Is this an application of the divergence theorem ?

**hatsoff** · October 4th 2011, 05:55 AM

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

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

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

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

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 !

**Danny** · October 4th 2011, 06:52 AM

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

$\int_a^b \left(\dfrac{\partial R}{\partial y} + \dfrac{\partial S}{\partial x}\right) dx = 0$

$\int_a^b \dfrac{\partial R}{\partial y} dx + S|_a^b = 0$

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

**hatsoff** · October 4th 2011, 06:55 AM

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

And of course, thank you !

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