∫ Ω ρ D u i D t d V = ∫ Ω ∇ j σ i j d V + ∫ Ω ρ g i d V ∫ Ω ( ρ D u i D t − ∇ j σ i j − ρ g i ) d V = 0 ρ D u i D t − ∇ j σ i j − ρ g i = 0 D u i D t − ∇ j σ i j ρ − g i = 0 {\displaystyle {\begin{aligned}\int _{\Omega }\rho {\frac {Du_{i}}{Dt}}\,dV&=\int _{\Omega }\nabla _{j}\sigma _{i}^{j}\,dV+\int _{\Omega }\rho g_{i}\,dV\\\int _{\Omega }\left(\rho {\frac {Du_{i}}{Dt}}-\nabla _{j}\sigma _{i}^{j}-\rho g_{i}\right)\,dV&=0\\\rho {\frac {Du_{i}}{Dt}}-\nabla _{j}\sigma _{i}^{j}-\rho g_{i}&=0\\{\frac {Du_{i}}{Dt}}-{\frac {\nabla _{j}\sigma _{i}^{j}}{\rho }}-g_{i}&=0\end{aligned}}}