What does the infinitesimal surface area element in flux represent?

[MaJellin]

OK so I took my multivariable calculus class last year and at that time, when we learned about flux I understood it as the sum of the outward facing components of every vector generated by a vector field at every point on a surface. F dot N integrated over the surface area of the surface. I took this to represent things such as fluid flow across a surface (the purpendicular components of the fluid's velocity at points on the surface summed over every point on the surface would provide the net velocity with which fluid entered or exited the surface).
Recently, however, I have been reviewing this and other topics for a physics course and I realized that flux would actually represent the sum of every purpendicular component multiplied by an infinitesimal surface area element dsa. I haven't a clue what this could represent physically. Could anyone help me out?

Also if anyone could shed some light on what electric flux represents that would help as well.

 

[Defennder]

Well the idea behind flux as you learned it before is the same. Replace 'fluid' with vector field and you have the abstract concept of flux through a surface. The flux through a surface would tell you how much of a particular vector field passes through that surface. For example if the surface is lying parallel to the vector field, the flux through it is zero. When you rotate the surface more and more such that more of it faces the incoming vector field, you are increasing the flux through the surface.

The electric flux can be interpreted as the amount of electric field which passes through a given surface. Yeah I know it's a sloppy definition, but it's quite intuitive if you think about it.

 

[MaJellin]

Replace 'fluid' with vector field and you have the abstract concept of flux through a surface. The flux through a surface would tell you how much of a particular vector field passes through that surface.

Yeah, this is actually the way I originally interpreted flux, I just gave the fluid flow as an example of something my original view of flux may have represented...another example could be wind velocities I suppose. I can't really think of any other physical interpretations and come to think of it, the wind and fluid only really have any meaning if they pass through a closed surface, or in 2D a closed loop, and even then the meaning seems ambiguous as no one really deals with fluid entering or exiting an area in terms of straight velocity. But I REALLY digress.

Anyway, as I was saying, I originally thought flux indicated the combined magnitudes of all the vectors exiting/passing through the surface but now I realize that this view does not account for the infinitesimal surface area element by which one must multiply the purpendicular component of each vector when summing them.

So as I stated in my original question, what could F dot N * dsa represent? Mathematically or physically?It might help to understand what I'm saying if I make a comparison. With the line integral for work (F dot T * ds) the ds represents the distance over which the force generated by the vector field acts. This fits the definition of work as parallel force multiplied by a displacement, thus the presence of ds makes sense. In the flux integral, be it of the line or surface variety, I can see no purpose for the ds or dsa which would give the resulting flux any sensical meaning.