I came across this, and I'm having difficulties understanding what the making means.
Let $U\subset \mathbb{R}^n $ an open subset, $f\in C^2 (U\to\mathbb{R}).$ For every $x\in U$ mark $H(x)_{ij}=\frac{\partial^2f}{\partial{x_i}\partial{x_j}}(x)$ . Assuming $x$ is a critical point, show that: $$<H(x)u,v>=\frac{\partial{f}^2}{\partial{u}\partial{v}}(x)$$
What I managed to get to so far (Left Hand Side): $$\begin{align} <H(x)u,v>=<\sum_{k=1}^{n}H(x)_{ik}u_k,v> = ... =\sum_{l=1}^{n}\sum_{k=1}^{n}\frac{\partial^2f}{\partial{x}_l\partial{x}_k}u_kv_l \end{align}$$
My questions:
- What does $\frac{\partial{f}^2}{\partial{u}\partial{v}}(x)$ mean in this context? a.k.a - what do I need to prove, and what does differantial according to a vector means?
- How/why do I use/need the fact that $x$ is a critial point in order to solve this?
Thank you,