## The Heat Equation in Spherical Polar Coordinates

In spherical polar coordinates the Heat Equation
$k \nabla^2 U =\frac{\partial U}{\partial t}$
becomes
$\frac{\partial}{\partial r}(r^2 \frac{\partial U}{\partial r}) + \frac{1}{r^2 sin \theta} \frac{\partial}{\partial \theta} (sin \theta \frac{\partial U}{\partial \theta}) + \frac{1}{r^2 sin^2 \theta} \frac{\partial^2 U}{\partial \phi^2} = \frac{1}{k} \frac{\partial U}{\partial t}$

If
$U=U(\theta , \phi)$
then
$\frac{\partial U}{\partial r} = \frac{\partial U}{\partial t} =0$

The above equation simplifies to
$\frac{\partial}{\partial \theta} (sin \theta \frac{\partial U}{\partial \theta}) + \frac{1}{ sin \theta} \frac{\partial^2 U}{\partial \phi^2} = 0$

If
$U=U(r ,t)$
then
$\frac{\partial U}{\partial \theta} = \frac{\partial U}{\partial \phi} =0$

The equation simplifies to
$\frac{1}{r^2} \frac{\partial}{\partial r}(r^2 \frac{\partial U}{\partial r}) = \frac{1}{k} \frac{\partial U}{\partial t}$