Laplacian

Posted October 18th, 2007 by Isoscel

Laplace equation in two dimension
$\Delta h=\frac{\partial ^2h}{\partial x^2}+\frac{\partial ^2h}{\partial y^2}=0$ can be written in polar coordinates $x=\rho \cos \phi\:y=\rho \sin \phi$ with inverse restriction expressed by $\rho(x,y)=\sqrt{x^2+y^2}\: and \:\phi(x,y)=arctan \frac {y}{x}$
let $h(x,y)=g(\rho (x,y),\phi (x,y))$

Then

$\frac{\partial h}{\partial x}=\frac{\partial g}{\partial \rho}(\rho,\phi)\frac{\partial \rho}{\partial x}+\frac{\partial g}{\partial \phi}(\rho,\phi)\frac{\partial \phi}{\partial x}$

$\frac{\partial h}{\partial y}=\frac{\partial g}{\partial \rho}(\rho,\phi)\frac{\partial \rho}{\partial y}+\frac{\partial g}{\partial \phi}(\rho,\phi)\frac{\partial \phi}{\partial y}$

$\frac{\partial ^2h}{\partial x^2}= \frac{\partial ^2g}{\partial \rho^2}(\rho,\phi){\frac{\partial \rho}{\partial x}}{\frac{\partial \rho}{\partial x}}+ \frac{\partial ^2g}{\partial \phi\partial \rho}(\rho,\phi)\frac{\partial \phi}{\partial x}\frac{\partial \rho}{\partial x}+ \frac{\partial ^2g}{\partial \rho\partial \phi}(\rho,\phi)\frac{\partial \rho}{\partial x}\frac{\partial \phi}{\partial x}+ \frac{\partial ^2g}{\partial \phi^2}(\rho,\phi)\frac{\partial \phi}{\partial x}\frac{\partial \phi}{\partial x}+$

$+\frac{\partial g}{\partial \rho}(\rho,\phi)\frac{\partial ^2 \rho}{\partial ^2x}+\frac{\partial g}{\partial \phi}(\rho,\phi)\frac{\partial ^2\phi}{\partial x^2}$

$\frac{\partial ^2h}{\partial y^2}= \frac{\partial ^2g}{\partial \rho^2}(\rho,\phi){\frac{\partial \rho}{\partial y}}{\frac{\partial \rho}{\partial y}}+ \frac{\partial ^2g}{\partial \phi\partial \rho}(\rho,\phi)\frac{\partial \phi}{\partial y}\frac{\partial \rho}{\partial y}+ \frac{\partial ^2g}{\partial \rho\partial \phi}(\rho,\phi)\frac{\partial \rho}{\partial y}\frac{\partial \phi}{\partial y}+ \frac{\partial ^2g}{\partial \phi^2}(\rho,\phi)\frac{\partial \phi}{\partial y}\frac{\partial \phi}{\partial y}+$

$+\frac{\partial g}{\partial \rho}(\rho,\phi)\frac{\partial ^2 \rho}{\partial ^2y}+\frac{\partial g}{\partial \phi}(\rho,\phi)\frac{\partial ^2\phi}{\partial y^2}$

We have

$grad \rho =(\frac{x}{\sqrt{x^2+y^2}},\frac{y}{\sqrt{x^2+y^2}})\: grad \phi =(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2})$

If we suppose that our function is of class $C^2$ than mixed partial derivative are equal by Young or Schwartz theorem so we can write
(*)

$\frac{\partial ^2h}{\partial x^2}= \frac{\partial ^2g}{\partial \rho^2}(\rho,\phi){\frac{\partial \rho}{\partial x}}{\frac{\partial \rho}{\partial x}}+ 2\frac{\partial ^2g}{\partial \phi\partial \rho}(\rho,\phi)\frac{\partial \phi}{\partial x}\frac{\partial \rho}{\partial x}+ \frac{\partial ^2g}{\partial \phi^2}(\rho,\phi)\frac{\partial \phi}{\partial x}\frac{\partial \phi}{\partial x}+ \frac{\partial g}{\partial \rho}(\rho,\phi)\frac{\partial ^2 \rho}{\partial ^2x}+\frac{\partial g}{\partial \phi}(\rho,\phi)\frac{\partial ^2\phi}{\partial x^2}$

(**) $\frac{\partial ^2h}{\partial y^2}= \frac{\partial ^2g}{\partial \rho^2}(\rho,\phi){\frac{\partial \rho}{\partial y}}{\frac{\partial \rho}{\partial y}}+ 2\frac{\partial ^2g}{\partial \phi\partial \rho}(\rho,\phi)\frac{\partial \phi}{\partial y}\frac{\partial \rho}{\partial y}+ \frac{\partial ^2g}{\partial \phi^2}(\rho,\phi)\frac{\partial \phi}{\partial y}\frac{\partial \phi}{\partial y}+ \frac{\partial g}{\partial \rho}(\rho,\phi)\frac{\partial ^2 \rho}{\partial ^2y}+\frac{\partial g}{\partial \phi}(\rho,\phi)\frac{\partial ^2\phi}{\partial y^2}$

For a more condensed expression it is useful to introduce

$grad f=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})$ and inner product $<grad f,grad g>=\frac{\partial f}{\partial x}\frac{\partial g}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial g}{\partial y}$

Then summing up (*) and (**) we have
$\Delta h=<grad\rho ,grad\rho>\frac{\partial ^2g}{\partial \rho^2}(\rho,\phi)+2<grad\rho,grad\phi>\frac{\partial ^2g}{\partial \phi\partial \rho}(\rho,\phi)+<grad \phi,grad \phi>\frac{\partial ^2g}{\partial \phi^2}(\rho,\phi)+$
$+\Delta \rho\frac{\partial g}{\partial \rho}(\rho,\phi)+\Delta \phi\frac{\partial g}{\partial \phi}(\rho,\phi) $
We have $grad \rho =(\frac{x}{\sqrt{x^2+y^2}},\frac{y}{\sqrt{x^2+y^2}})\: grad \phi =(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2})$ so
$<grad\rho ,grad\rho>=1\:<grad\rho,grad\phi>=0\:<grad \phi,grad \phi>=\frac{1}{x^2+y^2}=\frac{1}{\rho ^2}$
$\Delta \rho =\frac{1}{\rho}\:\Delta \phi=0$ and finally

$\Delta h=\frac{\partial ^2g}{\partial \rho^2}(\rho,\phi)+\frac{1}{\rho ^2}\frac{\partial ^2g}{\partial \phi^2}(\rho,\phi)+\frac{1}{\rho}\frac{\partial g}{\partial \rho}(\rho,\phi)$

Laplacian

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