Chapter 17.8, Problem 45E

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.

45.     $T\left(x,y,z\right)=100e^{-x^2-y^2-z^2}$ D is the sphere of radius a centered at the origin.