Science Fair Project Encyclopedia
- This article is about "path integrals" in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman.
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.
may be defined by subdividing the interval [a, b] into a = t0 < t1 < ... < tn = b and considering the expression
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 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.
Consider the function f(z)=1/z, and let the contour C be the unit circle about 0, which can be parametrized by eit, with t in [0, 2π]. Substituting, we find
which can be also verified by the Cauchy integral formula.
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.
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
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
If a vector field F is the gradient of a scalar field G, that is,
which happens to be the integrand for the path integral of F on r(t). It follows that, given a path C , then
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.
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.
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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