Consider the laplace equation for a ball of radius r described in spherical coordinates (r, 0) 2 1 cot 72 0= n where is the zenith angle and assume u is independent on the azirnuth angle d. a) by separation of variables, derive two ordinary differential equations of r and w: = cos e given by 2 f" (r) +2rf (r) - n(n + 1) f, (r) = 0, (1 w2)g (w) - 2wg", (w) +n(n +1)g, (w) 0. (n 0,1,2,. .) b) find fn (r) and gn (w) satisfying g(1) = 1 for n = 1,3. c) show that for n m, gn (w)gm (w)dw = 0. d) assume the boundary condition is given by u(r, o) = cos 0. then determine u(r, ) inside the ball.

Which of the following represents an upside-down image?

Explain the difference between each pair of concepts. a. frequency and relative frequency b. percentage and relative frequency a. select the correct choice below. a. frequency is the total number of observations in a data set. relative frequency is the number of times a particular distinct value occurs. b. frequency is the number of times a particular distinct value occurs. relative frequency is the ratio of the frequency of a value to the total number of observations. c. frequency is the total number of observations in a data set. relative frequency is the ratio of the number of times a particular distinct value occurs to the frequency. d. frequency is the number of times a particular distinct value occurs. relative frequency is the ratio of the frequency of two different values.

Afeather of mass 0.001 kg falls from a height of 2 m. under realistic conditions, it experiences air resistance. based on what you know about friction, what can you say about the kinetic energy of the feather as it reaches the ground? acceleration due to gravity is g = 9.8 m/s2. a. ke < 0.0196 j b. ke = 00196 j c. je = 0 j d. ke > 0.0196 j