As a result of radioactive decay, heat is generated uniformly throughout the interior of the earth at a rate of around $30$ watts per cubic kilometer. (A watt is a rate of heat production.) The heat then flows to the earth's surface where it is lost to space. Let $\vec F(x,y,z)$ denote the rate of flow of heat measured in watts per square kilometer. By definition, the flux of $\vec F$ across a surface is the quantity of heat flowing through the surface per unit of time.

(a) Suppose that the actual heat generation is $28 \mbox{W}/\mbox{km}^3$. What is the value of $\mbox{div }\vec F$?
$\mbox{div }\vec F =$
(Include units.).

(b) Assume the heat flows outward symmetrically. Verify that $\vec F = \alpha\vec r$, where $\vec r = x\vec i+y\vec j + z\vec k$ and $\alpha$ is a suitable constant, satisfies the given conditions. Find $\alpha$.
$\alpha =$
(Include units.).

(c) Let $T(x,y,z)$ denote the temperature inside the earth. Heat flows according to the equation $\vec F =-k\,\mbox{ grad}\,T$, where $k$ is a constant. If $T$ is in ${}^\circ$C, then $k=28,000^\circ\mbox{C}/\mbox{km}$. Assuming the earth is a sphere with radius $6400$ km and surface temperature $20^\circ$C, what is the temperature at the center?
$T =$ (degrees C)