# 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 $c\colon I\to M$, such that

 $\displaystyle c^{\prime}(t)$ $\displaystyle=$ $\displaystyle(X\circ c)(t),\,\,\,\,\,\,\,\mbox{for all t in I}$ $\displaystyle c(0)$ $\displaystyle=$ $\displaystyle x.$

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

 $\frac{dc^{i}}{dt}(t)=X^{i}\circ c(t),\quad\mbox{for all t}.$
Title integral curve IntegralCurve 2013-03-22 15:16:31 2013-03-22 15:16:31 matte (1858) matte (1858) 5 matte (1858) Definition msc 53-00