Fluxacross

Fluxacross is a term used in mathematics and physics to describe the measurement of a vector field’s flux across a prescribed boundary or surface. It is typically expressed as a surface integral that encodes the amount of the field passing through the surface, taking into account the chosen orientation of the surface.

In three dimensions, the flux across a smooth oriented surface S for a vector field F is given by the surface integral ∬S F · n dS, where n is the unit normal vector to S, and dS is the differential area on S. The sign of the flux depends on the direction of n relative to the field. For a closed surface, the flux is related to the divergence of F by Gauss’ divergence theorem: ∬S F · n dS = ∭V div F dV, where V is the volume enclosed by S.

In two dimensions, when considering flux across the boundary C of a region D in the plane, the flux across C is ∬D div F dA and, equivalently, can be written as a line integral ∬C F · n ds, with n denoting the outward unit normal along C and ds the line element. This 2D formulation mirrors the 3D divergence theorem in a planar setting.

Fluxacross is central to applications in electromagnetism, fluid dynamics, heat transfer, and numerical methods such as finite-volume techniques, where fluxes across cell faces model transport processes. While not a formal standardized term in all texts, fluxacross is often used as a shorthand for the process of computing boundary or surface fluxes of vector fields. See also flux, surface integral, divergence theorem, and Stokes’ theorem.