Peter B Gilkey
202 Deady Hall,1-541-346-4717 (office phone) 1-541-346-0987
(fax) email: email@example.com
University of Oregon,
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
Grade: Will be based
Note: No class Memorial Day Monday May 28 2014.
25% on the homework
25% on the mid term Wednesday 30 April 2014
50% on the Final
Exam 10:15 Thursday, June, 12.
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.
Notes available in the class:
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,
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
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
Week 4 (21 Apr - 25 Apr 2014): Read 56-73.
The Riemann integral, sets of measure and content zero.
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
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,
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,
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.
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.''
Week 8 (19 May - 23 May 2014): Discussion of differential forms. See
supplementary notes Green's, Gauss's, Stokes
Week 9 (27 May - 30 May 2014): Finish discussion of Green's, Gauss's, and Stokes
(26 May 2014 is Memorial Day)
Week 10 (02 Jun - 06 Jun 2014): Discuss fundamental theorem of algebra,
Brauer Fixed point theorem, Can't comb hair on billiard ball theorem.
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
- Improper Integrals
- Change of variable Theorem
- Green's, Gauss's, Stokes Theorem
- 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.
- 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.
- 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.
- 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
Undergraduate Grading Standards
There are two important issues that this grading policy recognizes.
Rubric for applied courses:
(1) Mathematics is hierarchical. A student who is given a grade of C or
higher in a course must have mastery of that material that allows
the possibility of succeeding in courses for which that course is a
- (2) Some mathematics courses are primarily concerned with techniques
and applications. In such courses student success is measured by the
student's ability to model , successfully apply the relevant technique,
and bring the calculation to a correct conclusion. The department's
100-level courses and most calculus courses are examples in this category
although these are not the only examples. Other courses such as this one are
primarily concerned with theoretical structures and proof. In such courses
student success is measured by the student's ability to apply the theorems
and definitions in the subject, and to create proofs on his or her own using
the models and ideas taught during the course.
Many courses are partly hybrids incorporating both techniques
and applications, and some element of theory. Some lean more toward applications, others more toward theory.
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.
- A: Consistently chooses appropriate models, uses correct techniques,
and carries calculations through to a correct answer. Able to estimate error
when appropriate, and able to recognize conditions needed to apply models as
- B: Usually chooses appropriate models and uses correct
techniques, and makes few calculational errors. Able to estimate error when
prompted, and able to recognize conditions needed to apply models
- C: Makes calculations correctly or substantially correctly, but requires
guidance on choosing models and technique. Able to estimate error
when prompted and able to recognize conditions needed to apply
models when prompted.
- D: Makes calculations correctly or substantially correctly, but unable
to do modeling.
- F: Can neither choose appropriate models, or techniques, nor carry
Rubric for pure courses:
- A: Applies the important theorems from the course. Constructs
counterexamples when hypotheses are weakened. Constructs complete and coherent
proofs using the definitions, ideas and theorems from the course. Applies ideas
from the course to construct proofs that the student has not seen before.
- B: Applies the important theorems from the course. Constructs
counterexamples when hypotheses are weakened. Constructs complete and coherent
proofs using the definitions, ideas and theorems from the course.
- C: Applies the important theorems from the course when the application is direct. Constructs simple proofs using the de nitions when
there are very few steps between the de nitions and the conclusions.
Explains most important counterexamples.
- D: Can do some single step proofs and explain some counterexamples.
- F: Unable to do even single step proofs or correctly use de nitions.
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
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.
- A+ Truly outstanding work
- A Good Ph.D. or M.S./M.A. level work
- A- Clearly Ph.D. level work, but below average. Good at M.S./M.A. level
- B+ Work which is at the lower margin of acceptable Ph.D. level work,
but quite satisfactory at the M.S./M.A. level
- B Substandard at the Ph.D. level but satisfactory at the M.S./M.A. level
- B- Barely passing at the graduate level
- C+ or below. Unsatisfactory at the graduate level
Academic Misconduct: The University Student Conduct Code (available at
conduct.uoregon.edu) 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
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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.
The university is committed to providing an environment for work and learning that is free from unlawful discrimination, including sexual harassment (which includes sexual assault, intimate partner or dating violence, and gender-based stalking or bullying). The safety of persons who participate in university programs and activities is critical.
In support of these priorities and consistent with legal obligations, university employees have a responsibility to report instances of certain inappropriate conduct as outlined in this letter. The intent of this letter is to communicate these reporting responsibilities that apply to all UO employees and to make clear how to report and to whom.
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, firstname.lastname@example.org, is the contact person for questions about the duty to report discrimination and discriminatory harassment.
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;
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, email@example.com, 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.