integral curve

DefinitionSuppose $M$ is a smooth manifold, and $X$ is asmooth vector field on $M$ . Then an integral curve of $X$ througha 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 $c^{\\prime}(t)$ isthe tangent vector in $T_{c(t)}M$ represented by the curve.

Suppose $x^{i}$ are local coordinates for $M$ , $c^{i}$ are functionsrepresenting $c$ 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 $t$}.