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: Template:IMP/label Template:IMP/eq 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 $𝒜$ (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: Template:IMP/eq It is also known that: Template:IMP/eq Green formula allows to go from the surface integral to the volume integral: Template:IMP/eq Final equation is thus: Template:IMP/eq Let us now introduce various conservation laws.