Line integral

A line integral accumulates a scalar function or vector field along a curve. Its value depends on the curve and, for scalar functions, on how the curve is parametrized when the integral includes arc length.

Parametric curve: A curve described by coordinate functions of a parameter. Its coordinate functions let a curve be substituted into the integrand.
Vector field: A function assigning a vector to each point in a domain. Vector line integrals sample these assigned vectors along a curve.
Circulation: The line integral of a vector field around a closed curve. It measures the field’s accumulated tangential component around a loop.
Surface integral: An integral accumulating a function or field over a surface. It integrates over a two-dimensional surface rather than a one-dimensional curve.
Arc length: The distance measured along a curve. Scalar line integrals use arc length to weight values by distance traveled.
Scalar field: A function assigning a scalar value to each point in a domain. Scalar line integrals accumulate these pointwise values along a curve.
Green's theorem: A theorem relating a line integral around a simple closed plane curve to a double integral over its interior. It converts planar boundary integrals into area integrals of derivatives.
Volume integral: An integral accumulating a function over a three-dimensional region. Its domain is a volume, unlike the curve domain of a line integral.
Line integral of a scalar field: An integral that accumulates scalar-field values along a curve, usually weighted by arc length. It is the scalar-valued form, where orientation does not affect the result.
Derivative: A measure of how a function changes with respect to its input. Differentiating a curve’s parametrization gives the tangent used in vector integrals.

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