Chapter 15, Problem 9P

(a) Show that when Laplace’s equation $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial z^2}=0$is written in cylindrical coordinates, it becomes $\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}u}{\partial {\theta }^2}+\frac{{\partial }^{2}u}{\partial z^2}=0$

(b) Show that when Laplace’s equation is written in spherical coordinates, it becomes $\frac{{\partial }^{2}u}{\partial {\rho }^2}+\frac{2}{\rho }\frac{\partial u}{\partial \rho }+\frac{\cot \varphi }{{\rho }^2}\frac{\partial u}{\partial \varphi }+\frac{1}{{\rho }^2{\sin }^2\varphi }\frac{{\partial }^{2}u}{\partial {\theta }^2}=0$