## Hessian of radial functions

Suppose ${u=u(r)}$ is a radial function on ${\mathbb{R}^n}$, here ${r=|x|}$. $\displaystyle u_{x_i}=u'\frac{x_i}{r}$ $\displaystyle u_{x_ix_j}=u''\frac{x_ix_j}{r^2}+u'(\frac{\delta_{ij}}{r}-\frac{x_ix_j}{r^3})=\frac{u'}{r}\delta_{ij}+(\frac{u''}{r^2}-\frac{u'}{r^3})x_ix_j$

therefore $\displaystyle \det D^2u = \left(\frac{u'}{r}\right)^{n}\det[ \delta_{ij}+\frac{r}{u'}(u''-\frac{u'}{r})\frac{x_ix_j}{r^2}]$ $\displaystyle =\left(\frac{u'}{r}\right)^{n}[1+\frac{r}{u'}(u''-\frac{u'}{r})]=\left(\frac{u'}{r}\right)^{n-1}u''$

If we use the polar coordinates ${(r,\theta_1,\cdots, \theta_{n-1})}$, and ${g=dr^2+r^2\sum_{i=1}^{n-1}d\theta_i^2}$ and the following fact $\displaystyle \nabla_X\partial_r=\begin{cases}\frac{1}{r}X&\text{if } X\text{ is tangent to }\mathbb{S}^{n-1}\\0 \quad &\text{if } X=\partial_r\end{cases}$

then one can calculate the Hessian of ${u}$ under this coordinates $\displaystyle Hess (u)(\partial_r,\partial_r)=u''$ $\displaystyle Hess (u)(\partial_{\theta_i},\partial_r)=0$ $\displaystyle Hess (u)(\partial_{\theta_i},\partial_{\theta_j})=ru'\delta_{ij}.$

Then $\displaystyle \frac{\det Hess (u)}{{\det g}}=\left(\frac{u'}{r}\right)^{n-1}u''$

If the metric is $g=dr^2+\phi^2ds_{n-1}^2$, then we will have $\displaystyle \frac{\det Hess (u)}{{\det g}}=\left(\frac{u'\phi'}{\phi}\right)^{n-1}u''$

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