On the definition(s) of flux/current density There are three common definitions of flux. Citing Wikipedia:


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*In transport phenomena (heat transfer, mass transfer and fluid dynamics), flux is defined as the rate of flow of a property per unit area.
The surface integral of $j$ over a surface $S$, gives the proper flowing per unit of time through the surface.
$${\displaystyle {\frac {\mathrm {d} q}{\mathrm {d} t}=\iint _{S}\mathbf {j} \cdot \mathbf {\hat {n}} \,\mathrm{d}A=\iint _{S}\mathbf {j} \cdot \mathbf {A}}} $$
Here $A$ (and its infinitesimal) is the vector area, combination of the magnitude of the area through which the property passes, $A$, and a unit vector normal to the area, ${\displaystyle \mathbf {\hat {n}} }$. The relation is ${\displaystyle \mathbf {A} =A\mathbf {\hat {n}} }$.





*As a mathematical concept, flux is represented by the surface integral of a vector field
$${\displaystyle \Phi _{F}=\iint _{A}\mathbf {F} \cdot \mathrm {d} \mathbf {A} =\iint _{A}\mathbf {F} \cdot \mathbf {\hat{n}} \,\mathrm {d} A}$$

Here it seems to mix the role of $\mathbf{j}$ and $\mathbf{F}$, but the main point is clear. Then, under "Current density":


*The current density vector can be expressed as:


$${\displaystyle \mathbf {j} =\rho \mathbf {v}}$$

Evidently, these three definitions have to be equivalent to one another. But are they? Usually the consistency of 1. 2. and 3. is shown by means of the continuity equation, but this is a hypothesis that can't be made a priori.
Are there other ways?
 A: It looks like you're messing up things.
Definition of flux of a vector field.
You can define the flux of a vector field $\mathbf{f}(\mathbf{r})$ through a surface $S$, with local unit normal vector $\mathbf{\hat{n}}$ as
$\Phi_S(\mathbf{f}) := \displaystyle \int_{\mathbf{r} \in S} \mathbf{f}(\mathbf{r}) \cdot \mathbf{\hat{n}}(\mathbf{r})$,
where the differential $dS$ can be made implicit, since the value of the flux doesn't depend on the parametrization used to describe the surface, but only on the actual surface $S$.
Fluxes in balance equation of mechanics. The balance equations of mechanics, here only mass, momentum, and total energy for brevity, written for a steady control volume read
$\displaystyle \dfrac{d}{dt} \int_V \rho = - \oint_{\partial V} \rho \mathbf{u} \cdot \mathbf{\hat{n}} \qquad $ (mass)
$\displaystyle \dfrac{d}{dt} \int_V \rho \mathbf{u} = - \oint_{\partial V} \rho \mathbf{u} \mathbf{u} \cdot \mathbf{\hat{n}} + \int_V \rho \mathbf{g} + \oint_{\partial V}\mathbf{t_n} \qquad $ (momentum)
$\displaystyle \dfrac{d}{dt} \int_V \rho e^t = - \oint_{\partial V} \rho e^t \mathbf{u} \cdot \mathbf{\hat{n}} + \int_V \rho \mathbf{g} \cdot \mathbf{u} + \oint_{\partial V}\mathbf{t_n} \cdot \mathbf{u} - \oint_{\partial V} \mathbf{q} \cdot \mathbf{\hat{n}} \qquad $ (total energy)
where, for each quantity $f$, its flux across the boundary of the domain due to the convection of the medium reads
$\displaystyle \tilde\Phi_{\partial V}(f) = \Phi_{\partial V}(f\mathbf{u}) = \oint_{\partial V} f \mathbf{u} \cdot \mathbf{\hat{n}}$.
Try to have a look here:

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*https://basics.altervista.org/test/Physics/CM/BalanceEquations/main.html

*https://basics.altervista.org/test/Physics/CM/BalanceEquations/povIntegral.html
Electric current. In electromagnetism, we can define the electric current through a surface as the flux of electric charges across the surface.
The volume density of electric current, the current density $\mathbf{j}(\mathbf{r})$, is defined as the product of the charge density times the local average velocity of the charges $\mathbf{j}(\mathbf{r}) = \rho(\mathbf{r}) \mathbf{v}(\mathbf{r})$. Thus, thee electric current across a (steady) surface $S$ can be written as
$i_S = \displaystyle \int_S \mathbf{j} \cdot \mathbf{\hat{n}} $
Relying on the principle of the conservation of the electrical charge, we can write that the time derivative of the charge inside a volume $V$ is equal to the charge per unit time crossing its boundary,
$\dfrac{d Q_V}{dt} = - \displaystyle \displaystyle \oint_{\partial V} \mathbf{j} \cdot \mathbf{\hat{n}}$
