Tangent Space. Tangent Vector. Motion along a trajectory is described by position and velocity. Position uses an origin References the trajectory Displacement points along the trajectory. Tangent to the trajectory Velocity is also tangent. x 3. x 2. x 1. Tangent Plane.
Fibers can be associated with all points in a chart, and all charts in a manifold.
This is a tangent bundle.
Set is TQQVn
Visualize for a 1-d manifold and 1-d vector.Tangent Bundle
A tangent plane is independent of the coordinates. charts in a manifold.
Coordinates are local to a neighborhood on a chart.
Charts can align in different ways.
Locally the same bundle
Different manifold TQTwisted Bundles
Map from tangent space back to original manifold. charts in a manifold.
p = TQQ; (x, v) (x)
Projection map p
Map from one tangent space to another
f: UW; U, W open
f is differentiable
(x, v) (f(x), Df(x)v)
Tangent map Tf
Df(x) is the derivative offTangent Maps