Question

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 is called the conductivity, which has metric units ofJ/m-s-K or W/m-K. A temperature function T for a region D is given below. Find the net outward heat flux. A temperature function for a region D is given below Find the net outward heat flux

dS∫∫S **F**•**n** dS=−k∫∫S ∇T•**n** dS

across the boundary S of D. It may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that
k=1.

T(x,y,z)=70 e^{-x}^{^2-y^2-z^2}

D is the sphere of radius a centered at the origin.Step 1

Given that,

Step 2

We want to use the Divergence Theorem and evaluate a triple integral.

We know,

Step 3

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