Spring 2020: MATH 437 "Differential forms and their Applications"

Description: This course is an introduction to differential forms and their applications. The exterior differential calculus of Elie Cartan is one of the most successful and illuminating techniques for calculations. The fundamental theorems of multivariable calculus are united in a general Stokes theorem which holds for smooth manifolds in any number of dimensions. This course develops this theory and technique to perform calculations in analysis and geometry. This course is independent of Math 436, although it overlaps with it. The point of view is more abstract and axiomatic, beginning with the general notion of a topological space, and developing the tools necessary for applying techniques of calculus. Local coordinates are necessary, but coordinate-free concepts are emphasized. Local calculations relate to global topological invariants, as exemplified by the Gauss-Bonnet theorem.

Assignments: Homework (33%), Midterm (33%), Final (33%).

Classes: TuTh 2pm - 3:15pm, MTH 0303.

Textbook: 0) A. Hatcher, Notes on point-set topology (pdf), 1) M.P. do Carmo, Differential Forms and Applications, 2) K. Janich, Vector Analysis.

Office hours: Tu 1pm - 2pm and Th 3:25-4:25 in my office (MATH 4416), and by appointment.

Syllabus and detailed course policy: (pdf).

Exam dates: Midterm: April 7, Final: Monday, May 18 10:30am-12:30pm.

Maintenance of grades: see the ELMS page of this course.

Homeworks: assigned roughly every other week. Late homeworks will not be accepted. Group work encouraged, though solutions need to written up individually.

Course log and assigned homework:

• January 28, Basic point-set topology (Hatcher).


• January 30, Connectedness (Hatcher).


• February 4, Compactness (Hatcher). Homework from Hatcher (due on Feb 13): Chapter 1: 3, 6, Chapter 2: 3,5,6, Chapter 3: 3, 4, 9.


• February 6, No class.


• February 11, Quotient spaces (Hatcher).


• February 13, Differential forms (do Carmo).


• February 18, Differential forms (do Carmo). Homework (due on Feb 27): [Hatcher, Chapter 4: 1 (page 53)], [do Carmo, Chapter 1: 1,2,7,8,16, Chapter 3: 2, 17] + read the Wikipedia entry of the `long line')


• February 20, Differential manifolds and elementary properties (do Carmo).


• February 25, The tangent bundle (do Carmo).


• February 27, The inverse function theorem (Lecture Notes). Homework (due on March 10): [do Carmo, Chapter 3: 1, 3, 6, 7,15 (skip all questions on orientation)]


• March 3, Immersions, embeddings, and submanifolds (do Carmo).


• March 5, Differential forms and vectorfields on manifolds (do Carmo).


• March 10, Orientation of manifolds, integration on manifolds (do Carmo). Homework (due on March 31) [do Carmo, Chapter 3: 7,8,9 Chapter 4: 4,5,8,9])


• March 12, Manifolds with boundary. Stokes theorem.


• March 31, ...

Previous semesters: Fall 2019 MATH 135, Spring 2019 MATH 740, Fall 2018 MATH 868C, Spring 2017 MATH 430, Fall 2016 MATH 868D, Fall 2015 MATH 220, MATH 401, Spring 2015 MATH 461, Fall 2014 MATH 430.