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integral curve (Definition)

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

$\displaystyle c'(t)$ $\displaystyle =$ $\displaystyle (X\circ c)(t), \,\,\,\,\,\,\,$$\displaystyle \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'(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

$\displaystyle \frac{dc^i}{dt}(t) = X^i\circ c(t),$   for all $ t$$\displaystyle . $



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Cross-references: functions, local coordinates, tangent vector, open interval, curve, point, vector field, smooth, smooth manifold
There are 6 references to this entry.

This is version 2 of integral curve, born on 2005-05-17, modified 2005-05-18.
Object id is 7063, canonical name is IntegralCurve.
Accessed 2476 times total.

Classification:
AMS MSC: 53-00 (Differential geometry :: General reference works )
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