What is a laplacian of a laplacian?

[tiredryan]

Homework Statement

What is a laplacian of a laplacian?

Homework Equations

$$laplacian = \Delta = \nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$$

The Attempt at a Solution

Is the follow true?
$$\nabla^2\nabla^2=\nabla^4$$
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.

 

[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).