Integral curve

For a smooth manifold $M$, and a vector field $X$, an integral curve of $X$ is a curve $\gamma :J\to M$ that “follows” the vector field. More formally, ${\gamma }^{\prime }(t)=X_{\gamma (t)}\forall t\in J$ Thus, the direction (and magnitude) of the path matches the vector field at the point.

Maximal integral curve

An integral curve is maximal if there is no integral curve $\tilde{\gamma }:\tilde{J}\to M$ such that $J\subset \tilde{J}$. In other words, a maximal integral curve cannot be extended to an integral curve on a larger domain.

Properties

Rescaling lemma

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Translation lemma

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