Euler's Hydrodynamical Equation for Flow of Ideal Incompressible Fluid

Theorem

 * $\begin{cases} \dfrac {D \mathbf u} {D t} & = & - \nabla w + \mathbf g \\

\nabla \cdot \mathbf u & = & 0 \end{cases}$

where:
 * $\mathbf u$ denotes the flow velocity vector, with components in an $N$-dimensional space $u_1, u_2, \dots, u_N$
 * $\dfrac D {D t}$ denotes the material derivative in time
 * $\cdot$ denotes the dot product
 * $\nabla$ denotes the nabla operator, used to represent the specific thermodynamic work gradient (first equation), and the flow velocity divergence (second equation)
 * $\mathbf u \cdot \nabla$ is the convective derivative
 * $w$ is the thermodynamic work per unit mass, the internal source term
 * $\mathbf g$ denotes body acceleration per unit mass acting on the continuum.