Flux

Flux quantifies how much of a vector field pierces a given surface. For a uniform field E through a flat area A with unit normal n̂, the flux is Φ = E·n̂ A — the component of E along n̂ times the area. For curved surfaces and non-uniform fields, you break the surface into infinitesimal patches, compute E·n̂ dA for each, and add them all up. The result is a single number with units of field strength times area.

Intuitively, imagine the vector field as the velocity of a fluid flowing through space; the flux of that velocity through a surface is the volume of fluid crossing per unit time. For the electric field the "fluid" is abstract, but the bookkeeping is the same. Flux through a closed surface tells you whether more field is leaving than entering — which, by Gauss's law, is the same as asking how much charge is enclosed.

Flux is linear: the flux of a sum of fields equals the sum of their fluxes. It is signed: flux out of a closed surface (following the outward normal) is positive, flux in is negative. And it is the main quantity Gauss's law talks about — the whole power of that law comes from recognising that, for symmetric charge distributions, flux integrals collapse to E × (area) and solve themselves.