Chapter 6.8, Problem 426E

For the following exercises, Fourier’s law of heat transfer states that the heat flow vector $\text{F}$ at a point is proportional to the negative gradient of the temperature; that is, $\text{F}=-k\nabla T$ , which means that heat energy flows hot regions to cold regions. The constant $k>0$ is called the conductivity, which has metric units of joules per meter per second-kelvin or watts per meter-kelvin. A temperature function for region $D$ is given. Use the divergence theorem to find net outward heat flux ${\displaystyle {\iint }_s\text{F}\cdot \text{N}dS}=-k{\displaystyle {\iint }_s\nabla T\cdot \text{N}dS}$ across the boundary $S$ of $D$ where $k=1$ .

426. $T(x,y,z)=100+e^{-x}{}^{{}^2}-y^2-z^2$ ; $D$ is the sphere of radius $a$ centered at the origin.