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In mathematics, a path integral (also known as a line integral) is an integral where the function to be integrated is evaluated along a path or curve. Various different path integrals are in use. In the case of a closed path it is also called a contour integral.

1 Complex analysis

The path integral is a fundamental tool in complex analysis. Suppose U is an open subset of C, γ : [a, b] → U is a rectifiable curve and f : U → C is a function. Then the path integral

may be defined by subdividing the interval [a, b] into a = t₀ < t₁ < ... < t_n = b and considering the expression

The integral is then the limit as the distances of the subdivision points approach zero.

If γ is a continuously differentiable curve, the path integral can be evaluated as an integral of a function of a real variable:

When γ is a closed curve, that is, its initial and final points coincide, the notation

is often used for the path integral of f along γ.

Important statements about path integrals are given by the Cauchy integral theorem and Cauchy's integral formula.

Because of the residue theoremComplex analysis Theorems The residue theorem in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. It generalizes the Cauchy integral theorem, one can often use contour integrals in the complex plane to find integrals of real-valued functions of a real variable. See Residue theoremComplex analysis Theorems The residue theorem in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. It generalizes the Cauchy integral theorem for an example which uses the theorem, or Cauchy's integral formula for an example which uses the Cauchy integral formula.

1.1 Example

Consider the function f(z)=1/z, and the contour C the unit circle about 0, which can be parametrized by e^it, t ∈ [0, 2π]. Substituting, then:

which can be also verified by the Cauchy integral formula.

2 Vector calculus

In qualitative terms, the integrand of a path integral in vector calculus can be thought of as a measure of the effect of a given vector fieldIn mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. Vector fields are often used in physics to model for example the speed and direction of a moving fluid throughout space, or the along a given curve.

2.1 Definition

For some scalar fieldA scalar field associates a single number (or scalar to every point in space. Scalar fields are often used in physics, for instance to indicate the temperature distribution throughout space, or the air pressure. Definition A scalar field is a function : o f : Rⁿ → R, the path (or line) integral on a curve C, parametrized as r(t) with t ∈ [a, b], is defined by

Similarly, for a vector fieldIn mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. Vector fields are often used in physics to model for example the speed and direction of a moving fluid throughout space, or the F : Rⁿ → Rⁿ, the path integral on a curve C, parametrized as r(t) with t ∈ [a, b], is defined by