# integral curve

Definition
Suppose $M$ is a smooth manifold^{}, and $X$ is a
smooth vector field on $M$. Then an integral curve of $X$ through
a point $x\in M$ is a curve $c:I\to M$, such that

${c}^{\prime}(t)$ | $=$ | $(X\circ c)(t),\text{for all}t\text{in}I$ | ||

$c(0)$ | $=$ | $x.$ |

Here $I\subset \mathbb{R}$ is some open interval of $0$, and ${c}^{\prime}(t)$ is the tangent vector in ${T}_{c(t)}M$ represented by the curve.

Suppose ${x}^{i}$ are local coordinates for $M$, ${c}^{i}$ are functions^{}
representing $c$ in these local coordinates, and
$X={X}^{i}\frac{\partial}{\partial {x}^{i}}$. Then the condition on $c$
is

$$\frac{d{c}^{i}}{dt}(t)={X}^{i}\circ c(t),\text{for all}t.$$ |

Title | integral curve |
---|---|

Canonical name | IntegralCurve |

Date of creation | 2013-03-22 15:16:31 |

Last modified on | 2013-03-22 15:16:31 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 5 |

Author | matte (1858) |

Entry type | Definition |

Classification | msc 53-00 |