Let $B_1(0)$ be the unit ball in $\mathbb{R}^n,f\in C^2(B_1(0))$,Prove that
(1)If $\sum_{i,j}x_ix_j\frac{\partial^2 f}{\partial x_i\partial x_j}=0$,$\nabla f(0)=0$, then $f$ is constant.
(2)If f satisfies $x_i \frac{\partial f}{\partial x_j}-x_j \frac{\partial f}{\partial x_i}=0,1\leq i,j\leq n$,then $f$ is constant on the sphere $\{|x|=\frac{1}{2}\}$.
$f$ can not attain its maximum or minimum in $B_1(0)$ because of the elliptic condition $\sum_{i,j}x_ix_j\geq 0$.The condition also reminds me of Taylor formula,but it doesn't work.For (2),I may use Green's formulas if it involves $x_i \frac{\partial f}{\partial x_i}\cdots$