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

Syllabus (version 1 ) Math 415 (CRN 33473) and Math 515 (CRN 33483) Introductory Analysis III Spring 2018

  • Calculus on Manifolds
  • MWF 1300-1350 DEA 102; Problem session F 1500-1550 DEA 209
  • Office hours Monday, Wednesday, Friday 0900-0950
  • Text: The text is a Aspects of Differential Geometry I (Synthesis Lectures on Mathematics and Statistics February 2015, 154 pages, (doi:10.2200/S00632ED1V01Y201502MAS015)) by Gilkey, Park, and Vazquez-Lorenzo. It is available from Morgan & Claypool. Since the UO subscribes to Morgan & Claypool, any UO Student or Faculty Member can download the PDF file for free. If after trying the PDF file you really want a hard copy, you can either order one direct from Morgan & Claypool or AMAZON.COM. But I suggest you work with the PDF file first. An auxiliary reference is Spivak: Calculus on manifolds (paperback) (Benjamin/Cummings Publishing Company). This is not required.
  • 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. 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. PDF files as available are posted; latex files will be emailed. Latexed homework is preferred to handwritten.
    1. Week 1
  • Grade: Will be based
    1. 25% on the homework
    2. 25% on the mid term Wednesday 2 May 2018
    3. 50% on the Final Exam 1445 Wedmesday 13 June 2018
    4. An incomplete can be assigned when the quality of work is satisfactory but a minor yet essential requirement of the course has not been completed for reasons acceptable to the instructor (NOTE: this grade requires a contract to be completed). According to faculty legislation, final exams may not be given early under any circumstances. Your final grade will be assigned on the basis of the total point score of 400 points. You must bring your photo ID to all exams. You may bring a 3x5 inch index card with any formulas on it to any exam if you wish.
  • If you have a notification letter from the Accessible Education Center, please give me a copy as soon as possible so the relevant accommodations can be finalized.
    Teaching Associate: Ekaterina Puffini. Academic Calendar

  • 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. We conclude this section with the change of variables theorem.
    3. The final section of the course deals with the generalized Stoke's theorem and differential forms. 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. 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

      Note Learning outcomes are brief statements identifying the major skills, abilities, and concepts a student is expected to acquire from your course. The word "outcomes" can be used interchangeably with "goals" or "objectives" as long as the abilities in question are meaningfully evaluated using exams, papers, and other accepted means. The point is to make your expectations more transparent by articulating what may be only implicit in your course description, lesson topics, and assignments. Three to six short sentences or bullet points will suffice. Active verbs (evaluate, analyze, demonstrate, etc.) concretize expectations better than vague ones (appreciate, study, learn, etc.).

      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, to examine conservation laws, and to invent new verbs such as "concretize".

    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 see also

    Ethical Standards

    From the President's Office 2 May 2014: The University of Oregon is a community of scholars dedicated to the highest standards of academic inquiry, learning, and service. We are also committed to the highest standards of ethics as we work to fulfill our mission. We all share responsibility for ensuring that we conduct our transactions in ways that are ethical, honest, and reflect sound fiduciary practices. To accomplish this, it is important that all UO employees review, understand, and consistently practice the standards included in the following laws, rules, and policies including:
  • 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 "
    1. ORS Chapter 244, which codifies ethics and conflict of interest policies that you are required to follow as you conduct University of Oregon business. See the guide for public officials here.
    2. The Oregon University System (OUS) has a responsibility to prevent and detect fraud, waste, and abuse and to hold accountable those who engage in it. The OUS Fraud, Waste, and Abuse policy sets forth guidelines for reporting known or suspected fraud, waste, or abuse within any OUS institution.
    3. If you are aware of fraud, waste, or abuse occurring at the UO or within the OUS, matters can be reported to campus management, OUS Internal Audit Division, or OUS Financial Concerns Hotline. Additional information is also available on the UO Business Affairs and UO vice president for finance and administration webpages.
    4. The OUS information security policy and UO information security policy set forth your responsibilities relating to the security of electronic information systems and confidentiality of data. A more comprehensive listing of state laws and rules that guide our operations is available here.
    5. The UO will continue a similar focus on these important issues under our new governance structure. We will communicate any changes to reporting protocols after July 1. We are all responsible for understanding and complying with ORS 244, applicable government regulations and policies. We also have a responsibility to raise compliance and ethics concerns through established channels. I appreciate your commitment to integrity and honesty, as it is an essential element in maintaining an ethical and secure UO workplace environment for everyone.

    Statement on Final Exams

  • 1. In the week preceding final examination during fall, winter, and spring terms: No examination worth more than 20% of the final grade will be given, with the exception of make-up examinations. No final examinations will be given under any guise. No work that will be evaluated for grades/credit will be due unless it has been clearly specified on the class syllabus within the first two weeks of the term.
  • 2. Take-home examinations will be due no earlier than the day of the formally assigned final examination for the class in question.

    This action clarifies and extends earlier faculty legislation (1911 Faculty Assembly archives) prohibiting the giving of final examinations earlier than officially scheduled.

    In addition, you should be aware of the Faculty Advisory Council's statement on students with multiple exams:

    Examination schedules are listed each term in the Time Schedule. Students who are scheduled to take more than three examinations within one calendar day may take the additional examination(s) as makeup examination(s) later in the examination week. The instructor(s) of record for the course(s) beyond the third examination, counting in the order the examination(s) are scheduled, will arrange for (a) makeup examination(s).

    The following procedures were approved by the Undergraduate Council to address rare circumstances of competing exam times. Students with examination conflicts may contact the Office of Academic Advising for assistance.

    In the case of two examinations scheduled at the same time, the course with the largest enrollment must provide an alternate examination. For conflicts between regular courses and combined examinations, the combined examination course must provide the alternate examination. For combined examinations with conflicts, the largest combined enrollment course must provide the alternative examination.

    Questions and concerns regarding this policy should be directed first to the relevant instructor, then the department head, and finally the dean if necessary. You may also find reference to the policy on the Academic Affairs website. If additional input is needed, please contact

    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.
  • Discrimination and Discriminatory Harassment: Oregon law requires that all university employees with credible evidence that any form of prohibited discrimination by or against students, faculty members, staff members, or visitors to our campus is occurring have a duty to report that information. "Prohibited discrimination" includes: Discrimination on the basis of age, disability, national origin, race, marital status, religion, gender, gender identity, gender expression or sexual orientation; and Discriminatory harassment, including all forms of sexual harassment. Reports are to be made to the employee’s supervisor or to the Office of Affirmative Action and Equal Opportunity (OAAEO) at 541-346-3123; or via email to the Office of Affirmative Action and Equal Opportunity. Any UO supervisor who has been notified of credible evidence that prohibited discrimination is occurring has a duty to report that to the OAAEO. Penelope Daugherty, Director of OAAEO and Title IX Coordinator, 541-346-2971,, is the contact person for questions about the duty to report discrimination and discriminatory harassment.

    I am a student directed employee. Students experiencing any form of prohibited discrimination or harassment, including sex or gender based violence, may seek information on,,, or or contact the non-confidential Title IX office (541-346-8136), AAEO office (541-346-3123), or Dean of Students offices (541-346-3216), or call the 24-7 hotline 541-346-SAFE for help. I am also a mandatory reporter of child abuse.

    Any student who has experienced sexual assault, relationship violence, sex or gender-based bullying, stalking, and/or sexual harassment may seek resources and help at To get help by phone, a student can also call either the UO’s 24-hour hotline at 541-346-7244 [SAFE], or the non-confidential Title IX Coordinator at 541-346-8136. From the SAFE website, students may also connect to Callisto, a confidential, third-party reporting site that is not a part of the university. Students experiencing any other form of prohibited discrimination or harassment can find information at or or contact the non-confidential AAEO office at 541-346-3123 or the Dean of Students Office at 541-346-3216 for help. As UO policy has different reporting requirements based on the nature of the reported harassment or discrimination, additional information about reporting requirements for discrimination or harassment unrelated to sexual assault, relationship violence, sex or gender based bullying, stalking, and/or sexual harassment is available at Discrimination & Harassment. Specific details about confidentiality of information and reporting obligations of employees can be found at Mandatory Reporting of Child Abuse UO employees, including faculty, staff, and GEs, are mandatory reporters of child abuse. This statement is to advise you that your disclosure of information about child abuse to a UO employee may trigger the UO employee’s duty to report that information to the designated authorities. Please refer to the following links for detailed information about mandatory reporting: Mandatory Reporting of Child Abuse and Neglect.

    The instructor of this class, as a Student Directed Employee, will direct students who disclose sexual harassment or sexual violence to resources that can help and will only report the information shared to the university administration when the student requests that the information be reported (unless someone is in imminent risk of serious harm or a minor). The instructor of this class is required to report all other forms of prohibited discrimination or harassment to the university administration.

    Safe Ride is an assault prevention shuttle that works to provide free, inclusive, and accessible alternatives to traveling alone at night for UO students, faculty, and staff. We are a schedule-ahead service and riders can (1) call once we open to schedule a ride with a dispatcher or (2) leave a voicemail on the day of their ride request. We do not call riders ahead of time to confirm due to capacity constraints, but riders are always welcome to call us to double-check that their ride was scheduled. We are a feminist, ‘for-the-students/by-the-students’ organization and operate out of the Women’s Center in EMU 12F. Operating hours: Spring term Summer term Fall/Winter term Sunday - Thursday | 7p - midnight Friday + Saturday | 7p - 2a Sunday - Thursday | 9p - midnight Friday + Saturday | 9p - 2a Sunday - Thursday | 6p - midnight Friday + Saturday | 6p - 2a Policy and rules: 1. We are a schedule-ahead service, we do not call ahead, and we can only wait for riders for 5 minutes at their pick-up time and location. 2. We only give rides to groups of 3 or less to prioritize groups that are at higher risk. 3. We are a free service and do not accept tips.

    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  1 April 2018 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