We suppose we don't know the constant of integration
in the Normal density, call it *c*.
As the coordinates are independent the joint density distribution
is the product of the marginal densities:

First we note that this is the same value, whatever the point (

What is the density distribution of the distance to the center, for a random point taken from this joint density? Here is its picture in 3 d:

From which we get :

this has to integrate to one: