Relation between second order partial derivative and gradient

Question

Suppose that $h \in C^2(V)$ ($h$ is two times continuously differentiable in the set $V$), where $V \subset \mathbb{R}^n$ is an open subset. So, if $p \in V$, and $v \in \mathbb{R}^n$ is a non-zero vector, we can prove that $$\frac{\partial h(p)}{\partial v} = \langle \nabla h(p),v \rangle.$$ Now, can I relate $\frac{\partial^2h}{\partial v^2}$ to $\nabla h(p)$? How?

Answer 1

If you mean $d^2/dt^2(f(P+vt))$ then it equals $$ v^TH(P)v $$ Where $H(P)$ is the Hessian matrix of second derivatives.