Let $A$ be $n \times n$ Hermitian matrix with its component $a_{ij}$. Let $v$ be a $n$ dimensional column matrix with its component $v_i$. Let $a_{ij} \in C^{\infty} ( \Bbb R^n)$ and $v_i \in W^{1,2}( \Bbb R^n)$ . Then I want to prove that $ \int_{\Bbb R^n} \sum_{j=1}^n \partial_j (A v \cdot \bar v) = 0 $ where $\partial_j = \frac{\partial}{\partial x_j} $ and $Av \cdot \bar v = \sum_{i,j=1}^n a_{ij} \bar{v_i} v_j. $
Do I need some more conditions for the function $a_{ij}$ ?