(Solved): Fourier's Law of heat transfer (or heat conduction) states that the heat flow vector F at a point i ...

Fourier's Law of heat transfer (or heat conduction) states that the heat flow vector

at a point is proportional to the negative gradient of the temperature; that is,

, which means that heat energy flows from hot regions to cold regions. The constant

is called the conductivity, which has metric units of

. A temperature function

for a region

is given below. Find the net outward heat flux

across the boundary

of

. It may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that

.

is the sphere of radius a centered at the origin.