path integral

The path integral is a generalization of the integral that is very useful in theoretical and applied physics. Consider a vector field $\overrightarrow{F}:{ℝ}^n\to {ℝ}^m$ and a path (http://planetmath.org/PathConnected) $\gamma \subset {ℝ}^n$. The path integral of $\overrightarrow{F}$ along the path $\gamma$ is defined as a definite integral. It can be constructed to be the Riemann sum of the values of $\overrightarrow{F}$ along the curve $\gamma$. Thusly, it is defined in terms of the parametrization of $\gamma$, mapped into the domain ${ℝ}^n$ of $\overrightarrow{F}$. Analytically,

$${\int }_{\gamma }\overrightarrow{F}\cdot 𝑑\overrightarrow{x}={\int }_a^b\overrightarrow{F}(\overrightarrow{\gamma }(t))\cdot 𝑑\overrightarrow{x}$$

where $\overrightarrow{\gamma }(a),\overrightarrow{\gamma }(b)$ are elements of ${ℝ}^n$, and  $d\overrightarrow{x}=⟨\frac{d{x}_1}{dt},\mathrm{\cdots },\frac{d{x}_n}{dt}⟩dt$  where each $x_i$ is parametrized into a function of $t$.

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

$$\underset{\mathrm{\Delta }t\to 0}{lim}\sum _a^b\overrightarrow{F}(\overrightarrow{\gamma }(t_i))\cdot {\overrightarrow{\gamma }}^{\prime }(t_i)\mathrm{\Delta }t$$

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).

$${\int }_{\gamma }\overrightarrow{F}\cdot 𝑑\overrightarrow{x}={\int }_a^b\overrightarrow{F}(\overrightarrow{\gamma }(t))\cdot {\overrightarrow{\gamma }}^{\prime }(t)𝑑t$$

Note that the path integral only exists if the definite integral exists on the interval $[a,b]$.

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 $\mathrm{\Gamma }$ 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:

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

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

Title	path integral
Canonical name	PathIntegral
Date of creation	2013-03-22 12:16:14
Last modified on	2013-03-22 12:16:14
Owner	slider142 (78)
Last modified by	slider142 (78)
Numerical id	19
Author	slider142 (78)
Entry type	Definition
Classification	msc 81S40
Classification	msc 46T12
Synonym	line integral
Related topic	ComplexIntegral
Related topic	ContourIntegral
Related topic	RealAndImaginaryPartsOfContourIntegral