Felületelem

Felületelem is a term used in differential geometry and numerical methods to denote an infinitesimal element of a surface embedded in three-dimensional space. It represents the small patch of area on a two-dimensional surface, and can be used to compute surface areas, fluxes, and related quantities.

For a parametric surface r(u, v), the oriented surface element is given by dS_vector = (∂r/∂u × ∂r/∂v) du dv, and the magnitude of the area element is dS = ||∂r/∂u × ∂r/∂v|| du dv. Equivalently, using the first fundamental form with E = ∂r/∂u · ∂r/∂u, F = ∂r/∂u · ∂r/∂v, G = ∂r/∂v · ∂r/∂v, the scalar area element is dS = sqrt(EG − F^2) du dv. The orientation of the element is determined by the direction of the cross product ∂r/∂u × ∂r/∂v.

In practice, felületelemek appear in surface integrals ∫∫_S f dS and in physics and engineering applications such as calculating surface areas, fluxes, or boundary integrals. In the finite element method, a felületelem refers to a 2D surface element (for example, a triangular or quadrilateral patch) used to approximate curved surfaces, define shape functions, and perform numerical integration over a surface. The accuracy depends on the element type, mesh resolution, and how well the parameterization or mapping from a reference element captures the surface geometry.