# vector field along a curve

Let $M$ be a differentiable manifold and $\gamma :[a,b]\to M$ be a differentiable^{} curve in $M$. Then a vector field along $\gamma $ is a differentiable map $\mathrm{\Gamma}:[a,b]\to TM$, the tangent bundle^{} of $M$, which projects to $\gamma $ under the natural projection $\pi :TM\to M$. That is, it assigns to each point ${t}_{0}\in [a,b]$ a vector tangent^{} to $M$ at the point $\gamma (t)$, in a continuous^{} manner. A good example of a vector field along a curve is the speed vector $\dot{\gamma}$. This is the pushforward of the constant vector field $\frac{d}{dt}$ by $\gamma $, i.e., at ${t}_{0}$, it is the derivation $\dot{\gamma}(f)={\frac{d}{dt}(f\circ \gamma )|}_{t={t}_{0}}$.

Title | vector field along a curve |
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Canonical name | VectorFieldAlongACurve |

Date of creation | 2013-03-22 13:58:55 |

Last modified on | 2013-03-22 13:58:55 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 4 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 53B05 |