Let $\displaystyle R,S:u(D)\to\mathbb{R}$ be differentiable for some $\displaystyle u : D\subseteq\mathbb{R}^2\to\mathbb{R}$ (note that $\displaystyle u$ is not necessarily continuous) such that

$\displaystyle \frac{\partial}{\partial y}R(u(x,y))+\frac{\partial}{\partial x}S(u(x,y))=0$.

According to my PDE textbook (p18), the "conservation law" (???) implies that for any $\displaystyle a<b$ and $\displaystyle y$ satisfying $\displaystyle [a,b]\times\{y\}\subset D$, we have

$\displaystyle 0=\frac{d}{dy}\int_a^b R(u(x,y))\;dx+S(u(b,y))-S(u(a,y))$.

If you guys have some spare time, can you perhaps enlighten me as to what "conservation law" the author is referring ? My professor mentioned something about the divergence theorem, but I don't see how that helps. Maybe I'm just missing something about how to apply the divergence theorem.

Any help would be much appreciated. Thanks !