# Homework Help: What is a laplacian of a laplacian?

1. Mar 10, 2010

### tiredryan

1. The problem statement, all variables and given/known data
What is a laplacian of a laplacian?
2. Relevant equations
$$laplacian = \Delta = \nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$$
3. The attempt at a solution
$$\nabla^2\nabla^2=\nabla^4$$
Also is this true?
$$\nabla^2\nabla^2=\frac{\partial^4}{\partial x^4}+\frac{\partial^4}{\partial y^4}+\frac{\partial^4}{\partial z^4}$$
Or does it include additional terms?
$$\nabla^2\nabla^2=\frac{\partial^4}{\partial x^4}+\frac{\partial^4}{\partial y^4}+\frac{\partial^4}{\partial z^4} +\frac{\partial^2}{\partial x^2}\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial x^2}\frac{\partial^2}{\partial z^2} +\frac{\partial^2}{\partial y^2}\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\frac{\partial^2}{\partial z^2} +\frac{\partial^2}{\partial z^2}\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial z^2}\frac{\partial^2}{\partial y^2}$$

Thanks.

Last edited: Mar 10, 2010
2. Mar 10, 2010

### tiny-tim

Hi tiredryan!

The second one (and the order of eg ∂x∂y doesn't matter, so you can cut out three of them, and double the matching three).