# 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 of​J/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 dSacross the boundary S of D. It may be easier to use the Divergence Theorem and evaluate a triple integral. Assume thatk=1. ​T(x,y,z)=70 e-x^2-y^2-z^2    D is the sphere of radius a centered at the origin.

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 of​J/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  Fn 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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