I am a student of mathematics, currently I am taking a course of PDE's, and I found myself having some troubles with some partial derivatives, I would thanks a lot any help, here is the problem:
Prove that if $u=u(x,y,z)$ satisfy the laplace equation ($\nabla u =0$), then $$ \nabla \left[r^{-1}u(\frac{x}{r^2},\frac{y}{r^2},\frac{z}{r^2})\right]=0, \quad r=\sqrt{x^2+y^2+z^2}$$ is true. To do this I take a function $$w(x,y,z)=r^{-1}u(\frac{x}{r^2},\frac{y}{r^2},\frac{z}{r^2})$$ and started to do derivates, the derivate of a product and chain rule, the problem that I am having is with the $\frac{\cdot}{r^2}$ thing, I think that it must be something like the chain rule but with an another thing that I am not seeing, I am just confused about this, thanks to everyone that wants to help me.
This is the Kelvin transformation law for the Laplacian. Probably the simplest brute force method for verifying it is to expand the Laplacian $\nabla^2 f (r,\theta, \phi)=0$ in spherical coordinates, and then see what happens when you make the transformation $r\to \tilde r=1/r$..
The other brute force method you outlined also works .. eventually.