Integral curve

Integral curves are known by various other names, depending on the nature and interpretation of the differential equation or vector field.

Let X be a vector field on M of class Cr−1 and let p ∈ M. An integral curve for X passing through p at time t0 is a curve α : J → M of class Cr−1, defined on an open interval J of the real line R containing t0, such that The above definition of an integral curve α for a vector field X, passing through p at time t0, is the same as saying that α is a local solution to the ordinary differential equation/initial value problem It is local in the sense that it is defined only for times in J, and not necessarily for all t ≥ t0 (let alone t ≤ t0).

In the above, α′(t) denotes the derivative of α at time t, the "direction α is pointing" at time t. From a more abstract viewpoint, this is the Fréchet derivative: In the special case that M is some open subset of Rn, this is the familiar derivative where α1, ..., αn are the coordinates for α with respect to the usual coordinate directions.

The same thing may be phrased even more abstractly in terms of induced maps.

The curve α induces a bundle map α∗ : TJ → TM so that the following diagram commutes: Then the time derivative α′ is the composition α′ = α∗ o ι, and α′(t) is its value at some point t ∈ J.