6709 MTG 5990 01 Calculus on manifolds (3) TR time 1700-1815 Room GC 285 (contact: rukim@fiu.edu).
Fall 2002

Description: Topics include:

  • Point set topology including Hausdorff spaces, metric spaces; axioms of countability; compactness and paracompactness, connectedness.
  • Differential calculus on euclidean n-space;
  • Differentiable manifolds and their (co)tangent bundles;
  • Tensor bundles and tensor calculus including Cartan's calculus of differential forms;
  • Integration on manifolds including Stokes' Theorem.
  • The final examination is scheduled for R, 1530-1815
  • Prerequisites: MTG 4302 (Topology) and MAS 3105 (Linear Algebra)

  • References:
  • Ralph Abraham and Jerold E. Marsden, Foundations of Mechanics (second edition), Addison-Wesley 1987.
  • R. Abraham; J.E. Marsden; T. Ratiu, Manifolds, tensor analysis and applications, second edition, Springer 1988.
  • (Textbook) Michael Spivak, Calculus on manifolds: a modern approach to classical theorems of advanced calculus, Reading, Massachusetts, Addison-Wesley, 1965.
  • James Munkres, Topology, second edition, Prentice Hall, NJ., 2000.


  • This is the first of a series of two courses for the beginning graduate student preparing for a qualifying examination in Geometry/Topology.
  • After a successful completion of MTG 5990, the student should be ready to take a second year graduate course in Differential Geometry, Algebraic Topology and/or Differential Topology.
  • The first half of the course provides a deeper coverage of point set topology than the undergraduate topology course MTG 4302. The second half deals with the actual calculus on manifolds.