A line integral, also known as a path integral, is a type of integral in calculus that aggregates a function's values along a curve or path in a multidimensional space. It extends the concept of definite integrals over intervals to integrals over curves in ℝⁿ or on manifolds.
Definition
Given a smooth (continuously differentiable) parametrized curve $ \mathbf{r} : [a,b] \to \mathbb{R}^n $ and a scalar field $ f : \mathbb{R}^n \to \mathbb{R} $, the line integral of $ f $ along the curve is defined by
$$ \int_{C} f(\mathbf{r}) , ds = \int_{a}^{b} f(\mathbf{r}(t)) , |\mathbf{r}'(t)| , dt, $$
where $ ds = |\mathbf{r}'(t)| dt $ represents an infinitesimal element of arc length.
For a vector field $ \mathbf{F} : \mathbb{R}^n \to \mathbb{R}^n $, the line integral (often called the work integral) is
$$ \int_{C} \mathbf{F} \cdot d\mathbf{r} = \int_{a}^{b} \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) , dt, $$
where $ d\mathbf{r} = \mathbf{r}'(t) dt $ and the dot denotes the Euclidean inner product.
Types
- Scalar line integral – integrates a scalar function over a curve, weighting by arc length.
- Vector line integral – integrates the component of a vector field tangent to the curve; interpretable as work done by a force field along a path.
- Complex line integral – integrates a complex-valued function over a contour in the complex plane, central to complex analysis.
Properties
- Additivity: If a curve $ C $ is partitioned into sub‑curves $ C_1, C_2, \dots, C_k $, then $\int_C = \sum_{i=1}^{k} \int_{C_i}$.
- Reparametrization invariance: The value of a line integral depends only on the geometric path and orientation, not on the specific parametrization, provided the parametrization preserves orientation and smoothness.
- Fundamental theorem for line integrals: If a vector field $ \mathbf{F} $ is conservative (i.e., $ \mathbf{F} = abla \phi $ for some scalar potential $ \phi $), then $\int_{C} \mathbf{F} \cdot d\mathbf{r} = \phi(\mathbf{r}(b)) - \phi(\mathbf{r}(a))$, making the integral path‑independent.
Applications
- Physics: Calculation of work done by a force field, circulation of fluid velocity fields, electromotive force in electromagnetism.
- Engineering: Evaluation of line loads and distributed forces along structural members.
- Mathematics: Integral theorems such as Green’s theorem, Stokes’ theorem, and the divergence theorem relate line integrals to surface and volume integrals, forming the backbone of vector calculus.
- Complex analysis: Cauchy’s integral theorem and formula use complex line integrals to evaluate analytic functions.
Historical Note
The concept originated in the development of calculus of variations and potential theory in the 18th and 19th centuries, with contributions from mathematicians such as Euler, Lagrange, and later Gauss and Green, who formalized the relationship between line integrals and area integrals.
