Context: I'm writing an essay on the subject of hydrodynamics which deals with numerical methods to solve a system of partial differential equations (shallow water equations or Saint-Venant equations). These can be written as a system of two equations or, alternatively, in a single line using vector notation:
\begin{equation} \frac{\partial U}{\partial t}+\frac{\partial F}{\partial U}\frac{\partial U}{\partial x} = S(U) \end{equation}
Where $F$ and $U$ are two vector functions with two components each. It's not important for the scope of this question to know what they represent physically, the important part is that both $F$ and $U$ are functions of $x$ and $t$ ($F=F(x,t)$, $U=U(x,t)$), but $F$ can be written as well as a function of the components of $U$ ($F=F(U)$). I want to write the system in an even more compact way using the Jacobian:
\begin{equation} \frac{\partial U}{\partial t}+J\frac{\partial U}{\partial x} = S(U) \end{equation}
Where $J$ is the Jacobian matrix of the vector function $F$ not with respect to the independent variables $(x,t)$, but with respect to the components of $U$: i.e., $J$ is the matrix of all the partial derivatives of the components of $F$ with respect to $U$:
\begin{equation} J =\frac{\partial F}{\partial U} = \begin{vmatrix}\frac{\partial F_1}{\partial U_1} && \frac{\partial F_1}{\partial U_2} \\ \frac{\partial F_2}{\partial U_1} && \frac{\partial F_2 }{\partial U_2}\end{vmatrix} \end{equation}
Which is the correct notation I should use to write it? $J(F)$? $J(F(U))$? $J_U(F)$? I'm really confused because the Jacobian matrix is usually calculated using $(x,y,z)$ as independent variables, but this time I want to signal that I'm deriving with respect to another set of variables!
