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# 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 $t$}.$ |

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## Mathematics Subject Classification

53-00*no label found*

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