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Applying Gauss' Law to other situations

  1. Nov 7, 2013 #1
    Hey everyone,

    I'm not sure if this belongs in the math or physics section of this forum, but I figure since my question is more related to the mathematical manipulation of what I am dealing with, I figured I would ask it here and then if it has to be moved, it can be.

    My question has to do with applying gauss' law to a differential equation I am dealing with. The differential equation is the partial derivative of some function of x,y,z,t with respect to t is equal to a constant times dell^2 of that function plus another constant times the same function. The case I am considering is when the time derivative is equal to zero. So I have:

    D*dell^2(n) + C*n = 0

    So I am thinking I am basically dealing with a divergence of the vector dell(n) (n is a scalar function). However, I'm not sure how to apply that logic to the C*n part. Can I just subtract it to the other side and take the triple volume integral? Or does the divergence theorem not apply? I think it does because I am basically dealing with a diffusion of particles out from a spherical surface. The only difference is there are particles being generated inside of the sphere as well. That's the C*n term.
     
  2. jcsd
  3. Nov 7, 2013 #2

    UltrafastPED

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    Can you type in the equation using LaTex or the special symbols?
     
  4. Nov 7, 2013 #3

    pasmith

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    If you integrate your PDE over a volume, then you can indeed replace the [itex]\nabla^2 n[/itex] term by an integral of [itex]\nabla n[/itex] over the boundary surface, but the integrals of the other terms will remain as volume integrals.

    Usually such conservation PDEs are obtained by looking at [itex]\frac{d}{dt} \int_V n\,dV[/itex] and then using the divergence theorem to replace the surface term expressing the flux of [itex]n[/itex] out of [itex]V[/itex] with a volume integral.
     
  5. Nov 7, 2013 #4
    Sorry about that, I am typing from my smart phone, and I don't know how to use LaTex from this interface. Mainly because I'm not familiar with LaTex. If you know any good sources where I can learn it, I'd be willing to use it.
     
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