In physics one frequently handles the change of a property with time by considering properties that do not change with time. For example, when two particles collide elastically, the momentum and the energy of each particle may change. However, this change can be found from the consideration that the total momentum and energy of the system are conserved. Often in physics, such conservation laws are the main ingredients for describing a system. In this chapter we deal with conservation laws for continuous systems. These are systems in which the physical properties are a continuous function of the space coordinates. Examples are the motion in a fluid or solid, and the temperature distribution in a body. The introduced conservation laws are not only of great importance in physics, they also provide worthwhile exercises in the use of vector calculus introduced in the previous chapters.

General form of conservation laws

In this section a general derivation of conservation laws is given. Suppose we consider a physical quantity Q. This quantity could denote the mass density of a fluid, the heat content within a solid or any other type of physical variable. In fact, there is no reason why Q should be a scalar, it could also be a vector (such as the momentum density) or a higher order tensor. Let us consider a volume V in space that does not change with time. This volume is bounded by a surface ∂V. The total amount of Q within this volume is given by the integral ∫VQdV.