Is this an application of the divergence theorem ?

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 !

Re: Is this an application of the divergence theorem ?

The conservation law is the first equation you give. As for the second part, integrate wrt x giving

.

Re: Is this an application of the divergence theorem ?

Oh, that was easy.... sorry for not noticing that before I bothered you guys with it.

And of course, thank you !