# path integral

The *path integral* is a generalization of the integral that is very useful in theoretical and applied physics. Consider a vector field $\overrightarrow{F}:{\mathbb{R}}^{n}\to {\mathbb{R}}^{m}$ and a path (http://planetmath.org/PathConnected) $\gamma \subset {\mathbb{R}}^{n}$. 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 ${\mathbb{R}}^{n}$ of $\overrightarrow{F}$. Analytically,

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

where $\overrightarrow{\gamma}(a),\overrightarrow{\gamma}(b)$ are elements of ${\mathbb{R}}^{n}$, and $d\overrightarrow{x}=\u27e8\frac{d{x}_{1}}{dt},\mathrm{\cdots},\frac{d{x}_{n}}{dt}\u27e9dt$ 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).

$${\int}_{\gamma}\overrightarrow{F}\cdot \mathit{d}\overrightarrow{x}={\int}_{a}^{b}\overrightarrow{F}(\overrightarrow{\gamma}(t))\cdot {\overrightarrow{\gamma}}^{\prime}(t)\mathit{d}t$$ |

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 |