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  
\[\frac{1}{r^2} \frac{\partial}{\partial r}(r^2 \frac{\partial U}{\partial r}) = \frac{1}{k} \frac{\partial U}{\partial t} \]

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