Let $x=(x_1,x_2,...)\in\mathbb R^n$. $f(x)$ is a multi-variate real function.
Consider a new orthonormal basis $y=(y_1,y_2,...)$ generated by directional vectors $V=(v_1,v_2,...)$ where each $v_1,v_2\in\mathbb R^n$. Obviously the coordinate of the point $x$ in the old system becomes $y=Vx$ in the new system. Let the graph of function $f(x)$ be identical with $g(y)$, then we have:
$$\frac{\partial g(y)}{\partial y_1}=v_1\cdot\nabla f(x)$$
$$\nabla g(y)=V\nabla f(x).$$
Let $\partial^2 f(x)$ denote the Hessian matrix of $f(x)$, the second partial derivative:
$$g_{y_1y_1}=v_1^T\partial^2f(x)v_1.$$
The question is, how to express $g_{y_1y_2}$ and $\partial^2 g(y)$ in term of $f(x)$?
One might conjecture that $g_{y_1y_2}=v_1^T\partial^2 f(x)v_2$, and $\partial^2 g(y)=V\partial^2f(x) V$?