Lineintegral

A line integral (often written line integral or contour integral in complex contexts) is an integral where a scalar or vector field is evaluated along a curve in space. It generalizes the definite integral to integration over a one-dimensional path and is defined for piecewise-smooth parameterized curves r(t), t = a..b.

For a scalar field f(x) measured along a curve C, the line integral is written ∫_C f ds and computed by parameterization as ∫_a^b f(r(t)) |r′(t)| dt, representing quantities like mass along a wire with linear density f. For a vector field F, the line integral takes the form ∫_C F · dr = ∫_a^b F(r(t)) · r′(t) dt; this computes work done by a force field or circulation of a fluid along C. Orientation of C matters for vector line integrals and for integrals over closed loops.

Important properties include the fundamental theorem for line integrals: if F = ∇φ (a gradient field), then ∫_C F · dr = φ(r(b)) − φ(r(a)), so the integral is path-independent in simply connected domains where curl F = 0. Closed line integrals relate to curl via Green’s theorem (planar case) and Stokes’ theorem (surface generalization), linking circulation around a loop to a surface integral of curl. Line integrals also appear as contour integrals in complex analysis, with applications across physics and engineering including mechanics, electromagnetism, and fluid dynamics.