Editing Euler equations (fluid dynamics) (section)

===Conservation form===
{{See also|conservation equation|}}
The incompressible Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively:
<math display="block">
\mathbf y = \begin{pmatrix}\rho  \\  \rho \mathbf u  \\0\end{pmatrix}; \qquad {\mathbf F} = \begin{pmatrix}\rho \mathbf u \\ \rho \mathbf u \otimes \mathbf u + p \mathbf I\\\mathbf u\end{pmatrix}.
</math>

Here <math>\mathbf y</math> has length <math>N+2</math> and <math>\mathbf F</math> has size <math>(N+2)N</math>.{{efn|In 3D for example <math>\mathbf y</math> has length 5, <math>\mathbf I</math> has size 3×3 and <math>\mathbf F</math> has size 5×3, so the explicit forms are:
<math display="block">
{\mathbf y}=\begin{pmatrix}\rho \\  \rho u_1 \\ \rho u_2  \\ \rho u_3  \\0\end{pmatrix}; \quad
{\mathbf F}=\begin{pmatrix}\rho u_1 & \rho u_2 & \rho u_3 \\
\rho u_1^2 + p &  \rho u_1u_2 &  \rho u_1u_3
\\  \rho u_1 u_2 & \rho u_2^2 + p &  \rho u_2u_3
\\ \rho u_3 u_1 &  \rho u_3 u_2 &  \rho u_3^2 + p
\\ u_1 & u_2 & u_3 \end{pmatrix}.
</math>
}}
In general (not only in the Froude limit) Euler equations are expressible as:
<math display="block">
\frac {\partial}{\partial t}\begin{pmatrix}\rho  \\  \rho \mathbf u  \\0\end{pmatrix}+ \nabla \cdot \begin{pmatrix}\rho \mathbf u\\\rho \mathbf u \otimes \mathbf u + p \mathbf I\\ \mathbf u\end{pmatrix} = \begin{pmatrix}0 \\  \rho \mathbf g \\ 0 \end{pmatrix}.
</math>