What is a laplacian of a laplacian?

[tiredryan]

Homework Statement

What is a laplacian of a laplacian?

Homework Equations

<br /> laplacian = \Delta = \nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}<br />

The Attempt at a Solution

Is the follow true?
<br /> \nabla^2\nabla^2=\nabla^4<br />
Also is this true?
<br /> \nabla^2\nabla^2=\frac{\partial^4}{\partial x^4}+\frac{\partial^4}{\partial y^4}+\frac{\partial^4}{\partial z^4}<br />
Or does it include additional terms?
<br /> \nabla^2\nabla^2=\frac{\partial^4}{\partial x^4}+\frac{\partial^4}{\partial y^4}+\frac{\partial^4}{\partial z^4}<br /> <br /> +\frac{\partial^2}{\partial x^2}\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial x^2}\frac{\partial^2}{\partial z^2}<br /> <br /> +\frac{\partial^2}{\partial y^2}\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\frac{\partial^2}{\partial z^2}<br /> <br /> +\frac{\partial^2}{\partial z^2}\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial z^2}\frac{\partial^2}{\partial y^2}<br />

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).