Integral curve

An integral curve called in mathematics a defined on a differentiable manifold curve, which is closely related to a given smooth vector field on this manifold. Thus, for example, electrical field lines integral curves of the vector field associated electrical dar. Clearly there is a small Styrofoam ball is moving in the ideal case the integral curves of the vector field, which is approximately determined by the flow of a river.

Definition

Be a smooth vector field on a manifold of dimension and an arbitrary point. Then is called a smooth curve on an open interval with integral curve of by when

Or, in other words, the tangent of which is the same as given by vector at this point in any position.

Existence

In local coordinates, the problem reduces to a system of ordinary differential equations:

Where and are the smooth functions are. So together with the boundary condition is a classical initial value problem and the set of Picard - Lindelöf thus guarantees a unique solution in a neighborhood of. Since you call solutions of differential equations often ' Integral ', here is the term ' integral curve ' near.

Local River

There exists a unique maximal local flow with the domain to any smooth vector field. This is the unique maximal integral curve is with and for all. If the compact manifold, then the flow is global, so it applies to all and.