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
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