Let x be a point in three dimensional space and let
The heat flux J, in the body is defined as a vector which has the following property.

where n is the unit normal in the desired direction. Thus if V is a three dimensional body,

where n is the unit exterior normal.
Fourier’s law of heat conduction states that the heat flux J satisfies J = −k∇
chemical reaction taking place. Then the divergence theorem can be used to verify the following equation for u. Such an equation is called a reaction diffusion equation.
 (27.6) 
Take an arbitrary V for which the divergence theorem holds. Then the time rate of change of the heat in V is

where, as in the preceding example, this is a physical derivation so the consideration of hard mathematics is not necessary. Therefore, from the Fourier law of heat conduction,


Since this holds for every sample volume V it must be the case that the above reaction diffusion equation 27.6 holds. Note that more interesting equations can be obtained by letting more of the quantities in the equation depend on temperature. However, the above is a fairly hard equation and people usually assume the coefficient of thermal conductivity depends only on x and that the reaction term f depends only on x and t and that ρ and c are constant. Then it reduces to the much easier equation
 (27.7) 
This is often referred to as the heat equation. Sometimes there are modifications of this in which k is not just a scalar but a matrix to account for different heat flow properties in different directions. However, they are not much harder than the above. The major mathematical difficulties result from allowing k to depend on temperature.
It is known that the heat equation is not correct even if the thermal conductivity did not depend on u because it implies infinite speed of propagation of heat. However, this does not prevent people from using it.