Chapter 15.P, Problem 9P

To determine

(a)

To show:

The 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 as $\frac{{\partial }^{2}u}{{\partial r}^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}u}{{\partial \theta }^2}++\frac{{\partial }^{2}u}{{\partial z}^2}=0$

To determine

(b)

To show:

The Laplace equation is written in the spherical co-ordinates as

$\frac{{\partial }^{2}u}{{\partial \rho }^2}+\frac{2}{\rho }\frac{\partial u}{\partial \rho }+\frac{\cot \varphi }{{\rho }^2}\frac{\partial u}{\partial \varphi }+\frac{1}{{\rho }^2}\frac{{\partial }^{2}u}{\partial {\varphi }^2}+\frac{1}{{\rho }^2{\sin }^2\varphi }\frac{{\partial }^{2}u}{{\partial \theta }^2}=0$