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# 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 $\gamma\subset\mathbb{R}^{n}$. 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 $[a,\,b]$.

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:

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 $S:P\rightarrow\vec{F}$.

## Mathematics Subject Classification

81S40*no label found*46T12

*no label found*

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