Generic Syllabus for Math 4/513
Course Title: Introduction to analysis I Math 413 and 513
Text: Rudin (Principles of mathematical analysis) 3rd Edition
Tentative Homework Assignments. The homework will trail the lectures slightly
to avoid having material covered on a Friday to be due the subsequent Monday.
- Week 1:
Read Chapter I of Rudin. The real and complex number systems.
Chapter 1: # 1, 2, 4, 5, 8, 9, 13, 15, 17, 18.
- Week 2: Read Cha II of Rudin. Set Theory and Basic Topology
Chapter 2: # 2, 3, 4, 5, 6 ,8, 9, 10, 11, 16. .
- Week 3: Start reading Chapter 3. Numerical sequences and series.
Chapter 2: #18, 19, 20, 21,
22, 23, 24, 25, 26.
- Week 4: Continue reading Chapter 3.
Chapter 3: #1, 2, 3, 4, 5.
- Week 5: Continue reading Chapter 3.
Chapter 3: # 6, 7, 8, 9, 10, 11.
- Week 6: Finish reading Chapter 3. Exam
Chapter 3: # 14, 16, 17, 19.
- Week 7: Begin reading Chapter 4. Continuity
Chapter 4: 1, 2, 3, 4, 6, 7, 8, 9, 10, 11
- Week 8: Finish reading Chapter 4. Start reading Chapter 5 Differentiation
Chapter 4: 12, 14, 15, 16, 18.
Chapter 5: 1, 2, 3, 4, 5 ,6
- Week 9: Continue reading Chapter 5 Differentiation. Thanksgiving holiday
Chapter 5: 9, 15, 16, 26.
- Week 10: Finish reading Chapter 5. Review the material of the course and put things
in context.
Course Goals
This course deals with the differential calculus. We begin Week 1 in Chapter I
with a discussion of the real and complex number systems to lay a good foundation for later material.
In Week 2, we turn to Chapter II and discuss set theory (finite, countable, and uncountable sets).
We introduce metric spaces and their basic topology and discuss compactness. Chapter 3 of
Rudin takes several weeks. We discuss numerical sequences and series, convergent
sequences, Cauchy sequences, upper and lower limits, series with non-negative terms,
the root and ratio tests, power series, summation, absolute convergence, addition and multiplication
of series, rearrangements, and so forth. Chapter 4 of Rudin again takes several weeks. We discuss
continuity, the limit of a function, continuous functions, continuity and compactness, continuity
and connectedness, infinite limits and limits at infinity. We complete the course with Chapter 5
and deal with differentiation, the mean value theorem, the continuity of derivatives, Taylor's theorem.
The final week of the course is left empty; it is likely that one falls behind just a bit on this syllabus
and time is left free to one does not have to rush through the material but rather can in fact spend
the extra lecture necessary to deal with particularly thorny issues. It is not necessary to delve too
deeply into multi-variable differentiation in Chapter 5 as this will be dealt with in Math 4/515.
Mathematics Department
Undergraduate Grading Standards
November 2011
There are two important issues that this grading policy recognizes.
-
(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.
Rubric for applied courses:
- 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.
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:
- 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.
Academic dishonesty
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 www.libweb.uoregon.edu/guides/plagiarism/students.