# Applying Green's Formula in 2D

1. Sep 19, 2010

### Somefantastik

for

$$\Delta w = \frac{ \partial^{2} w }{x_{1}^{2}} + \frac{ \partial^{2} w }{x_{2}^{2}}$$

and $$\nabla = \left(\frac{\partial}{\partial x_{1}},\frac{\partial}{\partial x_{2}}\right)$$

$$\int_{\Omega} \nabla v \nabla w d \Omega = \int_{\Gamma} v \frac{\partial w}{\partial n} d \Gamma - \int_{\Omega}v \Delta w d \Omega$$

where $$\Gamma$$ is the boundary of $$\Omega$$

so if I have

$$- \int_{\Omega}v \Delta w d \Omega$$

I can apply Green's thm to get

$$- \int_{\Omega}v \Delta w d \Omega = \int_{\Omega} \nabla v \nabla w d \Omega - \int_{\Gamma} v \frac{\partial w}{\partial n} d \Gamma$$

But what if I'm starting with

$$\int_{\Omega} v \left( \frac{\partial^{2} w }{\partial x_{1}^{2}} + k \frac{\partial^{2} w }{\partial x_{2}^{2}} \right) d \Omega$$

where k is some scalar? I'm thrown off by only one of the dimensions being scaled.