# Laplacian Differentiation

• Apr 24th 2009, 05:18 PM
Laplacian Differentiation
Let u(x, y) be a twice differentiable function. Show that in polar coordinates $x = r \cos(\theta), y = r \sin(\theta)$ the Laplacian of u takes the form...
$u_{xx} + u_{yy} = u_{rr} + \frac{1}{r} u_{r} + \frac{1}{r^2} \frac{{\partial}^2 u}{{\partial} {\theta}^2}$.

Should be an easy Q but i cant remember how to do the second derivative.

$u_{x} = u_{r} \frac{\partial r}{\partial x} + u_{\theta} \frac{\partial \theta}{\partial x}$ = $u_{r} \frac{x}{r} + u_{\theta} \frac{-y}{r^2}$.

But then how do you take the second derivative?
• Apr 24th 2009, 06:04 PM
Calculus26
http://www.mathhelpforum.com/math-he...12bc8738-1.gif = http://www.mathhelpforum.com/math-he...87deb5a7-1.gif

replace x = rcos(theta) y = rsin(theta)

This won't be pleasant but:

Uxx = (Ux)x so in http://www.mathhelpforum.com/math-he...12bc8738-1.gif replace U with Ux

similarly for Uyy