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Homework Help: What is a laplacian of a laplacian?

  1. Mar 10, 2010 #1
    1. The problem statement, all variables and given/known data
    What is a laplacian of a laplacian?
    2. Relevant 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]
    3. 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: Mar 10, 2010
  2. jcsd
  3. Mar 10, 2010 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    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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