For two independent variables, the Laplace's equation in cartesian coordinates is


\begin{displaymath}\nabla^{2}\phi = \frac{\partial^{2}\phi }{\partial x^{2}}+\frac{\partial^{2}\phi }{\partial y^{2}} = 0 \notag
\end{displaymath}  

For polar coordinates $(r,\theta)$, Laplace's equation assume the form


\begin{displaymath}\nabla ^2\phi = \frac{1}{r}\frac{\partial\phi }{\partial r}\l...
...{1}{r^2}\frac{\partial^2 \phi}{\partial \theta ^2} = 0, \notag
\end{displaymath}  

where $x = r\cos\theta$ and $y=r\sin\theta$.

For three independent variables, Laplace's equation in the cartesian $(x,y,z)$ coordinate system is


\begin{displaymath}\frac{\partial^{2} \phi}{\partial x^{2}}+\frac{\partial^{2} \...
...ial y^{2}}+\frac{\partial^{2} \phi}{\partial z^{2}} = 0 \notag
\end{displaymath}  


 
Figure 1: cartesian coordinate system
\begin{figure}
\centering\epsfig{file=lfig106.eps,height=1.8in,clip=}\centering \notag
\end{figure}

For Cylindrical $(r,\theta ,z)$ coordinate system, Laplace's equation has the form


\begin{displaymath}\nabla ^2\phi = \left( {\underbrace {\frac{\partial ^2\phi
}{...
...al \theta ^2} +
\frac{\partial ^2\phi }{\partial z^2}} \right)
\end{displaymath}

where $x = r\cos\theta$ and $y=r\sin\theta$.


 
Figure 2: Cylindrical coordinate system
\begin{figure}
\centering\epsfig{file=/afs/athena.mit.edu/course/other/fluids-mo...
...ows/WavesDef/DefWeb/lfig107.eps,height=1.8in,clip=}\centering\notag
\end{figure}

For Spherical $(r,\theta ,\varphi)$ coordinate system, Laplace's equation has the form


\begin{displaymath}\nabla ^2\phi = \left( {\underbrace {\frac{\partial ^2\phi }{...
...2\theta }\frac{\partial ^2\phi }{\partial
\varphi ^2}} \right)
\end{displaymath}


 
Figure 3: Spherical coordinate system.
\begin{figure}
\centering\epsfig{file=lfig108.eps,height=1.8in,clip=}\centering\notag
\end{figure}



Timothy S Choe
2003-03-15