Heat conduction with variable thermal conductivity 

(a) For steady-state heat conduction in solids, Eq. 11.2-5 becomes (∆ ∙ q) = 0, and insertion of Fourier's law gives (∆ ∙ k∆ T) = 0. Show that the function F = fkdT + const. satisfies the Laplace equation V²F = 0, provided that k depends only on T. 

(b) Use the result in (a) to solve Problem 10B.12 (part a), using an arbitrary function k(T).