Lecture 09 - Differential structures: the pivotal concept of tangent vector spaces (Schuller's Geometric Anatomy of Theoretical Physics)

Video Lecture:

Summary of contents: the space of smooth maps on a manifold; smooth curves on a manifold; directional derivative operator; tangent vectors at a point and tangent space at a point; proof that the sum of tangent vectors is a tangent vector; alternative definitions of tangent space (via equivalence classes of smooth curves, derivations at a point on germs of functions, and physical tangent vectors); algebras over an algebraic field; associative, unital and commutative algebras; Lie algebras, Lie bracket and Jacobi identity; commutator; derivations on an algebra; detailed examples; proof that derivations on a algebra constitute a Lie algebra; proof of equality of manifold dimension and tangent space dimension: dim M = dim TpM; coordinate-induced basis of tangent spaces; change of coordinates under a change of coordinate-induced bases.

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Lecture Notes for this lecture: