Let denote the heat flux vector and denote the outward unit normal to the boundary , at the point . Then represents the flow of heat per unit cross‐sectional area per unit time crossing a surface element. Thus,
is the amount of heat per unit time flowing into across the boundary . Here, represents the element of surface area. The minus sign reflects the fact that if more heat flows out of the domain than in, the energy in decreases. Finally, in general, the heat production is determined by external sources that are independent of the temperature. In some cases, (such as an air conditioner controlled by a thermostat), it depends on temperature itself, but not on its derivatives. Hence, in the presence of a source (or sink), we denote the corresponding rate at which heat is produced per unit volume by so that the source term becomes
Now, the law of conservation of energy takes the form
(1.5.8)
Applying the Gauss divergence theorem to the integral over , we get
where denotes the divergence operator. In the sequel, we shall use the following simple result:
Lemma 1.1
Let be a continuous function satisfying for every domain . Then .
Proof:
Let us assume to the contrary that there exists a point where . Assume without loss of generality that . Since is continuous, there exists a domain (maybe very small) , containing , and an , such that , for all . Therefore, we have , which contradicts the assumption.
This is the basic form of our heat conduction law. The functions and are unknown and additional information of an empirical nature is needed to determine the equation for the temperature . First, for many materials, over a fairly wide but not too large temperature range, the function depends nearly linearly on , so that
(1.5.11)
Here, , called the specific heat, is assumed to be constant. Next, we relate the temperature to the heat flux . Here, we use Fourier's law but, first, to be specific, we describe the simple facts supporting Fourier's law:
1 i. Heat flows from regions of high temperature to regions of low temperature.
2 ii. The rate of heat flux is small or large accordingly as temperature changes between neighboring regions are small or large.
To describe these quantitative properties of heat flux, we postulate a linear relationship between the rate of heat flux and the rate of temperature change. Recall that if is a point in the heat‐conducting medium and is a unit vector specifying a direction at , then the rate of heat flow at in the direction is and the rate of change of the temperature is , the directional derivative of the temperature. Since
