Solve Laplace's equation inside a 60° wedge of radius a, 1 ³u = 0, r2 a02 ди Osrsa, 0s0s r år ar subject to the boundary conditions u(r,0) = 0, u(r,) = 0, u(a,0) = f(). = 0, u(a,0) = f(0). %3D

5. A surface S is described by the equation f(x, y, z) = n where f (x, y, z) = yz + nx²y+ yz³ (a) Find the equation of the tangent plane to the surface S at the point P(1,1,0). (b) Find the directional derivative of the function f(x, y, z) at the point P(1, 1,0) and in the direction u = ni-j+nk

2. In spherical coordinates, (r, 0, ø), the gradient takes the form 1 df φ. r sin 0 dø 1ðf Vf (r, 0, 4) se -î + dr r d0 Determine the gradient of the following functions (a) f (r) = r. (b) ƒ (r) = Aº_" . (c) ƒ (r, 0, 4) = Ac©os(2m0)

Show that the function u = 9(ry) + /TY v(y/x) provides the uation ‚²u y? 0. dy?