dimension. We say that $\mathbf{u} \in \mathbb{R}^{d+2}$ is the state and
$\mathbb{f}(\mathbf{u}) \in \mathbb{R}^{(d+2) \times d}$ is the flux of
the system. In the case of Euler's equations the state is given by
-$\textbf{u} = [\rho, \textbf{m},E]^{\top}$: where $\rho \in \mathbb{R}^+$
+$\textbf{u} = [\rho, \textbf{m}^\top,E]^{\top}$: where $\rho \in \mathbb{R}^+$
denotes the density, $\textbf{m} \in \mathbb{R}^d$ is the momentum, and $E
\in \mathbb{R}^+$ is the total energy of the system. The flux of the system
$\mathbb{f}(\mathbf{u})$ is defined as
all $\mathbf{x} \in \Omega$ and $t \geq 0$ where
@f{align}
\mathcal{B} = \big\{ \textbf{u} =
- [\rho, \textbf{m},E]^{\top} \in \mathbb{R}^{d+2} \, \big |
+ [\rho, \textbf{m}^\top,E]^{\top} \in \mathbb{R}^{d+2} \, \big |
\
\rho > 0 \, ,
\