overflateintegral

The term overflateintegral is the Norwegian word for surface integral, a mathematical operation that extends the concept of multiple integration to integration over a two‑dimensional surface in three‑dimensional space. Surface integrals are used to calculate fluxes of scalar or vector fields through a surface, and they are fundamental in many branches of physics and engineering.

A scalar surface integral computes the integral of a scalar function f over a surface S. If S is parameterized by a mapping r(u,v) from a region D in the uv‑plane, the integral is ∫∫_S f dS, where dS = |r_u × r_v| du dv is the element of surface area determined by the cross product of the partial derivatives of r. For a vector field F, the surface integral (often called the flux integral) is written as ∬_S F · n dS, where n is a unit normal vector to the surface. The direction of n determines the orientation; reversing n changes the sign of the value.

Surface integrals are closely linked to several fundamental theorems. Stokes’ theorem relates the flux of the curl of a vector field through a surface to the circulation of the field along its boundary. The divergence theorem, also called Gauss’ theorem, equates the flux of a vector field through a closed surface to the volume integral of its divergence inside the surface. These theorems provide powerful tools for converting surface integrals into volume integrals and vice versa.

Applications are widespread. In electromagnetism, the electric flux through a surface relates to the enclosed charge via Gauss’ law. Fluid dynamics uses surface integrals to compute the net flow of fluid out of a region. In computer graphics, surface integrals are employed in shading models that compute light reflected over 3D surfaces. The concept also appears in differential geometry, where the area of a parametric surface and curvature calculations are expressed as surface integrals.