How to find difference.

Question

If $H:\mathbb R^n\to\mathbb R$ is convex.

Let $H(p)=\frac{1}{r}|p|^r$ where $|p|=\sqrt{p_1^2+..+p_n^2}$ and $1<r<\infty$.

I need to find $\nabla_pH$ but i confuse where $|p|$.

Please help for detail.

Answer 1

Note that $\nabla_p=\frac {\partial }{\partial p_1}\vec e_1+\frac {\partial }{\partial p_2}\vec e_2+\frac {\partial }{\partial p_3}\vec e_3+\ldots +\frac {\partial }{\partial p_n}\vec e_n$ and $|p|^r=(p_1^2+p_2^2+p_3^2+\ldots p_n^2)^{r/2}$. Just take the partials in the usual manner.