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 requires , and vice versa (from calculus the direction of maximal growth of a function is given by its gradient), our linear relation takes the form , with . Since specifies any direction at , this is equivalent to the assumption
(1.5.12)
which is Fourier's law. The positive function is called the heat conduction (or Fourier) coefficient. Let now and and insert (1.5.11) and (1.5.12) into (1.5.10) to get the final form of the heat equation:
(1.5.13)
The quantity is referred to as the thermal diffusivity (or diffusion) coefficient. If we assume that is constant, then the final form of the heat equation would be
(1.5.14)
1.5.3 The Wave Equation
The third equation in (1.5.2) is the wave equation: . Here, represents a wave traveling through an
