**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.

**Instructor:**
PHILIPPE RUKIMBIRA

**Objectives**:

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.

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