Padintegralen

Padintegralen, also known as line integrals, are a class of integrals in vector calculus that evaluate a function along a curve in a multidimensional space. They generalize the concept of a definite integral from a one‑dimensional interval to an arbitrary path, allowing the accumulation of quantities such as mass, work, or flux along a trajectory.

There are two primary types of padintegralen. A scalar line integral integrates a scalar field f over a curve C with respect to arc length, expressed as ∫C f ds. This measures the total value of the field encountered while moving along C, weighted by the infinitesimal distance ds. A vector line integral, also called a work integral, involves a vector field F and a parametrized curve r(t), t∈[a,b], and is written ∫C F·dr = ∫a^b F(r(t))·r′(t) dt. This computes the work done by the field on a particle traveling along the curve.

When the vector field is conservative, the line integral depends only on the endpoints of C, and its value can be obtained from a potential function φ such that F=∇φ; the fundamental theorem for line integrals then gives ∫C F·dr = φ(r(b))‑φ(r(a)). For non‑conservative fields, the integral generally depends on the specific path.

Padintegralen appear in physics, engineering, and mathematics. In electromagnetism they represent circulation of electric or magnetic fields; in fluid dynamics they quantify flux across streamlines; in differential geometry they provide tools for defining work, circulation, and circulation theorems. Computationally, line integrals are evaluated by parametrizing the curve and applying standard integration techniques or numerical quadrature when analytic solutions are unavailable.