Consider the following relations between cylindrical coordinates (r, 0, z) and spherical coordinates (p, 0, 0):...

 

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Consider the following relations between cylindrical coordinates (r, 0, z) and spherical coordinates (p, 0, 0): r = p sin o, z = p cos, 0 = 0. Under the change of coordinates from cylindrical to spherical, let u(r, 0, z) = v(p, 0, 0). (a) Show that if the spherical coordinates are considered to be functions of the cylindrical coordinates, then the following relations hold. Pr = sin o, 1 = cos o, P 0 = 0 = 0. or = (b) Use the chain rule to show that Up Ug Pz = cos o, 1 O = Uz = P VA Pe = 0, sin o, de 0, 1 up sin + v6 cos , P 1 Up cos ovo sin o. P (c) Let u(r, 0, z) satisfy the PDE (partial differential equation) ure = Uzz. Find the corresponding PDE for v(p, 0,0). Consider the following relations between cylindrical coordinates (r, 0, z) and spherical coordinates (p, 0, 0): r = p sin o, z = p cos, 0 = 0. Under the change of coordinates from cylindrical to spherical, let u(r, 0, z) = v(p, 0, 0). (a) Show that if the spherical coordinates are considered to be functions of the cylindrical coordinates, then the following relations hold. Pr = sin o, 1 = cos o, P 0 = 0 = 0. or = (b) Use the chain rule to show that Up Ug Pz = cos o, 1 O = Uz = P VA Pe = 0, sin o, de 0, 1 up sin + v6 cos , P 1 Up cos ovo sin o. P (c) Let u(r, 0, z) satisfy the PDE (partial differential equation) ure = Uzz. Find the corresponding PDE for v(p, 0,0).

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