Laplace's Equation in Cylindrical Coordinates
\[f\]
is \[\nabla^2 = \frac{\partial^2 f}{\partial x^2}+ \frac{\partial^2 f}{\partial y^2}+ \frac{\partial^2 f}{\partial z^2}\]
.In cylindrical polar coordinates
\[(r, \theta , z)\]
with \[r= \sqrt{x^2 +y^2} , \: \theta = tan^{-1} (\frac{y}{x}) , \: z=z\]
Hence
\[\frac{\partial r}{\partial x} = \frac{\partial}{\partial x}( \sqrt{x^2 +y^2}) = \frac{x}{\sqrt{x^2 +y^2}} = \frac{x}{r}= cos \theta , \: \frac{\partial r}{\partial y} = \frac{\partial}{\partial y}( \sqrt{x^2 +y^2}) = \frac{y}{\sqrt{x^2 +y^2}} = \frac{y}{r}= sin \theta\]
\[\theta = tan^{-1} (\frac{y}{x}) \rightarrow tan \theta = \frac{y}{x} \]
Differentiating:
\[ sec^2 \theta \frac{\partial \theta}{\partial x} =- \frac{y}{x^2} \rightarrow \frac{\partial \theta}{\partial x} = - \frac{y}{x^2 sec^2 \theta} = - \frac{y}{x^2 (1+ tan^2 \theta )} = - \frac{y}{x^2 (1+ y^2/x^2)}= - \frac{y}{r^2} = - \frac{sin \theta}{r}\]
\[ sec^2 \theta \frac{\partial \theta}{\partial y} = \frac{1}{x} \rightarrow \frac{\partial \theta}{\partial y} = \frac{1}{x sec^2 \theta} = \frac{1}{x (1+ tan^2 \theta )} = \frac{x}{x^2 (1+ y^2/x^2)}= \frac{cos \theta}{r} \]
and of course
\[\frac{\partial z}{\partial x}=0.\]
Hence
\[\frac{\partial f}{\partial x} = cos \theta \frac{\partial f}{\partial r} - \frac{sin \theta}{r} \frac{\partial f}{\partial \theta}\]
\[\frac{\partial f}{\partial y} = sin \theta \frac{\partial f}{\partial r} + \frac{cos \theta}{r} \frac{\partial f}{\partial \theta}\]
It follows that
\[\frac{\partial }{\partial x} = cos \theta \frac{\partial }{\partial r} - \frac{sin \theta}{r} \frac{\partial }{\partial \theta} , \: \frac{\partial }{\partial y} = sin \theta \frac{\partial }{\partial r} + \frac{cos \theta}{r} \frac{\partial f}{\partial \theta}\]
and
\[\begin{equation} \begin{aligned} \frac{\partial^2 f}{\partial x^2} &= \frac{\partial}{\partial x} (\frac{\partial f}{\partial x}) \\ &= (cos \theta \frac{\partial }{\partial r} - \frac{sin \theta}{r} \frac{\partial }{\partial \theta}) (cos \theta \frac{\partial f}{\partial r} - \frac{sin \theta}{r} \frac{\partial f}{\partial \theta}) \\ &= cos^2 \theta \frac{\partial^2 f}{\partial r^2} + \frac{cos \theta sin \theta}{r^2} \frac{\partial f}{\partial \theta} - \frac{cos \theta sin \theta}{r} \frac{\partial^2 f}{\partial r \partial \theta} + \frac{sin^2 \theta}{r} \frac{\partial f}{\partial r} -\frac{sin \theta cos \theta}{r} \frac{\partial^2 f}{\partial \theta \partial r} \\ &+ \frac{ sin \theta cos \theta}{r^2} \frac{\partial f}{\partial \theta} + \frac{sin^2 \theta}{r^2} \frac{\partial^2 f}{\partial r^2} \\ &= cos^2 \theta \frac{\partial^2 f}{\partial r^2} + \frac{sin^2 \theta}{r} \frac{\partial f}{\partial r} -\frac{2sin \theta cos \theta}{r} \frac{\partial^2 f}{\partial \theta \partial r} + \frac{ 2sin \theta cos \theta}{r^2} \frac{\partial f}{\partial \theta} + \frac{sin^2 \theta}{r^2} \frac{\partial^2 f}{\partial \theta^2} \end{aligned} \end{equation}\]
\[\begin{equation} \begin{aligned} \frac{\partial^2 f}{\partial y^2} &= \frac{\partial}{\partial y} (\frac{\partial f}{\partial t}) \\ &= (sin \theta \frac{\partial }{\partial r} + \frac{cos \theta}{r} \frac{\partial f}{\partial \theta}) (sin \theta \frac{\partial f}{\partial r} + \frac{cos \theta}{r} \frac{\partial f}{\partial \theta}) \\ &= sin^2 \theta \frac{\partial^2 f}{\partial r^2} - \frac{sin \theta cos \theta}{r^2} \frac{\partial f}{\partial \theta} - \frac{sin \theta cos \theta}{r} \frac{\partial^2 f}{\partial r \partial \theta} + \frac{cos^2 \theta}{r} \frac{\partial f}{\partial r} +\frac{cos \theta sin \theta}{r} \frac{\partial^2 f}{\partial \theta \partial r} \\ &- \frac{ cos \theta sin \theta}{r^2} \frac{\partial f}{\partial \theta} + \frac{cos^2 \theta}{r^2} \frac{\partial^2 f}{\partial r^2} \\ &= sin^2 \theta \frac{\partial^2 f}{\partial r^2} + \frac{cos^2 \theta}{r} \frac{\partial f}{\partial r} -\frac{2cos \theta sin \theta}{r} \frac{\partial^2 f}{\partial \theta \partial r} + \frac{ 2cos \theta sin \theta}{r^2} \frac{\partial f}{\partial \theta} + \frac{cos^2 \theta}{r^2} \frac{\partial^2 f}{\partial \theta^2} \end{aligned} \end{equation}\]
Then
\[\begin{equation} \begin{aligned} \nabla^2 f& = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}+ \frac{\partial^2 f}{\partial z^2} \\ &= cos^2 \theta \frac{\partial^2 f}{\partial r^2} + \frac{sin^2 \theta}{r} \frac{\partial f}{\partial r} -\frac{2sin \theta cos \theta}{r} \frac{\partial^2 f}{\partial \theta \partial r} + \frac{ 2sin \theta cos \theta}{r^2} \frac{\partial f}{\partial \theta} + \frac{sin^2 \theta}{r^2} \frac{\partial^2 f}{\partial \theta^2} \\ &+ sin^2 \theta \frac{\partial^2 f}{\partial r^2} + \frac{cos^2 \theta}{r} \frac{\partial f}{\partial r} -\frac{2cos \theta sin \theta}{r} \frac{\partial^2 f}{\partial \theta \partial r} + \frac{ 2cos \theta sin \theta}{r^2} \frac{\partial f}{\partial \theta} + \frac{cos^2 \theta}{r^2} \frac{\partial^2 f}{\partial \theta^2} + \frac{\partial^2 f}{\partial z^2} \\ &= \frac{\partial^2 f}{\partial r^2} + \frac{1}{r} \frac{\partial f}{\partial r} + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2} + \frac{\partial^2 f}{\partial z^2} \\ &= \frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial f}{\partial r}) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2} + \frac{\partial^2 f}{\partial z^2} \end{aligned} \end{equation}\]
