Radial function ($f(x)=g(|x|)$)

A radial function depends only on the distance from the origin: $f(x)=g(|x|)$. Differential operators then reduce to radial formulas such as $\Delta f=g^{\prime \prime }+(n-1)g^{\prime }/r$.

A radial profile is a one-variable function $\bar{u}$ representing a radial function $u$ by $u(x)=\bar{u}(|x|)$. On an annulus $a<|x|<b$, with $a>0$, the spherical coordinate system identifies its Lp norm with the weighted profile norm through $\int |u{|}^p=|S^{n-1}|{\int }_a^b|\bar{u}(r){|}^p{r}^{n-1}dr$. Thus equality of two radial functions almost everywhere identifies their profiles almost everywhere in the ordinary one-dimensional sense.