NationMaster - Encyclopedia: Path integral

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. Euclid, detail from The School of Athens by Raphael. ... In calculus, the integral of a function is a generalization of area, mass, volume and total. ... Partial plot of a function f. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...

1 Complex analysis
- 1.1 Example
2 Vector calculus
3 Quantum mechanics
4 See also
5 External links

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 Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ...

$int_gamma f(z),dz$

may be defined by subdividing the interval [a, b] into a = t₀ < t₁ < ... < t_n = b and considering the expression In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

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

The integral is then the limit of this sum, as the lengths of the subdivision intervals approach zero. In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes larger and larger; or the behavior of a sequences elements, as their index becomes larger and larger. ...

If γ is a continuously differentiable curve, the path integral can be evaluated as an integral of a function of a real variable: In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

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

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 path integral of f along γ.

Important statements about path integrals are the Cauchy integral theorem and Cauchy's integral formula. In mathematics, the Cauchy integral theorem in complex analysis, named after Augustin Louis Cauchy, is an important statement about path integrals for holomorphic functions in the complex plane. ... In mathematics, Cauchys integral formula, named after Augustin Louis Cauchy, is a central statement in complex analysis. ...

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). 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. ... 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. ...

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

$oint_C f(z),dz = int_0^{2pi} {1over e^{it}} ie^{it},dt = iint_0^{2pi} e^{-it}e^{it},dt$

$=iint_0^{2pi},dt = i(2pi-0)=2pi i$

which can be also verified by the Cauchy integral formula. Cauchys integral formula is a central statement in complex analysis. ...

Vector calculus

In qualitative terms, a path integral in vector calculus can be thought of as a measure of the effect of a given vector field along a given curve. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

Definition

For some scalar field f : Rⁿ → R, the path integral on a curve C, parametrized as r(t) with t ∈ [a, b], is defined by In mathematics and physics, a scalar field associates a scalar to every point in space. ...

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

Path integrals are independent of parametrization r(t), and also, because they depend only on the element of arc length, are independent of the direction of the parametrization r(t).

For a vector field F : Rⁿ → Rⁿ, the line integral on a curve C, parametrized as r(t) with t ∈ [a, b], is defined by Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

$int_C mathbf{F}(mathbf{x})cdot,dmathbf{x} = int_a^b mathbf{F}(mathbf{r}(t))cdotmathbf{r}'(t),dt.$

Line integrals are independent of parametrization, but they do depend on the direction of the parametrization r(t). Specifically, a change of direction in parametrization changes the sign of the line integral.

Path independence

If a vector field F is the gradient of a scalar field G, that is, In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ...

$nabla G = mathbf{F},$

then the derivative of the composition of G and r(t) is In mathematics, the derivative is defined as the instantaneous rate of change of a function. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...

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

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

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

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

For this reason, a vector field which is the gradient of a scalar field is called 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.

Relationship with the path integral in complex analysis

Viewing complex numbers as 2D vectors, the path integral in 2D of a vector field corresponds to the real part of the path 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 path integral being zero. In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary and sufficient condition for a function to be holomorphic. ... This article is about the cURL command line tool. ... Stokes Theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...

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. This article or section is in need of attention from an expert on the subject. ... For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ... This article may be confusing for some readers, and should be edited to enhance clarity. ... The word probability derives from the Latin probare (to prove, or to test). ... Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation, that is, magnitude of the maximum disturbance in the medium during one wave cycle. ... In particle physics, scattering is a class of phenomena by which particles are deflected by collisions with other particles. ...

External links

Solved problems on path integrals

Categories: Complex analysis | Vector calculus

Results from FactBites:

Path integral - Wikipedia, the free encyclopedia (708 words)
	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.
	The integral is then the limit of this sum, as the lengths of the subdivision intervals approach zero.
	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.

More results at FactBites »

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