Chapter 17.8, Problem 42E

Heat transfer Fourier’s Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = –k▿T. which means that heat energy flows from hot regions to cold regions. The constant k > 0 is called the conductivity, which has metric units of J/m-s-K. A temperature function for a region D is given. Find the net outward heat flux ${\displaystyle {\iint }_{S}F\cdot ndS=-k{\displaystyle {\iint }_S\nabla T\cdot ndS}}$across the boundary S of D. In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k = 1.

42.    T(x, y, z) = 100 + x² + y² + z²;

D = {(x, y, z): 0 ≤ x ≤ 1, 0 ≤ 1, 0 ≤ z ≤ 1}