Radial Laplacian - Knowlpedia

For $u(x)=F(r)$ , $r=|x|>0$ , in $\mathbb R^n$ , the radial Laplacian is

\Delta u=F''(r)+\frac{n-1}{r}F'(r).

This is the scalar Laplacian restricted to radial functions.

Calculation and the origin

Using $\partial_i r=x_i/r$ , differentiate $\partial_i u=F'(r)x_i/r$ and sum. At $r=0$ , the displayed singular coefficients require the regularity of the underlying Cartesian function. If $F(r)=H(r^2)$ with smooth $H$ , the formula extends to the origin and has value $2nH'(0)$ . A cylindrical radial scalar independent of its axial coordinate uses the transverse dimension $n=2$ .