# 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

$\frac{d}{dt}\int_D A_idv+\int_{\partial D} \alpha_{ij} n_jd\sigma=\int_D a_idv$

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:

$\frac{d}{dt}\int_D A_idv=\int_D \frac{d}{dt} A_idv$

It is also known that:

$\frac{dA_i}{dt}=\frac{\partial A_i}{\partial t}+(A_iu_j)_{,j}$

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

$\int_{\partial D} \alpha_{ij}n_jd\sigma=\int_{D} \alpha_{ij,j}dv$

Final equation is thus:

$\frac{\partial A_i}{\partial t}+(A_iu_j+\alpha_{ij})_{,j}=a_i$

Let us now introduce various conservation laws.