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Thread: Is this an application of the divergence theorem ?

  1. #1
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    Is this an application of the divergence theorem ?

    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 !
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  2. #2
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    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

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

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

    $\displaystyle \dfrac{d}{dy} \int_a^b R dx + S(u(b,y) - S(u(a,y)) = 0$.
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  3. #3
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    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 !
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