line integral

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

The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulas in physics (for example, $W=\vec F\cdot\vec d$ ) have natural continuous analogs in terms of line integrals ( $W=\int_C \vec F\cdot d\vec s$ ). The line integral finds the work done on an object moving through an electric or gravitational field, for example.

Vector calculus

In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given vector field along a given curve.

Scalar definition

For some scalar field f : U $\to$ R defined on an open subset U of Rⁿ, the line integral on a curve C, parametrized as r(t)∈U with t $\isin$ [a, b] is defined by

$\int_C f\, ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\, dt.$

where

f is the scalar field being integrated,

r(t): [a, b]

\to

C is a bijective parametrization of the curve C and

r(a) and r(b) give the endpoints of C.

The symbol ds is heuristically interpreted as an elementary arc length. Because they depend only on the element of arc length, line integrals of scalar fields are independent of the parametrization r(t).

Vector definition

For a vector field F : U ⊆ Rⁿ $\to$ Rⁿ, the line integral on a curve C, parametrized as r(t)∈U with t ∈ [a, b], is defined by

$\int_C \mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt.$

Line integrals of vector fields are independent of the parametrization r(t) in absolute value, but they do depend on its orientation. Specifically, a reversal in the orientation of the parametrization changes the sign of the line integral.

Path independence

Main article: gradient theorem

If a vector field F is the gradient of a scalar field G, that is,

$\nabla G = \mathbf{F},$

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

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

which happens to be the integrand for the line integral of F on r(t). It follows that, given a path C , then

$\int_C \mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt = \int_a^b \frac{dG(\mathbf{r}(t))}{dt}\,dt = G(\mathbf{r}(b)) - G(\mathbf{r}(a)).$

In words, the integral of F over C depends solely on the values of G in the points r(b) and r(a) and is thus independent of the path between them.

For this reason, a line integral of a vector field which is the gradient of a scalar field is called path independent.

Applications

The line 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 line integral of F on C.

Relationship with the line integral in complex analysis

Viewing complex numbers as 2D vectors, the line integral in 2D of a vector field corresponds to the real part of the line integral of the conjugate of the corresponding complex function of a complex variable.

Due to the Cauchy-Riemann equations the curl of the vector field corresponding to the conjugate of a holomorphic function is zero. This relates through Stokes theorem both types of line integral being zero.

Complex analysis

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

$\int_\gamma f(z)\,dz$

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

$\sum_{1 \le k \le n} f(\gamma(t_k)) ( \gamma(t_k) - \gamma(t_{k-1}) ).$

The integral is then the limit of this sum, as the lengths of the subdivision intervals approach zero.

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

$\int_\gamma f(z)\,dz =\int_a^b f(\gamma(t))\,\gamma\,'(t)\,dt.$

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

$\oint_\gamma f(z)\,dz$

is often used for the line integral of f along $γ$ .

Important statements about contour integrals are 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).

Example

Consider the function f(z)=1/z, and let the contour C be the unit circle about 0, which can be parametrized by e^it, with t in $[0,2π]$ . Substituting, we find

$\oint_C f(z)\,dz = \int_0^{2\pi} {1\over e^{it}} ie^{it}\,dt = i\int_0^{2\pi} e^{-it}e^{it}\,dt$

$=i\int_0^{2\pi}\,dt = i(2\pi-0)=2\pi i$

which can be also verified by the Cauchy integral formula.

Quantum mechanics

The "path integral formulation" of quantum mechanics actually refers not to path integrals in this sense but to functional integrals, that is, integrals over a space of paths, of a function of a possible path. However, path integrals in the sense of this article are important in quantum mechanics; for example, complex contour integration is often used in evaluating probability amplitudes in quantum scattering theory.

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