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

1. ## 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:

. . . . .$\Delta f\, =\, \dfrac{\partial^2 f}{\partial x^2}\, +\, \dfrac{\partial^2 f}{\partial y^2}$

. . . . .\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}

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.