Math 550: Calculus on Manifolds

  Spring 2014


The goal of this course is to develop the theory of integration of functions of several variables. The plan is to begin with Riemann integral on Rn, followed by the development of differential n-forms, culminating in Stokes Theorem on manifolds and the Poincare Lemma.  At this degree of generality, Stokes Theorem unifies the classical integral formulas of vector calculus into a single coherent formula.  Green's theorem, the Divergence Theorem, and the Fundamental Theorem of Calculus are all special cases.  Applications and more advanced topics will be presented as time permits.
Instructor David Klein
Santa Susana Hall,  Room 127  
Phone: (818) 677-7792
email: david.klein@csun.edu, web page: www.csun.edu/~vcmth00m
Office Hours: to be announced
Class Meetings              Mondays & Wednesdays, 5:00 to 6:15 p.m., Chaparral Hall 5208
Office hours
 M Tu W 10:45 to 11:15 & by appointment
Prerequisite          
Math 450
Grading There will be a midterm and a final exam, each contributing approximately 50% to the final grade for the course.  Plus grades (+) and minus grades (–) will be assigned. 

Final Exam: Wednesday, December 10, 2014, 5:30 to 7:30 p.m.
Homework Most assigned problems will not be collected for evaluation, but it is important that you do them.  The midterm and final exams will be largely based on homework problems. Student collaboration is encouraged, but you should understand everything you turn in.  The final exam will be based on homework assignments and other material from the textbook and lectures.
 
Textbooks
Differential Forms and Applications, by Manfredo do Carmo

Calculus on Manifolds, by Michael Spivak



Additional References

Vector Calculus, Linear Algebra, and Differential Form: A Unified Approach, by Hubbard and Hubbard

Principles of Mathematical Analysis, by Walter Rudin

Introduction to Analysis, by William Wade


Assignments

Assignments will be posted here.

Review

Spivak, pg 39, Do problem 2-37

Chap 3 of Spivak

Sect 3.1, pg 49, Do problems 3-1, 3-3, 3-5, 3-6
Sect 3.2, pg 52, Do problems 3-9, 3-10, 3-13
Sect 3.3, pg 56, Do problems 3-14, 3-15, 3-16, 3-20, 3-21, 3-22
Sect 3.4, pg 61-62, Do all *-problems
Read Section 3.5 and do review problem 2-26* from Chapter 2 (pg 29)
Sect 3.6, pg 73, Do problem 3-41

Chap 1 of do Carmo

pg 10, Do problems 2, 4, 5, 8, 9, 10, 11, 12, 13, 14

Chap 2 of do Carmo

pg 27, Do problems 1, 2a

Chap 3 of do Carmo

pg 50, Do problems 2, 3, 7

Chap 4 of do Carmo

pg 69, Do problems 1, 2, 3, 5, 6, 7, 8, 9



Note: This syllabus is subject to change during the semester.  Any changes will be posted here and announced in class.