Flux Terminology: Scalar Flux vs. Flux Density in Different Contexts

[etotheipi]

Forgive me if a similar thread has been posted before... I was doing some questions and I just noticed an apparent discrepancy in how the term "flux" is thrown around.

In the context of surface integrals, the scalar result is usually termed the flux whilst the vector field is termed the flux density, i.e. in $$\phi = \iint_S \mathbf{F} \cdot d\mathbf{S}$$ the $\phi$ would be flux through $S$ and $\mathbf{F}$ the flux density. This is also the terminology used in electromagnetism.

However in other contexts we appear to use flux to denote the vector field; e.g. we let the volumetric flux $\mathbf{v}$ be the velocity, and call the flow rate $$\frac{dV}{dt} = \iint_S \mathbf{v} \cdot d\mathbf{A}$$ The same goes for mass flow rates, current densities, and so on. Is one of these old usage/preferred over the other? To me it seems it would be more consistent to use the first approach (fluxes for the scalars and flux densities for the vector fields), but I'm not sure which is preferred. I wondered whether anyone could clarify - thanks!

 

[A.T.]

. I was doing some questions and I just noticed an apparent discrepancy in how the term "flux" is thrown around... To me it seems it would be more consistent to use the first approach (fluxes for the scalars and flux densities for the vector fields)

Could it be that "density" was simply omitted sometimes?

 

[etotheipi]

Could it be that "density" was simply omitted sometimes?

It could well be, I'm not too sure. The question itself was regarding Fick's law of diffusion where you have $\mathbf{J} = - D\nabla \phi$ and $\mathbf{J}$ is usually called the diffusion flux vector, even though it is measured per unit area per unit time. Maybe they're just being sloppy?