the Laplacian operator, V = Or2 + dy2 using the polar coordinate: V2 =. 1 a 1 r or r2 202 Now we will apply the change of coordinate system from Cartesian to spherical. Recall that I = psin y cos 0; y = psin y sin 0; 2 = p cos . (9) (a) (15 points) Apply the chain rule to express 8/Op, 8/dy, and 8/20 using a/Or, a/dy, and 8/0z. (b) (20 points) Solve algebraically for 0/Oz, 0/dy, and 0/Oz with a/Op, a/dy, and a/de when p 0 and sing * 0. (Hint: you can use method of elimination to reduce the number of variables.) (c) (10 points) Express O'/Or with p, 4, 0, and their partials. (d) (10 points) Express 2/dy? with p, ip, 0, and their partials. (e) (10 points) Express "/0z' with p, ip, 0, and their partials. (f) (5 points) Express the Laplacian operator using spherical coordinate V2 ay2 az2