olve the following Legendre polynomials and spherical harmonics below. As a reminder, P0(x) = 1, P1(x) = x, P2(x) = 1 2 (3x2 1). You may use results for spherical coordinates as found in your LP and SPH tutorials. (a) Show explicitly that P2(x) satisfies the Rodrigues formula. (b) Using relationships between the Legendre polynomials, associated Legendre functions, and spherical harmonics, find Y 1 2 (, ). (c) The surface temperature on a sphere of radius a is held at = 0 cos2 . Find the steady state temperature distribution inside the sphere. (d) Consider a hemisphere of radius a. The equatorial plane z = 0 is held at zero Celsius, while the curved half-sphere surface is held at (a, , ) = 3 4 0 cos (1 + sin cos ). Find the steady-state temperature distribution everywhere inside the hemisphere