1.2.7 Euler equations: from integral to differential form II

Merging the two integrals gives

$$\int_{V} \left( \frac{\partial \rho}{\partial t} + \ensu... ...ath$\scriptscriptstyle v$}}} \right) dV = 0 \enspace .$$ (19)

Because this is true for all (even small) volumes we conclude that the integrand has to be zero and get the differential form of the mass transport equation,

$$\frac{\partial \rho}{\partial t} + \ensuremath{\mathchoi... ...}} {\mbox{\boldmath$\scriptscriptstyle v$}}} = 0 \enspace .$$ (20)

This works the same for the energy equation and requires only a generalization of the Gauß theorem to be applicable to the momentum equation.