Introduction to Mathematical Physics/Continuous approximation/Conservation laws

Integral form of conservation laws
A conservation law\index{conservation law} is a balance that can be applied to every connex domain strictly interior to the considered system and that is followed in its movement. such a law can be written:

Symbol $$\frac{d}{dt}$$ represents the particular derivative (see appendix chapretour). $$A_i$$ is a scalar or tensorial\footnote{ $$A_i$$ is the volumic density of quantity $${\mathcal A}$$ (mass, momentum, energy ...). The subscript $$i$$ symbolically designs all the subscripts of the considered tensor. } function of eulerian variables $$x$$ and $$t$$. $$a_i$$ is volumic density rate provided by the exterior to the system. $$\alpha_{ij}$$ is the surfacic density rate of what is lost by the system through surface bording $$D$$.

Local form of conservation laws
Equation eqcon represents the integral form of a conservation law. To this integral form is associated a local form that is presented now. As recalled in appendix chapretour, we have the following relation:

It is also known that:

Green formula allows to go from the surface integral to the volume integral:

Final equation is thus:

Let us now introduce various conservation laws.