Editing Thermal conduction (section)

===Differential form===
The differential form of Fourier's law of thermal conduction shows that the local [[heat flux]] density <math>\mathbf{q}</math> is equal to the product of [[thermal conductivity]] <math>k</math> and the negative local temperature gradient <math>-\nabla T</math>. The heat flux density is the amount of energy that flows through a unit area per unit time.
<math display="block">\mathbf{q} = - k \nabla T,</math>
where (including the [[SI]] units)
* <math>\mathbf{q}</math> is the local heat flux density, [[Watt|W]]/m<sup>2</sup>,
* <math>k</math> is the material's [[thermal conductivity|conductivity]], W/(m·[[Kelvin|K]]),
* <math>\nabla T</math> is the temperature gradient, K/m.

The thermal conductivity <math>k</math> is often treated as a constant, though this is not always true. While the thermal conductivity of a material generally varies with temperature, the variation can be small over a significant range of temperatures for some common materials. In [[Anisotropy|anisotropic]] materials, the thermal conductivity typically varies with orientation; in this case <math>k</math> is represented by a second-order [[tensor]]. In non-uniform materials, <math>k</math> varies with spatial location.

For many simple applications, Fourier's law is used in its one-dimensional form, for example, in the {{math|''x''}} direction:
<math display="block">q_x = - k \frac{dT}{dx}.</math>

In an isotropic medium, Fourier's law leads to the [[Heat equation#Specific examples|heat equation]]
<math display="block">\frac{\partial T}{\partial t} = \alpha\left(\frac{\partial^2T}{\partial x^2} + \frac{\partial^2T}{\partial y^2} + \frac{\partial^2T}{\partial z^2}\right)</math>
with a [[Heat equation#Fundamental solutions|fundamental solution]] famously known as the [[heat kernel]].