integral curve
IntegralCurve
2005-05-17 13:57:48
2005-05-18 14:04:46
Definition
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{\bf Definition}
Suppose $M$ is a smooth manifold, and $X$ is a
smooth vector field on $M$. Then an {\bf integral curve} of $X$ through
a point $x\in M$ is a curve $c\colon I\to M$, such that
\begin{eqnarray*}
c'(t) &=& (X\circ c)(t), \,\,\,\,\,\,\,\mbox{for all $t$ in $I$}\\
c(0) &=& x.
\end{eqnarray*}
Here $I\subset \sR$ 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
$$
\frac{dc^i}{dt}(t) = X^i\circ c(t), \quad \mbox{for all $t$}.
$$