Correct way to write a symbol for the Jacobian matrix

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:

$$$$\frac{\partial U}{\partial t}+\frac{\partial F}{\partial U}\frac{\partial U}{\partial x} = S(U)$$$$

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:

$$$$\frac{\partial U}{\partial t}+J\frac{\partial U}{\partial x} = S(U)$$$$

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 $$U$$:

$$$$J =\frac{\partial F}{\partial U} = \begin{vmatrix}\frac{\partial F_1}{\partial U_1} && \frac{\partial F_1}{\partial U_2} \\ \frac{\partial F_2}{\partial U_1} && \frac{\partial F_2 }{\partial U_2}\end{vmatrix}$$$$

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!

• I like just $\frac{\partial F}{\partial U}$. It’s the most informative notation. Feb 17, 2021 at 16:01

I usually write $$J_F (U)$$