# 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:

eqcon

${\displaystyle {\frac {d}{dt}}\int _{D}A_{i}dv+\int _{\partial D}\alpha _{ij}n_{j}d\sigma =\int _{D}a_{i}dv}$

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

## 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:

${\displaystyle {\frac {d}{dt}}\int _{D}A_{i}dv=\int _{D}{\frac {d}{dt}}A_{i}dv}$

It is also known that:

${\displaystyle {\frac {dA_{i}}{dt}}={\frac {\partial A_{i}}{\partial t}}+(A_{i}u_{j})_{,j}}$

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

${\displaystyle \int _{\partial D}\alpha _{ij}n_{j}d\sigma =\int _{D}\alpha _{ij,j}dv}$

Final equation is thus:

${\displaystyle {\frac {\partial A_{i}}{\partial t}}+(A_{i}u_{j}+\alpha _{ij})_{,j}=a_{i}}$

Let us now introduce various conservation laws.