# Introduction to Mathematical Physics/Continuous approximation/Momentum conservation

We assume here that external forces are described by ${\displaystyle f}$ and that internal strains are described by tensor ${\displaystyle \tau _{ij}}$.

${\displaystyle {\frac {d}{dt}}\int _{D}\rho u_{i}dv+\int _{\partial D}\tau _{ij}n_{j}d\sigma =\int _{D}f_{i}dv}$

This integral equation corresponds to the applying of Newton's law of motion\index{momentum} over the elementary fluid volume as shown by figure figconsp.

Momentum conservation law corresponds to the application of Newton's law of motion to an elementary fluid volume.}
figconsp

Partial differential equation associated to this integral equation is:

${\displaystyle {\frac {\partial }{\partial t}}(\rho u_{i})+(\rho u_{i}u_{j})_{,j}+\tau _{ij,j}=f_{i}}$

Using continuity equation yields to:

${\displaystyle \rho ({\frac {\partial }{\partial t}}u_{i}+u_{j}u_{i,j})+\tau _{ij,j}=f_{i}}$

Remark: Momentum conservation equation can be proved taking the first moment of Vlasov equation. Fluid momentum ${\displaystyle {\bar {p}}}$ is then related to repartition function by the following equality:

${\displaystyle {\bar {p}}=\int pdpf(r,p,t)}$

Later on, fluid momentum is simply designated by ${\displaystyle p}$.