Equation of heat conduction, vague text book

[Somefantastik]

Hey all,

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

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

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.

TIA

 

[Somefantastik]

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.

 

[statdad]

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?

 

[Somefantastik]

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.

 

[statdad]

The hint is to let f be positive at some point P and then f acts like some ball centered at P.

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

 

[Somefantastik]

stupid question-

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

 

[statdad]

"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 $$\mathcal{P}$$ 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).

 

[Somefantastik]

Ok, that helps a little. Thanks so much :)