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Thread: Third-order Laplacian from 2nd-order: delta-f = (partial^2 f)/(partial x^2) + (...

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    Third-order Laplacian from 2nd-order: delta-f = (partial^2 f)/(partial x^2) + (...

    Hi, I am trying to find the third-order Laplacian. The second order is given:

    . . . . .[tex]\Delta f\, =\, \dfrac{\partial^2 f}{\partial x^2}\, +\, \dfrac{\partial^2 f}{\partial y^2}[/tex]

    . . . . .[tex]\begin{align}\Delta f\, &=\, \dfrac{1}{4}\, \dfrac{\partial}{\partial r}\, \left(r\, \dfrac{\partial f}{\partial r}\right)\, +\, \dfrac{1}{r^2}\, \dfrac{\partial^2 f}{\partial \theta^2} \\ \\&=\, \dfrac{\partial^2 f}{\partial r^2}\, +\, \dfrac{1}{r}\, \dfrac{\partial f}{\partial r}\, +\, \dfrac{1}{r^2}\, \dfrac{\partial^2 f}{\partial \theta^2}\end{align}[/tex]

    How can I find the third order Laplacian, where the differential operator in the first eqn. is not the second derivative, but to the third?

    I take I could exponentiate the first order Laplacian to the third order, however I cannot find the first order Laplacian either.
    Last edited by stapel; 03-07-2018 at 05:12 PM. Reason: Typing out the text in the graphics; creating useful subject line.

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