This may seem a basic question about surface integrals, but I'm reading Div, Grad, Curl, and All That, and it prompted me some soul-searching.
In most textbooks, they start discussion of a surface integral by saying take a small patch, $S$, of the surface and imagine a force acting through it.
This force acts at some angle relative to the normal vector, as the book illustrations invariably show.
Then, say the books, the formula for the flux of that small area, $S$, is the dot product of the Force and the Normal vector. And of course as $S$ gets infinitesimally small, integration becomes the solution.
So my intuition question, and apologies for the elementary nature of this. Is the reason you take the dot product of the normal and the force because we are projecting the force on to the normal? And is this because the actual force acting on $S$ is not F itself, since it is an angle (kind of like the incline plane problem in basic physics) to the normal, but the projection of F onto N?
In other words: in a simple incline plane problem, the force of someone pulling up the hill has a vertical and a horizontal component, and to find the amount of force in the horizontal direction you can take the projection of F onto the horizontal plane (right?). So in the case of a surface integral, the force in the normal direction, which I am really interested in, is the dot product of F and the normal vector. And that is why you integrate F dot N ds in surface integrals. And, as the vertical force in an inclined plane problem is a bit of a "waste," so, too, flux will be less the greater the angle between the force and the normal vector.
How wrong is that?
Best Answer
I think you're missing the main point in the physical interpretation here. We're finding the flux of the vector field across the surface. We want to find the (rate of) fluid flow of $\vec F$ across the surface (think of $\vec F$ as some variant of a velocity vector field of the fluid). Fluid that flows tangent to the surface contributes no flux. Thus, we want the normal component of $\vec F$, multiplied by area, to figure out how much fluid goes across. (If it helps, see part of my YouTube video. There is a small glitch, unfortunately, in the discussion, which I did correct a few minutes later.)