Definitions, Theorems, Examples, Topics for Asmar 2nd edition Chapter 4: 4.1 only 4.1 ==== SUMMARY of RESULTS The Laplace equation u_xx + u_yy =0 can be expressed in polar coordinates, cylindricl coordinates or spherical coordinates. The new equations, resp., are: u[r][r] + (1/r)u[r] + (1/r^2)u[theta][theta] = 0 u[rho][rho] + (1/rho)u[rho] + (1/r^2)u[phi][phi] + u[z][z] = 0 u[r][r] + (2/r)u[r] + (1/r^2)(u[theta][theta] + cot(theta)u[theta] + (csc(theta))^2 u[phi][phi] = 0 DEF. Polar coordinates x = r cos(theta), y = r sin(theta) r^2 = x^2 + y^2, tan(theta) = y/x THEOREM. The Laplace equation u_xx + u_yy =0 in polar coordinates is the equation u[r][r] + (1/r)u[r] + (1/r^2)u[theta][theta] = 0 DEF. Cylindrical coordinates x = rho cos(phi), y = rho sin(phi), z = z THEOREM. The Laplace equation u_xx + u_yy =0 in cylindrical coordinates is the equation u[rho][rho] + (1/rho)u[rho] + (1/r^2)u[phi][phi] + u[z][z] = 0 DEF. Spherical coordinates x = r cos(phi) sin(theta), y = r sin(phi) sin(phi), z =r cos(theta) r^2 = x^2 + y^2 + z^2 THEOREM. The Laplace equation u_xx + u_yy =0 in cylindrical coordinates is the equation u[r][r] + (2/r)u[r] + (1/r^2)(u[theta][theta] + cot(theta)u[theta] + (csc(theta))^2 u[phi][phi] = 0 EXAMPLE 1. Use spherical coordinates to compute the Laplacian of the function f(x,y,z) = ln(r^2) = 2 ln(r). ANSWER: Laplacian of f(x,y,z) = 2/r^2.