PlanetMath: path integral

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path integral

(Definition)

The path integral is a generalization of the integral that is very useful in theoretical and applied physics. Consider a vector field $\vec{F}\!:\mathbb{R}^n\rightarrow\mathbb{R}^m$ and a path $\gamma\subset\mathbb{R}^n$ . The path integral of $\vec{F}$ along the path $\gamma$ is defined as a definite integral. It can be constructed to be the Riemann sum of the values of $\vec{F}$ along the curve $\gamma$ , aka the area under the curve $S:\gamma\rightarrow\vec{F}$ . Thusly, it is defined in terms of the parametrization of $\gamma$ , mapped into the domain $\mathbb{R}^n$ of $\vec{F}$ . Analytically,

$\displaystyle \int_\gamma \vec{F}\cdot d\vec{x} = \int_a^b \vec{F}(\vec{\gamma}(t))\cdot d\vec{x}$

where $\vec{\gamma}(a), \vec{\gamma}(b)$ are elements of $\mathbb{R}^n$ , and $d\vec{x}=\langle\frac{dx_1}{dt},\cdots,\frac{dx_n}{dt}\rangle dt$ where each

is parametrized into a function of

Proof and existence of path integral:
Assume we have a parametrized curve $\vec{\gamma}(t)$ with $t\in[a,b]$ . We want to construct a sum of $\vec{F}$ over this interval on the curve $\gamma$ . Split the interval into subintervals of size $\Delta t=(b-a)/n$ . Note that the arc lengths need not be of equal length, though the intervals are of equal size. Let be an element of the th subinterval. The quantity $\vert\vec{\gamma}'(t_i)\vert$ gives the average magnitude of the vector tangent to the curve at a point in the interval $\Delta t$ . $\vert\vec{\gamma}'(t_i)\vert\Delta t$ is then the approximate arc length of the curve segment produced by the subinterval $\Delta t$ . Since we want to sum $\vec{F}$ over our curve $\vec{\gamma}$ , we let the range of our curve equal the domain of $\vec{F}$ . We can then dot this vector with our tangent vector to get the approximation to $\vec{F}$ at the point $\vec{\gamma}(t_i)$ . Thus, to get the sum we want, we can take the limit as $\Delta t$ approaches 0.

$\displaystyle \lim_{\Delta t\rightarrow 0}\sum_a^b \vec{F}(\vec{\gamma}(t_i))\cdot\vec{\gamma}'(t_i)\Delta t$

This is a Riemann sum, and thus we can write it in integral form. This integral is known as a path or line integral (the older name).

$\displaystyle \int_\gamma \vec{F}\cdot d\vec{x} = \int_a^b \vec{F}(\vec{\gamma}(t))\cdot\vec{\gamma}'(t)dt$

Note that the path integral only exists if the definite integral exists on the interval

Properties:
A path integral that begins and ends at the same point is called a closed path integral, and is denoted with the summa symbol with a centered circle: $\oint$ . These types of path integrals can also be evaluated using Green's theorem.
Another property of path integrals is that the directed path integral on a path $\Gamma$ in a vector field is equal to the negative of the path integral in the opposite direction along the same path. A directed path integral on a closed path is denoted by summa and a circle with an arrow denoting direction.

Visualization Aids:

$\includegraphics[width=3.65in,height=3.65in]{pintegral}$

This is an image of a path $\gamma$ superimposed on a vector field $\vec{F}$ .

$\includegraphics[width=4in,height=3in]{bootleg}$

This is a visualization of what we are doing when we take the integral under the curve $S:P\rightarrow\vec{F}$ .

"path integral" is owned by slider142. [ full author list (3) | owner history (3) ]

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See Also: complex integral, contour integral

Other names:

line integral

Keywords:

integral of a vector field, integral over a vector field

Attachments:: area of plane region (Topic) by pahio

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Cross-references: image, arrow, opposite, negative, property, Green's theorem, types, circle, closed path, limit, approximation, tangent vector, range, segment, point, tangent to the curve, vector, average, length, arc lengths, size, subintervals, interval, sum, proof, function, domain, terms, area, curve, Riemann sum, definite integral, vector field, integral
There are 6 references to this entry.

This is version 15 of path integral, born on 2002-02-03, modified 2007-01-03.
Object id is 1700, canonical name is PathIntegral.
Accessed 24560 times total.

Classification:

AMS MSC:	46T12 (Functional analysis :: Nonlinear functional analysis :: Measure on manifolds)
	81S40 (Quantum theory :: General quantum mechanics and problems of quantization :: Path integrals)

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