# Balance Laws

## Abstract

The general, mathematical theory of balance laws expounded in this chapter has been designed to provide a unifying framework for the multitude of balance laws of classical continuum physics, obeyed by so-called “extensive quantities” such as mass, momentum, energy, etc. The ambient space for the balance law will be \( \mathbb{R}^k \), with typical point *X*. In the applications to continuum physics, \( \mathbb{R}^k \) will stand for physical space, of dimension one, two or three, in the context of statics; and for space-time, of dimension two, three or four, in the context of dynamics. The generic balance law will be introduced through its primal formulation, as a postulate that the production of an extensive quantity in any domain is balanced by a flux through the boundary; it will then be reduced to a field equation. It is this reduction that renders continuum physics mathematically tractable. It will be shown that the divergence form of the field equation is preserved under change of coordinates, and that the balance law, in its original form, may be retrieved from the field equation. The properties discussed in this chapter derive solely from the divergence form of the field equations and thus apply equally to balance laws governing equilibrium and evolution.

## Keywords

Weak Solution Shock Front Bounded Variation Borel Subset Lipschitz Continuous Function## Preview

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