# Understanding the manipulation of Laplacian

1. Jun 25, 2009

### h_cet

Hi;

I am trying to understand the rytov approximation... and when I was studying that, I could not understand a manipilation...

ΔeØ + k2eØ = 0

▼[▼ØeØ] + k2eØ = 0

2ØeØ + (▼Ø)2eØ+k2eØ = 0

I can not understand these manipilations... for a long time, I have searched the properties of laplecian but I could not find any propertiy in connection with laplecian...

So, please explain how this property is working...

be well...

2. Jun 25, 2009

### Mute

It's an application of the rule

$$\nabla \cdot (\phi \mathbf{F}) = \nabla\phi \cdot \mathbf{F} + \phi \nabla \cdot \mathbf{F}$$,

where phi is a scalar function and F is a vector. For your case, the scalar function is $\exp[\Theta]$ and the vector is $\nabla \Theta$.

This comes from treating the laplacian as div grad: grad acts on exp(Theta) to give (grad Theta) exp[Theta] by the chain rule, and then the div acts on (grad Theta) exp[Theta].