simple set of differential equations

Question

Can I write for a strictly convex, differentiable function $H$ from $\mathbb{R}^n\rightarrow\mathbb{R}$ $$ H(x)=\sum_{i=1}^{n}x_{i}\frac{\partial{}H}{\partial{}x_{i}}(x) $$ and if yes, why?

Answer 1

In general $H(x)=\sum_{i=1}^{n}x_{i}\frac{\partial{}H}{\partial{}x_{i}}(x)$ is false ! An example is a constant function $H \ne 0$.