# 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 $\Gamma:[a,b]\to TM$, 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 VectorFieldAlongACurve 2013-03-22 13:58:55 2013-03-22 13:58:55 bwebste (988) bwebste (988) 4 bwebste (988) Definition msc 53B05