Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 239: Differential Topology
Approved: 2015-08-21,

Suggested Textbook: (actual textbook varies by instructor; check your instructor)
An Introduction to Manifolds by Loring W. Tu
Search by ISBN on Amazon: 978-1-4419-7399-3

Suggested Schedule:

Lectures Sections (from Tu's book)
Week 1
§1 Smooth Functions on a Euclidean Space
§2 Tangent Vectors in R^n as Derivations
§3 The Exterior Algebra of Multicovectors
Week 2
§4 Differential Forms on R^n
§5 Manifolds
Week 3 §6 Smooth Maps on a Manifold
§8 The Tangent Space
Week 4 §9 Submanifolds
§12 The Tangent Bundle
Week 5 §13 Bump Functions and Partitions of Unity
§14 Vector Fields
Week 6 §17 Differential 1-Forms
§18 Differential k-Forms
§19 The Exterior Derivative
Week 7 §21 Orientations
§22 Manifolds with Boundary
Week 8 §23 Integration on Manifolds


For the remaining time, instructors are welcome to cover some of the following topics. Also, some instructors might go faster, and leave time for the following time-permitting topics.

  • §15 Lie Groups
  • §16 Lie Algebras
  • §24 De Rham Cohomology
  • Sard's theorem
  • Morse theory

Additional Notes:

Other references:
  • Introduction to Smooth Manifolds, by Jack Lee.
  • Notes on Manifolds, by Dmitry Fuchs.