Peter B Gilkey
202 Deady Hall,1-541-346-4717 (office phone) 1-541-346-0987 (fax) email:
Mathematics Department, University of Oregon, Eugene Oregon 97403 USA

Math 4/515 Introductory Analysis III Spring 2014 CRN 36240

Syllabus Version 3

  • Calculus on Manifolds
  • MWF 0900-0950 209 Deady; Problem session F 1500-1550 101 Peterson.
  • Office hours Monday, Wednesday, Friday 1000-1050
  • Text: Spivak: Calculus on manifolds (paperback). (Benjamin/Cummings Publishing Company). Any edition is fine - sometimes used copies of previous editions can be purchased more cheaply on the web.
  • Homework will be due each Monday on the material of the subsequent week. The Friday discussion hour is an opportunity for you to ask questions about the homework. The homework problems will be challenging and it is essential that you have thought about the homework before comming to the discussion hour. You should also feel free to ask questions regarding the lecture that have come up then (or during class of course). I will drop your 2 lowest homework scores in computing the homework average. This is to allow for life's little emergencies in case you have to miss turning in 1 or 2 homeworks. Late homework will not be accepted.
  • Grade: Will be based
    1. 25% on the homework
    2. 25% on the mid term Wednesday 30 April 2014
    3. 50% on the Final Exam 10:15 Thursday, June, 12.
  • Note: No class Memorial Day Monday May 28 2014.
    Teaching Associate: Ekaterina Puffini. Academic Calendar

  • Here are tentative reading and homework assignments. Note that the homework will "trail" the reading assignments and class lectures to give you a bit of time to digest it before it is due. I don't like to lecture on material Friday and then have homework on it due Monday. Subject to change
    1. Week 1 (31 March - 4 Apr 2014): Read 1-34. The orthogonal group, the oscillation, the derivative as the best linear approximation.
      Do 1.7, 1.10, 1.22, 1.30, 2.4, 2.5, 2.7.
    2. Week 2 (7 Apr - 11 Apr 2014): Read 34-45. The inverse function theorem.
      Do 2.12, 2.13., 2.21, 2.22, 2.23, 2.24, 2-25, 2-26. Also Extra problem
    3. Week 3 (14 Apr - 17 Apr 2014): Read 46-56. Finish the implicit function theorem. Start on integration theory.
      Do 2.29, 2.30, 2.31, 2.32, 2.35, 2.36, 2.37 [not part b], 2.38, 2.39
    4. Week 4 (21 Apr - 25 Apr 2014): Read 56-73. The Riemann integral, sets of measure and content zero. Integrable functions.
      Do 3.1-3.10.
    5. Week 5 (28 Apr - 2 May 2014): Read 56-73. Jordan measurability, Fubini's theorem.
      Do 1.18, 3.11, 3.12, 3.14, 3.15, 3.16, 3.17, 3.18, 3.19. Exam Wednesday 30 April 2014
    6. Week 6 (5 May - 9 May 2014): Finish reading 56-73. Partitions of unity, improper integrals, change of variables. See supplementary notes on Improper Integrals
      Do 3.13, 3-20, 3.21, 3.22, 3.23, 3.26, 3.28, 3.29, 3.36.
    7. Week 7 (12 May - 16 May 2014): Finish up change of variables theorem. See supplementary notes on Change of variable Theorem
      Do 1.17, 3.30, 3.31, 3.32, 3.33, 3.34, 3.37, 3.38
      1. Problem A1:  Prove or disprove the following assertion: ``Let U be a bounded open subset of R-n. Then the characteristic function of U is integrable in the extended sense over U.
      2. Problem A2:   Prove or disprove the following assertion: ``If U is any unbounded open subset of R-n, then the characteristic function of U is not integrable in the extended sense over U.''
    8. Week 8 (19 May - 23 May 2014): Discussion of differential forms. See supplementary notes Green's, Gauss's, Stokes Theorem
      Assignment #8
    9. Week 9 (27 May - 30 May 2014): Finish discussion of Green's, Gauss's, and Stokes Theorem.
      Assignment #9
    10. (26 May 2014 is Memorial Day)
    11. Week 10 (02 Jun - 06 Jun 2014): Discuss fundamental theorem of algebra, Brauer Fixed point theorem, Can't comb hair on billiard ball theorem.
    12. Week 11 (10 June - 14 June 2014) Final Exam 10:15 Thursday, 12 June 2014. Owing to faculty legislation, final exams may not be given early under any circumstances
  • Notes available in the class:
    1. Improper Integrals
    2. Change of variable Theorem
    3. Green's, Gauss's, Stokes Theorem
    4. Applications Fundamental Theorem Algebra, Brauer fixed point formula, combing the hair on a billiard ball.

    Course Goals

    This course builds on Math 4/513 and Math 4/514. As an exemplar: the uniform limit of continuous functions is continuous, a continuous function on a compact set attains its minimum and maximum values, closed bounded subsets of Rn are compact, etc. The course divides naturally into three (roughly) equal pieces.
    1. In the first piece, we discuss the notion of differentiation in the multi-variate context. The derivative is presented as the best linear approximation; it is then related to the Jacobian matrix of partial derivatives and to directional derivatives. The chain rule is established and the elementary properties of the derivative are derived. The inverse and implicit functions are established.
    2. We then turn to the theory of Riemann integration. The notion of upper and lower sums are introduced in the multi-variate context building on the theory introduced in Math 4/513 and Math 4/514. Sets of content zero and measure zero are introduced and it is shown that a function is Riemann integrable if and only if it is bounded and continuous except on a set of measure zero. Jordan measurability is defined in this context and Fubini's theorem is established (interchanging the order of integration is a bit complicated). Partitions of unity and compact exhaustions are presented and applied to discuss improper integrals; lecture notes are available to supplement Spivak's treatment. We conclude this section with the change of variables theorem; again, lecture notes are available to supplement Spivak's treatment. The change of variables theorem and Fubini's theorem is explored at length in Homework Assignment 8.
    3. The final section of the course deals with the generalized Stoke's theorem and differential forms. At this point Spivak's treatment is just a bit too abstract and requires algebraic sophistication not really appropriate to the level of this course. Course notes are available. The exterior algebra will be presented in terms of universal properties, but the properties are stressed not the algebraic foundations. The role of the determinant in relation to top degree forms is discussed as is the change of variables and exterior differentiation in the discussion of the integration theory of differential forms. The generalized Stokes theorem is proved. The relationship of this theorem to the classical Green's, Gauss's, and Stoke's theorem is explored at length in homework assignment 9. If time permits, the fundamental theorem of algebra, the Brauer fixed point theorem (any continuous map from the disk to itself has a fixed point), and the Billiard ball theorem are presented as applications of this material.

      Learning outcomes

      Students should be able to solve problems what involve giving proofs in the differential calculus. They should be able to present examples and counterexamples illustrating the relationship between various notions of the derivative. They should be able to state and solve problems that involve sets of content zero and measure zero. They should be able to use Fubini's theorem to evaluate certain multi-variable integrals. They should understand and be able to use partitions of unity and plateau functions in evaluating improper integrals and to use the change of variables theorem correctly. They should be able to use Green's, Gauss's, and Stokes's theorem to evaluate multi-variable integrals and to examine conservation laws.

    Mathematics Department Undergraduate Grading Standards November 2011

    There are two important issues that this grading policy recognizes. Rubric for applied courses: Modeling, in mathematical education parlance, means the process of taking a problem which is not expressed mathematically and expressing it mathematically (typically as an equation or a set of equations). This is usually followed by solving the relevant equation or equations and interpreting the answer in terms of the original problem.

    Rubric for pure courses:

    Many courses combine pure and applied elements and the rubrics for those courses will have some combination of elements from the two rubrics above. Detailed interpretation of the rubrics depends on the content and level of the course and will be at the discretion of instructors. Whether to award grades of A+ is at the discretion of instructors.

    Grades in graduate courses

    The faculty has reached basic agreement on the meaning of grades for graduate students in the 500- and 600-level courses: Faculty teaching 600-level courses shall have the option to use different (but functionally equivalent) assessment procedures to grade students who have been admitted to the Ph.D. program compared to students in the Master's/Pre-Ph.D. stage of the of the program.

    Academic dishonesty

    Academic Misconduct: The University Student Conduct Code (available at defines academic misconduct. Students are prohibited from committing or attempting to commit any act that constitutes academic misconduct. By way of example, students should not give or receive (or attempt to give or receive) unauthorized help on assignments or examinations without express permission from the instructor. Students should properly acknowledge and document all sources of information (e.g. quotations, paraphrases, ideas) and use only the sources and resources authorized by the instructor. If there is any question about whether an act constitutes academic misconduct, it is the students' obligation to clarify the question with the instructor before committing or attempting to commit the act. Additional information about a common form of academic misconduct, plagiarism, is available at

    Title IX

    Under Title IX, I have a duty to report relevant information. The UO is committed to providing an environment free of all forms of prohibited discrimination and sexual harassment, including sexual assault, domestic and dating violence and gender-based stalking. Any UO employee who becomes aware that such behavior is occurring has a duty to report that information to their supervisor or the Office of Affirmative Action and Equal Opportunity. The University Health Center and University Counseling and Testing Center can provide assistance and have a greater ability to work confidentially with students. Note: UO employees also have a duty to report child abuse. For those classes and/or processes in which students have historically reported information regarding child abuse, the language can be expanded to provide that notice as well by adding the following statement: All UO employees are required to report to appropriate authorities when they have reasonable cause to believe that any child with whom they come in contact has suffered abuse or any person with whom they come in contact has abused a child.

    Other information

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  • Child Abuse Under the Oregon Child Abuse Reporting Statutes, all UO employees have a duty to make a report to the Oregon Department of Human Services or a law enforcement agency when they have reasonable cause to believe any child with whom the employee comes in contact has suffered abuse or that any person with whom the employee comes in contact has abused a child. For instances that relate to UO authorized activities, UO employee are to report to the University of Oregon Police Department. For purposes of this reporting responsibility, a “child” is any “unmarried person who is under 18 years of age” and “abuse” includes, but is not limited to: assault of a child; physical injury to a child caused by other than accidental means; any mental injury to a child caused by cruelty to a child; rape of a child; sexual abuse; sexual exploitation; Š negligent treatment or maltreatment of a child; threat of harm to a child; buying or selling of a child; allowing a child on the premises where methamphetamine is being manufactured; and unlawful exposure to a controlled substance that subject a child to risk of harm. The duty of employees of public universities to report incidents of child abuse applies at all times, not just to those incidents occurring during working hours or on campus. For this purpose, university employees include all faculty and staff members, student workers, graduate teaching fellows, and temporary employees.
  • Under the law, reports must be made to the local office of the Department of Human Services or to a law enforcement agency in the county where the employee making the report is located at the time of the contact. Failure to report when required to do so is a Class A violation. Persons who make reports in good faith are immune from liability for making the report. For instances that relate to UO-authorized activities, UO employees are expected to make the report immediately to the UO Police Department at 541-346-2919. Karen Logvin, Director of Work/Life Resources in Human Resources, 541-346-2962,, is the initial point of contact for further questions related to the reporting of child abuse. In addition, you will find additional information and resources regarding mandatory reporting of child abuse and neglect at "

    To rest on the blue of the day, like an eagle rests on the wind, over the cold range, confident on its wings and its breadth.

    Web page spun on  31 March 2014 by Peter B Gilkey 202 Deady Hall, Department of Mathematics at the University of Oregon, Eugene OR 97403-1222, U.S.A. Phone 1-541-346-4717 of Deady Spider Enterprises