Introduction to Mathematical Physics/Continuous approximation/Conservation laws

Integral form of conservation lawsEdit

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

 

Symbol   represents the particular derivative (see appendix chapretour).   is a scalar or tensorial\footnote{   is the volumic density of quantity   (mass, momentum, energy ...). The subscript   symbolically designs all the subscripts of the considered tensor. } function of eulerian variables   and  .   is volumic density rate provided by the exterior to the system.   is the surfacic density rate of what is lost by the system through surface bording  .

Local form of conservation lawsEdit

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.