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Equation of heat conduction, vague text book

  1. Aug 26, 2008 #1
    Hey all,

    This is the equation of heat conduction in my PDE textbook:

    [tex]\int ^{t_{2}}_{t_{1}} \int\int\int_{A} [c \rho \frac{\partial u}{\partial t} - \nabla \dot \left( k \nabla u \right)]dxdydzdt = 0. [/tex]

    where c is specific heat, rho is density, A is the subregion bounded by a smooth closed surface S with exterior unit normal n.

    this integrand is continuous and valid for all subregions A and all intervals t1,t2, it follows that the integrand must be zero for all (x,y,z) in Ω, where Ω is the interior of a body.

    Can someone please explain this further? It probably involves some obscure Calculus theorems.

  2. jcsd
  3. Aug 27, 2008 #2
    let the integrand be f(x) for brevity's sake

    I ended up going this route:

    let f(x) > 0

    since an integral is just a sum and the sum of all positive values is always positive, then the integral must > 0.

    let f(x) < 0

    blah blah blah, sum of all negative values is always negative, then the integral must be < 0

    therefore, f(x) = 0 (identically)

    That's how I went about solving that problem but I was sort of hoping for a more robust explanation than that.
  4. Aug 27, 2008 #3


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    Your ideas would work if you could be sure the integrand were always positive (negative), but you need to have some proof of that - do you have any?
    Is the surface insulated, or isolated, so that no heat flows in our out?
  5. Aug 27, 2008 #4
    The book doesn't say, which is what makes me think it's some Cal trick from somewhere. The hint is to let f be positive at some point P and then f acts like some ball centered at P. I'm not sure what that is hinting to; I thought I did but now I'm not sure.
  6. Aug 27, 2008 #5


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    Just a thought - some details I left nebulous on purpose, so you can
    investigate/justify them.
    if you can assume that [tex] f [/tex] is positive at some point [tex] \mathcal{P} [/tex], in the interior of the region, since [tex] f [/tex] is continuous throughout you know there is a small ball [tex] \mathcal{B} [/tex] centered at [tex] \mathcal{P} [/tex] in which [tex] f [/tex] is positive. Since the integral is identically zero for all regions, it is zero over [tex] \mathcal{B} [/tex]. Since [tex] \mathcal{P} [/tex] is arbitrary, you can conclude ... (fill in the rest)
  7. Aug 27, 2008 #6
    stupid question-

    Why do I "know" there is a positive ball B centered at point P?
  8. Aug 27, 2008 #7


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    "Why do I "know" there is a positive ball B centered at point P?"

    Well, if I understand the problem, and remember my multi-var analysis well enough, since the integrand is continuous throughout the region of integration, the fact that it is positive at a point [tex] \mathcal{P} [/tex] means that there must be a small open set, which is an open ball in several dimensions, in which the integrand is always positive. If that's the case, all else follows (I hope).
  9. Aug 27, 2008 #8
    Ok, that helps a little. Thanks so much :)
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