Tangent Space

• Let be a smooth mapping of the domain of a vector space into a domain of a vector space. The image of the vector with origin under the mapping denoted is the velocity vector with which the moving point leaves the point when the moving point leaves the point with velocity . Thus

  Such that and also

  • does not depend on the motion provided leaves with velocity .
    • Intuition: This follows from the smoothness of the mapping.

• The set of velocity vectors leaving is a vector space called the tangent space. We denote this as .

  • The mapping is linear. In fact .
    Furthermore, let and be the bases of and . Then
    In other words, is the Jacobian and applying it to is equivalent to matrix-vector multiplication.
  • The set of points
    Is called the set of critical points.. Its image is the set of critical values.

• A tangent vector to a domain at a point is an equivalence class of smooth motions for which . The equivalence class is defined as follows:

  The distance between the points a and in any coordinate system is as . or more formally

Links §

• Ordinary Differential Equations by Arnold
• Differential Calculus