# path integral

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 (http://planetmath.org/PathConnected) $\gamma\subset\mathbb{R}^{n}$. 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 $\gamma$. Thusly, it is defined in terms of the parametrization of $\gamma$, mapped into the domain $\mathbb{R}^{n}$ of $\vec{F}$. Analytically,

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

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 $x_{i}$ is parametrized into a function of $t$.

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 $[a,\,b]$ into $n$ 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 $t_{i}$ be an element of the $i$th subinterval. The quantity $|\vec{\gamma}^{\prime}(t_{i})|$ gives the average  magnitude of the vector tangent  to the curve at a point in the interval $\Delta t$. $|\vec{\gamma}^{\prime}(t_{i})|\Delta t$ 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.

 $\lim_{\Delta t\rightarrow 0}\sum_{a}^{b}\vec{F}(\vec{\gamma}(t_{i}))\cdot\vec{% \gamma}^{\prime}(t_{i})\Delta t$

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).

 $\int_{\gamma}\vec{F}\cdot d\vec{x}=\int_{a}^{b}\vec{F}(\vec{\gamma}(t))\cdot% \vec{\gamma}^{\prime}(t)dt$

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

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: This is an image of a path $\gamma$ superimposed on a vector field $\vec{F}$. This is a visualization of what we are doing when we take the integral under the curve $S:P\rightarrow\vec{F}$.

Title path integral PathIntegral 2013-03-22 12:16:14 2013-03-22 12:16:14 slider142 (78) slider142 (78) 19 slider142 (78) Definition msc 81S40 msc 46T12 line integral ComplexIntegral ContourIntegral RealAndImaginaryPartsOfContourIntegral