Given a smooth function $F: \mathbb{R}^n \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$, we define a characteristic curve as a parametric path in the extended phase space:
\begin{align}
T : A &\to \mathbb{R}^n \times \mathbb{R} \times \mathbb{R}^n, \\
s &\mapsto (G(s), U(s), X(s)),
\end{align}
where $A \subset \mathbb{R}$ is an interval, satisfying the following system of Ordinary Differential Equations for $i, j = 1, \dots, n$:
\begin{align}
\dot{X}_j(s) &= \frac{\partial F}{\partial a_j}\bigg|_{T(s)} \\
\dot{G}_i(s) &= - \frac{\partial F}{\partial c_i}\bigg|_{T(s)} - G_i(s)\frac{\partial F}{\partial b}\bigg|_{T(s)} \\
\dot{U}(s) &= \sum_{j=1}^n G_j(s) \frac{\partial F}{\partial a_j}\bigg|_{T(s)}
\end{align}
Here, the arguments of $F$ are denoted by $(a, b, c)$ corresponding to the gradient slot ($a \in \mathbb{R}^n$), the value slot ($b \in \mathbb{R}$), and the position slot ($c \in \mathbb{R}^n$), respectively.
