Chapter 17.5, Problem 57E

Heat flux Suppose a solid object in ¡³ has a temperature distribution given by T(x, y, z). The heat flow vector field in the object is F = –k▿T, where the conductivity k > 0 is a property of the material. Note that the heat flow vector points in the direction opposite that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is ▿·F = –k ▿·▿T = –k▿²T (the Laplacian of T). Compute the heat flow vector field and its divergence for the following temperature distributions.

57.     $T\left(x,y,z\right)=100e^{-x^2+y^2+z^2}$