vector field along a curve

Let M be a differentiable manifold and γ:[a,b]M be a differentiableMathworldPlanetmathPlanetmath curve in M. Then a vector field along γ is a differentiable map Γ:[a,b]TM, the tangent bundleMathworldPlanetmath of M, which projects to γ under the natural projection π:TMM. That is, it assigns to each point t0[a,b] a vector tangentPlanetmathPlanetmath to M at the point γ(t), in a continuousMathworldPlanetmath manner. A good example of a vector field along a curve is the speed vector γ˙. This is the pushforward of the constant vector field ddt by γ, i.e., at t0, it is the derivation γ˙(f)=ddt(fγ)|t=t0.

Title vector field along a curve
Canonical name VectorFieldAlongACurve
Date of creation 2013-03-22 13:58:55
Last modified on 2013-03-22 13:58:55
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 4
Author bwebste (988)
Entry type Definition
Classification msc 53B05