Peter B Gilkey

202 Deady Hall,1-541-346-4717 (office phone) 1-541-346-0987 (fax) email: gilkey@uoregon.edu

Mathematics Department, University of Oregon, Eugene Oregon 97403 USA

- 25% on the homework
- 25% on the mid term Wednesday 29 April 2015
- 50% on the Final Exam 10:15 Friday 12 June 2015.
- 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. Any student getting at least a B on the final will receive at least a C- in the course; no student can pass the course unless they receive a grade of D or better on the final exam. 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.

Teaching Associate: Ekaterina Puffini. Academic Calendar

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.

- Week 1 (30 March - 3 Apr 2015): The orthogonal group, the oscillation, the derivative as the best linear approximation.
- Week 2 (6 Apr - 10 Apr 2015): The inverse function theorem.
- Week 3 (13 Apr - 17 Apr 2015): Start on integration theory.
- Week 4 (20 Apr - 24 Apr 2015): The Riemann integral, sets of measure and content zero. Integrable functions.
- Week 5 (27 Apr - 1 May 2015): Read 56-73. Jordan measurability, Fubini's theorem. Exam Wednesday 29 April 2015
- Week 6 (4 May - 8 May 2015): Partitions of unity, improper integrals, change of variables.
- Week 7 (11 May - 15 May 2015): Finish up change of variables theorem.
- Week 8 (18 May - 22 May 2015): Discussion of differential forms.
- Week 9 (27 May - 29 May 2014): Green's, Gauss's, and Stokes Theorem. (25 May 2014 is Memorial Day)
- Week 10 (01 Jun - 05 Jun 2014): Discuss fundamental theorem of algebra, Brauer Fixed point theorem, Can't comb hair on billiard ball theorem.

- 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

**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".

- (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 prerequisite.
- (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.

- 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 appropriate.
- 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 when prompted.
- 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 through calculations.

**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 of instructors.

- 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

- 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.
- 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.
- 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.
- 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.
- 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.

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 CouncilÕs 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 srviceprovost@uoregon.edu.

Web page spun on 11 October 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 Email:peter.gilkey.cc.67@aya.yale.edu of Deady Spider Enterprises |