flux integral
Hold a hoop in a stream and ask how much water passes through it per second. If you hold the hoop face-on to the current, lots flows through; turn it edge-on and almost nothing does. Flux is the answer to this question for any vector field and any surface: the net amount of the field that crosses the surface, counting only the component that actually goes through, not the part sliding along it.
The flux of a vector field F through an oriented surface S is the surface integral integral over S of F dot n dS, where n is the unit normal that gives the surface its chosen 'positive' side and dS is the area element. The dot product F dot n keeps only the part of F perpendicular to the surface — exactly the part that pierces it — so a field running parallel to the surface contributes zero flux. In parametrized form this is integral over the parameter region of F dot (r_u cross r_v) du dv, where the cross product supplies both the normal direction and the area scaling at once.
Flux is the physical meaning of half the great theorems and of much of physics. The flux of the electric field through a closed surface, by Gauss's law, equals the enclosed charge over epsilon_0; the flux of a fluid's velocity field through a surface is the volume crossing per unit time; heat flux measures energy crossing a boundary. The divergence theorem is precisely the statement that the total flux out of a closed surface equals the integral of the divergence inside — flux and divergence are the same idea at two scales.
The flux of the radial field F = (x, y, z) outward through the unit sphere is integral over S of F dot n dS. On the sphere F dot n = 1 (the field is already normal, of length 1), so the flux equals the surface area 4 pi — consistent with the divergence theorem, since div F = 3 times the ball's volume 4 pi/3 is also 4 pi.
The sphere's outward flux 4 pi matches the volume integral of the divergence — the divergence theorem in one line.
Flux depends on the orientation you pick: choosing the opposite normal flips its sign, and a non-orientable surface like a Mobius band has no consistent normal at all, so flux is simply undefined there. Always state which way the surface is oriented before reporting a flux.