6709 MTG 5990 01 Calculus on manifolds (3) TR
time 1700-1815 Room GC 285 (contact: email@example.com).
Description: Topics include:
Point set topology including Hausdorff spaces,
metric spaces; axioms of countability; compactness and
Differential calculus on euclidean n-space;
Differentiable manifolds and their (co)tangent bundles;
Tensor bundles and tensor calculus including Cartan's calculus of
Integration on manifolds including Stokes' Theorem.
The final examination is scheduled for R, 1530-1815
Prerequisites: MTG 4302 (Topology) and MAS 3105 (Linear Algebra)
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
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
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.