Introduction to Mathematical Physics/Continuous approximation/Conservation laws

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Integral form of conservation laws[edit]

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 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 laws[edit]

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.