Let T (x, y,z, t) denote the temperature at the point (x, y,z) of a solid object D at

Chapter 7, Problem 8

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Let T (x, y,z, t) denote the temperature at the point (x, y,z) of a solid object D at time t. We define the heat flux density H by H = kT . (The constant k is the thermal conductivity. Note that the symbol denotes differentiation with respect to x, y,z, not with respect to t.) The vector field H represents the velocity of heat flow in D. It is a fact from physics that the total heat contained in a solid body D having density and specific heat is D T dV. Hence, the total amount of heat leaving D per unit time is D T t dV. (Here we assume that and do not depend on t.) We also know that the heat flux may be calculated as D H dS. Exercises 610 concern these notions of temperature, heat, and heat flux densityIn the heat equation of Exercise 6, suppose that , , and k are all constant and the temperature T of the solid D does not vary with time. Show that then T must be harmonic, that is, that 2T = 0 at all points in the interior of D.