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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 $a_{ij}$ ?

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up vote 1 down vote accepted

In one dimension, your problem asks $$ \int_{-\infty}^\infty \frac{\partial }{\partial x} \left( a(x) |v(x)|^2 \right) \, dx = 0 $$ Improper integrals are "correctly" expressed as limits $$ \lim_{M \to \infty} \int_{-M}^M \frac{\partial }{\partial x} \left( a(x) |v(x)|^2 \right) \, dx = a(M)|v(M)|^2 - a(-M) |v(-M)|^2 \hspace{0.1in}=0$$

Perhaps $a(x),v(x)$ need to vanish near infinity or be well-defined on the Riemann-sphere.

In higher dimensions, you are integrating a total derivative, which should not depend on the endpoints. $$ \int_x^y d(Av\cdot \overline{v}) = A(y)|v(y)|^2 - A(x)|v(x)|^2$$ This is Fundamental Theorem of Calculus.

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