What is a laplacian of a laplacian?

  • Thread starter tiredryan
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  • #1
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Homework Statement


What is a laplacian of a laplacian?

Homework Equations


[tex]
laplacian = \Delta = \nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}
[/tex]

The Attempt at a Solution


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

Thanks.
 
Last edited:

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,832
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Hi tiredryan! :smile:

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). :wink:
 

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