Aluelement

Aluelement, often referred to in English as an area element, is a differential quantity that represents an infinitesimal area in a two‑dimensional region. It is the fundamental unit used when integrating over areas, for example to compute totals such as mass, probability, or charge distributed across a region.

In common coordinate systems, the area element takes specific forms. In Cartesian coordinates, the differential area is dA = dx dy. In polar coordinates, where points are described by (r, θ), the area element becomes dA = r dr dθ. For a surface in three-dimensional space described parametrically by a vector function r(u, v), the differential area is dA = |∂r/∂u × ∂r/∂v| du dv, which can also be written in terms of the first fundamental form as dA = sqrt(EG − F^2) du dv, where E, F, and G are metric coefficients derived from the parameterization.

The area element is used in double integrals over regions in the plane and on surfaces. If a region R in the plane carries a density function f(x, y), then the total quantity over R is given by the integral ∬R f(x, y) dA. This concept extends to physics and engineering for calculating quantities such as mass, charge, or probability when a density is defined over an area.

History and related ideas place the area element within the broader framework of measure theory and differential geometry, where it generalizes to surface and volume elements. It is complemented by volume elements dV in three dimensions and surface elements dS for curved surfaces.