Answer Happy

a 10. Let T(t, x, y, z) be a function with continuous second derivatives giving the temperature at time t at the point (1, y, z) of a solid occupying a region Q in R3. If the solid's heat capacity and mass density are denoted by constants c and p, the quantity cpT is called the solid's heat energy per unit volume. (a) Explain why - VT points in the direction of heat flow. (b) Let - KVT denote the energy flux vector. (The constant k is called the conductivity.) Assuming the Law of conservation of mass V.pF+ др = 0 at with F= -kVT and p = cpT, derive the diffusion (heat) equation at = KVPT at k where K = -> 0 is the diffusivity constant. cp Remark: V is the operator (en ), so when we apply it we consider t constant. = a a a