Editing Thermal conduction (section)

===Integral form===
By integrating the differential form over the material's total surface <math>S</math>, we arrive at the integral form of Fourier's law:

: {{oiint|intsubscpt=<math>\scriptstyle S</math>|integrand=<math>\mathbf q \cdot \mathrm{d}\mathbf{S}</math>}}<math>{}={}</math>{{oiint|preintegral=<math>-k</math>|intsubscpt=<math>\scriptstyle S</math>|integrand=<math>\nabla T \cdot \mathrm{d}\mathbf{S}</math>}}

where (including the [[SI]] units):
* {{oiint|preintegral=<math>\dot{Q}=</math>
|intsubscpt=<math>\scriptstyle S</math>|integrand=<math>\mathbf q \cdot \mathrm{d}\mathbf{S}</math>}} is the [[thermal power]] transferred by conduction (in W), time derivative of the transferred [[heat]] <math>Q</math> (in J),
* <math>\mathrm{d}\mathbf{S}</math> is an [[Vector area|oriented surface area element]] (in m<sup>2</sup>).
<!-- Missing image removed: [[Image:exponential Heat flow.svg|thumb|200px|exponential heat flow]] -->
The above [[differential equation]], when [[Integral|integrated]] for a homogeneous material of 1-D geometry between two endpoints at constant temperature, gives the heat flow rate as
<math display="block">Q = - k \frac{A\Delta t}{L} \Delta T,</math>
where
* <math>\Delta t</math> is the time interval during which the amount of heat <math>Q</math> flows through a cross-section of the material,
* <math>A</math> is the cross-sectional surface area,
* <math>\Delta T</math> is the temperature difference between the ends,
* <math>L</math> is the distance between the ends.

One can define the (macroscopic) [[thermal resistance]] of the 1-D homogeneous material:
<math display="block">R = \frac 1 k \frac L A </math>

With a simple 1-D steady heat conduction equation which is analogous to [[Ohm's law]] for a simple [[electric resistance]]:
<math display="block">\Delta T= R \, \dot{Q} </math>

This law forms the basis for the derivation of the [[heat equation]].