Peter B Gilkey
202 Deady Hall,1-541-346-4717 (office phone) 1-541-346-0987
(fax) email: firstname.lastname@example.org
University of Oregon,
Oregon 97403 USA
Math 4/515 Introductory Analysis III Spring 2013 34402 (M415) and 34410 (M515)
Syllabus Version 1
Calculus on Manifolds
MWF 1100-1150 (Deady 209) + F 1500-1550 (Peterson 103)
Office hours Monday, Wednesday, Friday 0900-0950
Text: Spivak: Calculus on manifolds (paperback). (Benjamin/Cummings Publishing
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 27 2013.
25% on the homework
25% on the mid term Wednesday 1 May 2013
50% on the Final
Exam 10:15 Thursday 13 June 2013.
Teaching Associate: Ekaterina Puffini. Academic Calendar
Here are tentative reading and homework assignments.
Notes available in the class:
Week 1 (1 Apr - 5 Apr 2013): Read 1-34. Do 1.7, 1.10, 1.22, 1.30, 2.4, 2.5,
Week 2 (8 Apr - 12 Apr 2013): Read 34-45. Do 2.12, 2.13., 2.21, 2.22, 2.23,
2.24, 2-25, 2-26. Also
Week 3 (15 Apr - 19 Apr 2013): Read 46-56. Do 2.29, 2.30, 2.31, 2.32, 2.35,
2.36, 2.37 [not part b], 2.38, 2.39
Week 4 (22 Apr - 28 Apr 2013): Read 56-73. Do 3.1-3.10.
Week 5 (29 Apr - 3 May 2013): Read 56-73. Do 1.18, 3.11, 3.12, 3.14, 3.15, 3.16,
3.17, 3.18, 3.19. Exam Wednesday May 01 2013
Week 6 (6 May - 10 May 2013): Do 3.13, 3-20, 3.21, 3.22, 3.23, 3.26, 3.28,
Week 7 (13 May - 17 May 2013): 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 (20 May - 24 May 2013): Assignment #8
Week 9 (25 May - 30 May 2013): Assignment #9
(27 May 2013 is Memorial Day)
Week 10 (03 Jun - 07 Jun 2013): TBA
Week 11 (10 June - 14 June 2013) Final
Exam Thursday 13 June 2013 10:15. Owing to faculty legislation, final
exams may not be given early under any circumstances
Undergraduate Grading Standards
There are two important issues that this grading policy recognizes.
- 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.
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
Course Goals This material was added 24 May 2013 and was not part of the original syllabus.
Still, as it perhaps will assist the students in their review, we felt it worth adding.
This is a rigorous introduction to the material necessary to understand multi-variable
calculus and calculus on manifolds.
- The differential calculus. The derivative is the "best linear approximation" to a map from Rm to Rn. This
will be discussed in connection with directional derivatives and the Jacobian matrix of partial derivatives. The
chain rule, inverse function theorem, and implicit function theorem will be proved from this point of view. This
typically lasts the first 3 weeks of the course.
- The integral calculus. We use the method of Riemann sums (upper sum, lower sum, upper integral, lower integral).
The notion of content zero and measure zero are introduced and a bounded function on a rectangle is shown to be
Riemann integrable if and only if the set of discontinuities has measure 0; the oscillation function plays a central role here.
Fubini's theorem is established. Partitions of unity and mesa functions are defined and the improper integral given in terms
of these concepts. The change of variables theorem is proved - this reprises the material of the course as the chain rule,
Fubini's theorem, and improper integrals all play a central role. This typically lasts the next 4 weeks of the course. Course
notes have been provided for improper integrals and for the change of variables theorem as the treatment in Spivak needs
just a bit of supplementing.
- "Integration by parts". In analysis, "Integration by parts" is often the terminology used to describe Green's, Gauss's, and
Stoke's theorem and there is a lot of truth to this.
The final 3 weeks of the course is devoted to the generalized Stoke's theorem.
At this stage, we leave the treatment in Spivak and turn to the course notes which are provided as Spivak is a bit too abstract.
We introduce the calculus of differential forms and prove the generalized Stoke's theorem. We then show that Green's, Gauss's,
and the classical Stoke's theorem of Math 282 (vector calculus) are all special cases of a single general result. And, in fact,
once the necessary notation is set up, the proof again reprises the material of the course (change of variables theorem,
mixed partials commute, chain rule, etc). If time permits, we will spend the last few lectures of the course discussing some
wonderful applications - the "you can't comb the hair on a billiard ball" theorem,
the fundamental theorem of algebra (every complex polynomial has at least one root), and the Brauer fixed point formula
(any continuous map from the disk to itself has a fixed point) are typical exemplars. The generalized Stokes theorem is in
many respects a segway to both differential geometry and algebraic topology and it is worth mentioning this briefly, but only
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.