Mathematics 430

Calculus on Manifolds

Catalog Description

Multivariable calculus including the inverse and implicit function theorems, manifolds, differential forms, and Stokes Theorem for compact, oriented k-manifolds. Prerequisites: Mathematics 210 (Linear Algebra) and 215 (Finite Mathematics).

Objectives of the course


Analysis on Manifolds,
by James R. Munkres, Westview Press, ISBN 0201315963, 1997 (Required)

Useful References

Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus,
by Michael Spivak, HarperCollins Publishers, ISBN 0805390219, 1965 (Reference only; not required)

Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (2nd Edition),
by John H. Hubbard and Barbara Burke Hubbard, Prentice Hall, ISBN 0130414085, 2001 (Reference only; not required)

Second Year Calculus : From Celestial Mechanics to Special Relativity (Undergraduate Texts in Mathematics / Readings in Mathematics),
by David M. Bressoud, Springer, ISBN 038797606X, 2001 (Reference only; not required)

These additional references relate well to this course.

Online Resources

Schedule for Spring 2006

The link (pdf) is to a file which was generated from an Excel spreadsheet. It can be displayed by your web browser or by an application such as Adobe Acrobat or Preview.


Much of our course will center on working through a substantial set of exercises taken from the Munkres text and from the online MIT OpenCourseWare materials written by Victor Guillemin which complement that text. Developing clear, clean, and elegant solutions to those problems, and explaining those solutions to each other, will take up much of our class time. You are then to write up your own versions of those solutions and hand them in as homework. The class discussions represent a community effort and may not extend to all of the exercises for which you are responsible. The homework for this course consists in writing a representation of your own understanding of the solutions to these exercises, whether discussed in class or not. Although the class sessions are collaborative and explorative, writing the homework solutions is an individual effort and is definitive. The goal is to engage and internalize the central mathematical issues of correctness, precision of expression, elegance, and style.

Course Notes and Related Materials


Chris Parrish
Office: Woods Laboratories 120

Location and Time

Tuesday, Thursday, 9:30 - 10:45 a.m.
Final Examination: Saturday, May 6, 2-4 pm
Woods Laboratories 134

Office Hours

Sewanee's tradition of cordial and constructive student-faculty relationships is one of its great strengths, and I am very happy to support that tradition. If you would like to talk with me, please make an appointment, either when you see me in class or in the hallways or in my office (WL120), or by email (, or by voice mail message (x1333).

Homework Day!

Homework is due on Thursdays!


Honor Code

The Honor Code applies to all examinations and written work produced in this course. Plagiarism is copying or imitating the language and thoughts of others, whatever the medium (written papers or computer programs). All work on the examinations must be done individually.


Attendance is required and is an important factor in doing well in the class. All assignments must be completed, and the student is responsible for making up any work missed due to absence. Late work will be accepted ONLY if appropriate arrangements have been made with the instructor PRIOR to the due date. The Dean's Office may be notified after two absences.