Wegintegralen

Wegintegralen, in Dutch mathematics, are integrals taken along a curve. They generalize the notion of summing quantities as one moves along a path in space and come in two main types: line integrals of scalar fields and line integrals of vector fields. Both use a parameterization of the curve C.

For a scalar field f: R^3 → R and a smooth curve C parameterized by r(t) for t in [a, b], the line integral of f with respect to arc length is ∫_C f ds = ∫_a^b f(r(t)) |r′(t)| dt. Here ds represents an infinitesimal element of arc length along C.

For a vector field F: R^3 → R^3 and the same parameterization, the line integral of F along C is ∫_C F · dr = ∫_a^b F(r(t)) · r′(t) dt. In coordinates, if F = (P, Q, R) and r′(t) = (dx/dt, dy/dt, dz/dt), then ∫_C P dx + Q dy + R dz, evaluated along the curve.

Orientation matters for vector line integrals: reversing the curve changes the sign. For scalar line integrals, the value typically depends on the function and the path length, not on an orientation, though care is needed when ds is defined with a sign in other contexts.

Key properties include that if F is conservative (F = ∇φ), then ∫_C F · dr = φ(final) − φ(initial). Line integrals relate to broader theorems such as Green’s, Stokes’, and the divergence theorem, connecting path integrals to areas and surfaces.

Applications arise in physics (work done by forces), engineering, and electromagnetism, among others. Computation typically uses a convenient parameterization of the curve and reduces to a single-variable integral.