Vektorflux

Vektorflux is a concept in vector calculus describing the net flow of a vector field through a surface, considering the surface's orientation. Given a vector field F defined on a region of space and an oriented surface S, the vektorflux through S is a scalar Φ defined by the surface integral Φ = ∬_S F · n dS, where n is the chosen unit normal to S and dS is the element of surface area.

To compute it when S is parameterized by r(u,v) for (u,v) in a domain D, the differential area vector is (r_u × r_v) du dv; hence Φ = ∬_D F(r(u,v)) · (r_u × r_v) du dv.

For a closed surface S bounding a volume V, Gauss's divergence theorem gives Φ = ∬∬_V (div F) dV, connecting vektorflux to the divergence of F.

For surfaces with boundary, related theorems describe other integrals: Stokes's theorem relates the surface integral of curl F to a line integral around the boundary ∂S, though that is a curl-based flux rather than F · n dS.

In application, vektorflux measures how much of the field passes through S, with applications in electromagnetism (electric flux), fluid dynamics (mass or volume flow through a surface), and more generally in any physical context where a field transfers quantity across a surface.