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Line integral

This is not about "path integrals" in the sense which was studied by Richard Feynman. See Functional integration.

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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.

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

<math>\int_\gamma f(z)\,dz<math>

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

<math>\sum_{1 \le k \le n} f\left( \;\gamma(t_k)\;\right) \left[ \; \gamma(t_k) - \gamma(t_{k-1}) \; \right]<math>

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:

<math>\int_\gamma f(z)\,dz

=\int_a^b f(\gamma(t))\,\gamma\,'(t)\,dt<math>

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

<math>\oint_\gamma f(z)\,dz<math>

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 theorem, one can often use contour integrals in the complex plane to find integrals of real-valued functions of a real variable. See Residue theorem for an example which uses the theorem, or Cauchy's integral formula for an example which uses the Cauchy integral formula.

Vector calculus

Definition

For some scalar field f : Rⁿ → R, the path (or line) integral on a curve C, parametrized as r(t) with t ∈ [a, b], is defined by

<math>\int_C f\ ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)| dt<math>

Similarly, for a vector field F : Rⁿ → Rⁿ, the path integral on a curve C, parametrized as r(t) with t ∈ [a, b], is defined by

<math>\int_C \mathbf{F}(\mathbf{x})\cdot\,d\mathbf{x} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt<math>

Path independence

Given

<math>\nabla G = \mathbf{F}<math>

then the derivative of the composition of G and r(t) is

<math>\frac{dG(\mathbf{r}(t))}{dt} = \nabla G(\mathbf{r}(t)) \cdot \mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)<math>

which is the integrand of the path integral of F on r. It then follows that

<math>\int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt = \int_a^b \frac{dG(\mathbf{r}(t))}{dt}\,dt = G(\mathbf{r}(b)) - G(\mathbf{r}(a))<math>

Notice that the value of the integral depends solely on the values of the points r(b) and r(a) and is thus independent of the path between the them, hence, path independence. Therefore, for any vector field F, if there exists a function G such that F is the gradient of G, then F is said to be path independent.

Applications

The path integral has many uses in physics. For example, the work done on a particle traveling on a curve C inside a force field represented as a vector field F is the path integral of F on C.

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