Change of variable for Laplace equation

• Nov 21st 2009, 05:22 PM
jkhayer
Change of variable for Laplace equation
How do I show that the change of variable:
x = rcosθ , y = rsinθ

converts the Laplace equation uxx + uyy = 0 to

urr + 1/r ur + 1/r2 uθθ = 0

Here is my understanding:

uxx is the second partial derivative of u with respect to x, also written as d2u / d2x, but what is the equation with u in it that I have to differentiate?

• Nov 22nd 2009, 08:08 AM
Jester
Quote:

Originally Posted by jkhayer
How do I show that the change of variable:

x = rcosθ , y = rsinθ

converts the Laplace equation uxx + uyy = 0 to
urr + 1/r ur + 1/r2 uθθ = 0

Here is my understanding:

uxx is the second partial derivative of u with respect to x, also written as d2u / d2x, but what is the equation with u in it that I have to differentiate?

Use the transformation rules

$
u_x = \cos \theta u_r - \frac{\sin \theta}{r} u_{\theta},\;\;\;u_y = \sin \theta u_r + \frac{\cos \theta}{r} u_{\theta}
$

and the operators

$
\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}{\partial \theta}.
$

Find $\frac{\partial}{\partial x} \left(u_x \right)$ and similarly for $\frac{\partial}{\partial y} \left(u_y \right)$ using the above, add and simplify.