overfladeintegrallet

Overfladeintegralet, or surface integral, is a generalization of integration to a surface in three-dimensional space. There are two main kinds: the surface integral of a scalar field and the surface integral (flux) of a vector field. Both rely on parametrizing the surface.

For a regular surface S parameterized by a vector function r(u,v) on a domain D in the uv-plane, the surface integral of a scalar function f is written as ∬_S f dS. Here dS = ||∂r/∂u × ∂r/∂v|| du dv is the element of surface area. If S is given as a graph z = g(x,y), then dS = sqrt(1 + g_x^2 + g_y^2) dx dy and ∬_S f dS = ∬_D f(x,y,g(x,y)) sqrt(1 + g_x^2 + g_y^2) dx dy.

For a vector field F, the surface integral (flux) through S is ∬_S F · n dS, where n is the unit normal to S. Equivalently, with a parametrization r(u,v), ∬_S F · n dS = ∬_D F(r(u,v)) · (∂r/∂u × ∂r/∂v) du dv, with the orientation determined by the chosen normal direction.

Orientation is essential: reversing the normal flips the sign of the result. For closed surfaces, the outward orientation is conventional. Fundamental theorems relate surface integrals to other quantities: the divergence theorem connects ∬_S F · n dS to a volume integral of div F, and Stokes’ theorem relates ∬_S curl F · n dS to a line integral around the boundary ∂S.

Applications span physics, engineering, geometry and computer graphics, including calculating surface areas, fluxes, and physical quantities like electric or fluid flux through surfaces. A common example is the surface area of a sphere, computed via a suitable parametrization.