# Change of variable in 4D space-time

I was reading Sidney Coleman's article "Fate of the false vacuum: Samiclassical theory" and i stumbled upon a change of variables that i can't seem to prove. The problem is this: trying to solve the equation (3.1)

$$\left( \frac{\partial^2}{\partial\tau^2}+\nabla^2\right)\phi = \frac{dU(\phi)}{d\phi}$$

he uses the variable $$\rho = (\tau^2+|x|^2)^{1/2}$$ to simplify the equation being $$\phi$$ invariant under four-dimensional Euclidian rotations. Under this change of variable the equation becomes

$$\frac{d^2\phi}{d\rho^2}+\color{red}{\frac{3}{\rho}}\frac{d\phi}{d\rho}.$$

I cannot get that red factor. Evaluating the change of variable as such

$$\frac{\partial}{\partial\tau}=\frac{\partial\rho}{\partial\tau}\frac{\partial}{\partial\rho}\\ \frac{\partial}{\partial x} = \frac{\partial\rho}{\partial x}\frac{\partial}{\partial\rho}$$

and to the same extent, evaluating the second partial derivatives, gets me a factor $$1/\rho$$ instead of $$3/\rho$$.

Why is that? How should I evaluate that change of variables?

The chain rule for partial derivatives is rather more complicated than you seem to think. Your expressions for the change of coordinates are not correct. Look at a derivation of the Laplacian in three dimensions to see how the $$(2/r)\partial_r$$ arises there.
If you want the Laplacian in $$n$$ dimensions, it is easiest to start from the action
$$\int r^{n-1} dr d\Omega \left(g^{\mu\nu}\partial_\mu \phi \partial_\nu \phi\right)$$ where $$d\Omega$$ refers to all the angular parts of the measure. The metric is $$ds^2= dr^2+r^{n-1}(Angle-parts)$$ The $$(n-1)/r$$ in the Laplacian comes from the $$n-1$$ in the $$r^{n-1} dr$$.