# laplace operator on a laplace

What is the result when Laplace operator is applied on Laplace operator, is it

$$\nabla^2(\nabla^2 r)= \nabla^4r = \frac{\partial^4 r}{\partial x^4} + \frac{\partial^4 r}{\partial y^4} + \frac{\partial^4 r}{\partial z^4}$$

or something else with more terms ?

Thanks

• I think we need to define what $\cdot$ means in this context. – manooooh Dec 12 '18 at 9:16

The way you put it is $$\nabla^2\left( \frac{\partial^2 r}{\partial x^2} + \frac{\partial^2 r}{\partial y^2} + \frac{\partial^2 r}{\partial z^2}\right)= \frac{\partial^4 r}{\partial x^4} + \frac{\partial^4 r}{\partial y^4} + \frac{\partial^4 r}{\partial z^4}+2\frac{\partial^4r}{\partial x^2\partial y^2} +2\frac{\partial^4r}{\partial x^2\partial z^2} +2\frac{\partial^4r}{\partial y^2\partial z^2}.$$