Can anyone give me a reference which explain the derivation of the partial differential operator expression for the laplacian on the euclidean n-dimensional space and on $S^n$ ?

One generally writes the laplacian on the n-dim euclidean space as a sum of a operator on the radial coordinate and $\frac{1}{r^2}$ times the laplacian on $S^n$.

And very often the laplacian on $S^n$ is written through a recursion relation.

I am looking for a reference which shows me the derivations of these.