- #1
Somefantastik
- 230
- 0
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.
TIA
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.
TIA