Felületelemekhez

Felületelemekhez is a term rooted in the concept of a “felületelem” (surface element), a fundamental idea in geometry and physics. A felületelem refers to an infinitesimally small piece of a surface embedded in three-dimensional space. These tiny elements are used to define and compute surface-related quantities such as area, mass distribution, and flux of vector fields.

Mathematically, for a smooth parametric surface r(u,v) with parameters (u,v) in a domain D, the area element is dS = |∂r/∂u × ∂r/∂v| du dv. The oriented surface element is the vector dS⃗ = (∂r/∂u × ∂r/∂v) du dv, which carries a normal direction. If a scalar function f is defined on the surface, its surface integral is ∬_S f dS. For a vector field F, the flux through the surface is ∬_S F · dS⃗.

A common example is the sphere of radius R, parameterized by r(θ, φ) = (R sinθ cosφ, R sinθ sinφ, R cosθ). Here dS = R^2 sinθ dθ dφ and dS⃗ = n dS with n = r̂ (the outward unit normal). This leads to practical formulas for areas and fluxes on spherical surfaces.

Felületelemekhez are widely used in calculations of surface integrals, mass and charge distributions on shells, and in the application of the divergence and Stokes theorems. In Hungarian mathematical texts, the term felületelemek szerepel a felületet érintő integrálások és geometriai összefüggések leírásában.