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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 t_{1},t_{2}, 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

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# Equation of heat conduction, vague text book

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