In $\mathbb{R}^n$ consider $f = f(r)$. Writing $\partial_j \equiv \partial/\partial x_j$ and $\partial_r \equiv \partial/\partial_r$, etc., we have that $\partial_j r = x_j/r$, so

$\partial_j f = (\partial_r f)(x_j r^{-1})$

and

$\partial_{jj} f = (\partial_{rr} f)(x_j^2 r^{-2}) + (\partial_r f)(r^{-1} - x_j^2 r^{-3})$.

Summing over $j$ and comparing with the Cartesian expression for $\Delta$ gives the decomposition into radial and spherical operators.

For a more general case, see the end of Chapter 2 of The Laplacian on a Riemannian manifold: an introduction to analysis on manifolds by Rosenberg. Unfortunately some of the relevant section (the Laplacian in exponential coordinates) is blocked both from Amazon and Google:

http://books.google.com/books?id=gzJ6Vn0y7XQC&dq=laplacian+on+a+riemannian+manifold&source=gbs_navlinks_s

In $\mathbb{R}^n$ consider $f = f(r)$. Writing $\partial_j \equiv \partial/\partial x_j$ and $\partial_r \equiv \partial/\partial_r$, etc., we have that $\partial_j r = x_j/r$, so

$\partial_j f = (\partial_r f)(x_j r^{-1})$

and

$\partial_{jj} f = (\partial_{rr} f)(x_j^2 r^{-2}) + (\partial_r f)(r^{-1} - x_j^2 r^{-3})$.

Summing over $j$ and comparing with the Cartesian expression for $\Delta$ gives the decomposition into radial and spherical operators. To be explicit you should consider $f = f(r, \omega)$, where $\omega \in S^{n-1}$.

For a more general case, see the end of Chapter 2 of The Laplacian on a Riemannian manifold: an introduction to analysis on manifolds by Rosenberg. Unfortunately some of the relevant section (the Laplacian in exponential coordinates) is blocked both from Amazon and Google:

http://books.google.com/books?id=gzJ6Vn0y7XQC&dq=laplacian+on+a+riemannian+manifold&source=gbs_navlinks_s

The end of Chapter 2 of The Laplacian on a Riemannian manifold: an introduction to analysis on manifolds by Rosenberg. Unfortunately some of the relevant section (the Laplacian in exponential coordinates) is blocked both from Amazon and Google:

http://books.google.com/books?id=gzJ6Vn0y7XQC&dq=laplacian+on+a+riemannian+manifold&source=gbs_navlinks_s

In $\mathbb{R}^n$ consider $f = f(r)$. Writing $\partial_j \equiv \partial/\partial x_j$ and $\partial_r \equiv \partial/\partial_r$, etc., we have that $\partial_j r = x_j/r$, so

$\partial_j f = (\partial_r f)(x_j r^{-1})$

and

$\partial_{jj} f = (\partial_{rr} f)(x_j^2 r^{-2}) + (\partial_r f)(r^{-1} - x_j^2 r^{-3})$.

Summing over $j$ and comparing with the Cartesian expression for $\Delta$ gives the decomposition into radial and spherical operators.

For a more general case, see the end of Chapter 2 of The Laplacian on a Riemannian manifold: an introduction to analysis on manifolds by Rosenberg. Unfortunately some of the relevant section (the Laplacian in exponential coordinates) is blocked both from Amazon and Google:

http://books.google.com/books?id=gzJ6Vn0y7XQC&dq=laplacian+on+a+riemannian+manifold&source=gbs_navlinks_s