Flux: Total, Per-Area, Vector & Density Explained

[Rasalhague]

Am I right in thinking that flux can mean either a surface integral of the dot product of a vector field with a unit vector perpendicular to a given surface (the total flux through a surface of area A):

$$\int \mathbf{F} \cdot \hat{\mathbf{N}} dA,$$

(e.g. electric flux, magnetic flux), or such a total divided by the area of the surface:

$$\frac{\int \mathbf{F} \cdot \hat{\mathbf{N}} dA}{A},$$

(e.g. volumetric flux)? The latter kind of flux seems sometimes to be expressed as a vector quantity, as in the Poynting vector and heat flux (also called heat flux density), although some sources only call the magnitude of the Poynting vector energy flux. Is this just a matter of convention or convenience? What happens if the surface isn't flat; which direction does the vector point? Have I even got the right formula for defining it? Is total flux ever expressed as a vector?

I gather that total flux, in the relavant contexts, may be called mass flow rate and volumetric flow rate, while the per-area flux is called mass flux and volumetric flux. Are these terms limited to certain instances of flux in the broader sense? What are electric and magnetic flow rate (are they synonymous with electric and magnetic flux, as normally defined, or do they refer to something else)?

I read in Wikipedia that energy flux can refer to either kind of flux, total or per-unit-area, the latter sometimes being called flux density, and I've seen the Poynting vector called an energy flux density vector. Does the term "flux density" applied to the magnetic B field have the same meaning as "flux density" when it's the per-unit-area kind of energy flux, or is it a flux density in the sense that Davis and Snider use the term in Introduction to Vector Analysis, § 3.7, where they define flux density of the flux

$$\mathbf{F} \cdot \hat{\mathbf{N}} ds$$

as the function F, or does flux density have some other sense when applied to the magnetic B field?

What is flow rate density? Why is density used for a "per unit area" quantity, as opposed to pressure; is it just a linguistic quirk/convention?

Are there general, unambiguous terms for these concepts? Are there other uses of the words flux and flux density, etc. in physics that I haven't covered.

 

[vanesch]

Flux is indeed a surface integral of a vector field, and is hence a scalar quantity, associated to a surface (actually, to a flux tube).

If you divide by the surface, you get a flux density, and in fact, the correct description of a flux density is nothing else but the original vector field you integrated over the surface.

However, people are sometimes sloppy with names, and sometimes they omit "density".

 

[Rasalhague]

If you divide by the surface, you get a flux density, and in fact, the correct description of a flux density is nothing else but the original vector field you integrated over the surface.

So is flux density effectively synonymous with vector field (and, in that case, why is the electric D field called electric flux density rather than the E field, given that electric flux is a surface integral of the E field), or can flux density also mean the scalar quantity obtained by diving the surface integral of a vector field by the total area of the surface?

$$\frac{\int \mathbf{F} \cdot \mathbf{\hat{N}} dA}{A}$$

This isn't the same as the original vector field F. It's a scalar and depends on the orientation of the surface at each point. Are you saying that this scalar quantity is sometimes loosely called flux density although the term flux density is more correctly applied to the vector field F itself?

 

[csco]

The flux density is

$$\mathbf{F} \cdot \mathbf{\hat{n}}$$

while

$$\frac{\int \mathbf{F} \cdot \mathbf{\hat{n}}dA}{A}$$

is the average flux density over the surface

 

[vanesch]

The flux density is

$$\mathbf{F} \cdot \mathbf{\hat{n}}$$

while

$$\frac{\int \mathbf{F} \cdot \mathbf{\hat{n}}dA}{A}$$

is the average flux density over the surface

Yes, and if you want to summarize the flux density "for all local surfaces with all possible orientations", you're back to the vector field itself.

 

[Rasalhague]

Thanks to you both for the answers. So, in the following example,

http://en.wikipedia.org/wiki/Heat_flux

if we were to use more rigorous terms, would the integral be called heat "flux", and the original vector field heat "flux density" when conceived of as a scalar valued function with values $$\mathbf{F} \cdot \hat{\mathbf{n}}$$ at each point on a given surface? Is this vector field $$\mathbf{\phi_{q}}$$ the gradient of a scalar temperature field? The illustation shows it orthogonal to a surface though, as if maybe $$\mathbf{\phi_{q}}$$ itself is only defined relative to some specified surface, but maybe that's just a coincidence.