The conservation law is the first equation you give. As for the second part, integrate wrt x giving
.
Let be differentiable for some (note that is not necessarily continuous) such that
.
According to my PDE textbook (p18), the "conservation law" (???) implies that for any and satisfying , we have
.
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 !