Math 414-514, Spring 2021 Syllabus

  1. Course Information
    1. Instructor
    2. Section
  2. Course Learning Outcomes
  3. Textbook
    1. Required textbook
    2. Additional texts
    3. Topics
  4. Grading
    1. Components of course grade
    2. Term Paper
    3. Midterm and final exams
    4. Homework
      1. Turning in written assignments
      2. Homework formatting
    5. Attendance
  5. Help
    1. Allowed resources
    2. Helpful Websites
    3. A note on collaboration
    4. University Resources
  6. Important Dates
  7. Other

Course Information

Course Number:
Math 414, Math 514
Course Title:
Real Analysis
Course Description:
Covers Riemann integration, the fundamental theorem of calculus, sequences and series of functions, multivariable calculus. Additional topics may include Fourier series, analysis of metric spaces, the Baire property, and advanced topology of Euclidean space.
Prerequisites:
MATH 275 and MATH 314.

Instructor

Instructor:
Zach Teitler [he/him/his]
Email:
zteitler@boisestate.edu
Website:
https://zteitler.github.io
Office:
MB 233A
Office Phone:
208-426-1086

Section

Section Number:
001
Meeting Times:
TuTh 10:30-11:45
Meeting Remotely:
We will meet remotely using Zoom. Zoom sessions may be recorded for students who are not able to attend.
Zoom meeting ID:
TBA

Course Learning Outcomes

By the end of this course, students will be able to:

  1. Demonstrate familiarity with introductory concepts, definitions, and theorems in real analysis, including:
    1. the Riemann integral and fundamental theorem of calculus;
    2. sequences and series of functions, including power series, as well as absolute and uniform convergence;
    3. analysis of metric spaces;
    4. aspects of multivariable calculus including the Jacobian and derivatives of functions between Euclidean spaces and the implicit and inverse theorems.

  2. Read and write rigorous proofs.

  3. Write mathematical text with high quality and good style, using appropriate technology.

Textbook

Required textbook

Basic Analysis: Introduction to Real Analysis, by Jiří Lebl, https://www.jirka.org/ra/

This book is available as a free PDF, a free web version, and on paper (not free, but inexpensive). There are two volumes: volume 1 (chapters 1–7) and volume 2 (chapters 8–11). We will mostly use volume 1, plus a small part of volume 2.

Additional texts

You do not need any textbooks besides the required one. But if you want to supplement with an additional text, I recommend:

Elementary Real Analysis, by Thomson, Bruckner, Bruckner, http://classicalrealanalysis.info/Elementary-Real-Analysis.php

You are welcome to use any book you like, including other free books available online, from the library, or from previous classes you may have taken.

Topics

We will cover topics from the following chapters:

Topic Lebl Chapter TBB Chapter
Continuous functions, Derivatives, Sequences 3.1-4, 4.1-2, 2.3  
Riemann integral 5 8
Sequences and series of functions 2.6, 4.3, 6 9
Metric spaces 7 13
Differentiation under the integral 9.1  
Partial derivatives and Jacobian 8.3, 4.4, 8.5 12

If time remains at the end of the course, we may be able to cover additional material such as analysis of Fourier series (Lebl: 11, TBB: 10).

Grading

Components of course grade

Course grades will be based on a holistic overall evaluation of the quality of each student’s work. The approximate weight of each component of the work is as follows.

Category Approximate weight
Term Paper 35%
Midterm and final exams 35%
Homework 30%

Term Paper

Each student will write a term paper (6–15 pages) about a topic of the student’s choice within real analysis.

  1. An initial topic proposal (1–2 pages) is due by the end of week 4.
  2. An outline is due by the end of week 6.
  3. A first draft is due by the end of week 9.
  4. A second draft is due by the end of week 12.
  5. The term paper is due by the end of week 15.

Midterm and final exams

A midterm exam in the 8th week of the semester, and the final exam, will be individual oral exams. The exams are intended to gauge mastery of course material, but also reflection on the course and its place in a larger context.

Homework

You are encouraged to work collaboratively on homework but you must turn in your own solutions.

Turning in written assignments

Homework submissions and grading will be paperless. You will turn in your homework by uploading PDFs to BlackBoard and/or GradeScope.

Homework formatting

Homework must be typed in LaTeX. LaTeX tutorials are available online, e.g., https://www.latex-tutorial.com and https://www.gnu.org/software/teximpatient/. You may wish to use a free online LaTeX system such as https://overleaf.com. (Overleaf includes a LaTeX tutorial.)

Use a new page (\newpage) for each problem. State which question you are answering (textbook section and exercise number) and the actual question. Then, start your answer in a new paragraph.

For legibility, use the 12pt option (\documentclass[12pt]{amsart}) and \linespread{2.4}. If you use figures, I recommend learning to use TikZ to generate high-quality figures within LaTeX. Alternatively you may use figures/plots generated in other programs such as Sage, Mathematica, Maple, or Inkscape, saved to PDF, and included in your document with commands like \includegraphics. It’s also fine to include hand-drawn figures that you scanned or photoed.

Attendance

Class attendance is required during the first week of the semester. After the first week, class attendance will be optional, not required, and not graded. On the other hand it is highly recommended. Attending class has many benefits such as:

  • Opportunities to ask questions and get immediate feedback (and then, ask followup questions)

  • Establish relationships with other students and with the instructor (might be helpful in case you decide to look for a study group, or research opportunities or a letter of recommendation)

Help

Allowed resources

In this 400/500 level class, you are encouraged to use the full array of resources at your disposal to learn the material “by any means necessary.” For homework assignments, you are encouraged (in fact, expected) to collaborate with your classmates and to ask me questions. Other resources (books, online sources, people outside the class) are highly recommended for clarifying class topics or for enrichment (but not for getting solutions to problems).

You are allowed to use things that you learn from a book, online source, or person outside the class that help you and your classmates to find your own solution for a problem. However, if you read a full solution, so that there’s little or nothing left for you and your classmates to figure out, then you may not turn in that solution for credit.

Helpful Websites

The Mathematics Stack Exchange (https://math.stackexchange.com) is a very useful question-and-answer site for undergraduate/graduate level mathematics. You are welcome to browse the Mathematics Stack Exchange and even post questions there. Hopefully it will help you learn and understand the material! However please remember that if you use that site to get a solution to a problem, then you can’t turn that solution in for credit.

Other helpful websites for basic information include Wikipedia and MathWorld (http://mathworld.wolfram.com).

A note on collaboration

Solving mathematical problems has three parts:

  1. The discovery phase.

    This is the time you spent trying to figure out how to solve the problems, and it often takes most of the time. You are welcome and encouraged to collaborate with other students in this phase. Collaboration is a healthy practice, and this is how mathematics is done in real life.

    This phase starts with working to understand what a problem or question is asking for. That might include reviewing material from previous textbook sections.

  2. The write-up phase.

    This consists of writing your solutions once you have an idea of how the problem can be solved. You should do this entirely by yourself. Be alone when you write your solutions. If you collaborate on this part, or you copy part of your solutions from somebody else, or you have notes written by somebody else in front of you when you write your solutions, you are hurting yourself by depriving yourself of an opportunity to learn and practice.

    If you need help in the write-up phase, talk to me! I can help you.

  3. The editing phase.

    This consists of editing and revising your write-up for clarity, organization, and presentation. At this stage it can be very helpful to get feedback and suggestions from other students.

University Resources

Boise State University’s The Basics web page has links to many forms of support, ranging from academic resources to family, living, and food resources.

The Graduate College has many resources for graduate students such as GradWell, the Graduate Student Success Center, and Graduate Writing Consultations, as wells as forms, deadlines, and graduation information for graduate students.

Boise State University’s Writing Center may be helpful.

You may reach out to me at any time if there’s anything I can help with or if there’s anything you think I should know.

Important Dates

Monday 1/11 First day of classes
Monday 1/18 Dr. Martin Luther King Jr./Idaho Human Rights Day. No classes.
Monday 1/25 Last day to register/add or to drop without a W
Monday 2/15 Presidents’ Day. No classes.
Friday 3/19 Last day to drop with a W or completely withdraw
  4/12-16 Spring Break. No classes.
Friday 4/30 Last day of instruction for regular classes
Tuesday 5/11 Grades due. (You will be able to see your grade by this date.)

Other

Respect for Diversity:
Students from all backgrounds and with all perspectives are welcome in this course. It is my intent that all students be well served by this course, that students’s learning needs be addressed both in and out of class, and that the diversity that students bring to this class be viewed as a resource, strength, and benefit. It is my intent to maintain a classroom atmosphere that is welcoming and respectful of diversity: gender, sexuality, disability, age, socioeconomic status, ethnicity, race, and culture. Your suggestions are encouraged and appreciated. Please let me know ways to improve the effectiveness of the course for you personally or for other students or student groups.
ADA Policy Statement:
Students with disabilities needing accommodations to fully participate in this class should contact the EAC. All accommodations must be approved through the EAC prior to being implemented. To learn more about the accommodation process, visit the EAC’s website at https://www.boisestate.edu/eac/new-students/.
Email:
In accordance with Boise State University Policy #2280, it is expected that you will receive and read emails sent to your boisestate.edu email address.
Communication:
Additional information and updates may be announced in class, sent by email, and/or posted on BlackBoard (https://blackboard.boisestate.edu/).
Academic Integrity:
Getting answers to homework or exam problems from unauthorized sources is a very serious form of academic misconduct.
Behavioral Expectations:
Every student has the right to a respectful learning environment. In order to provide this right to all students, students must take individual responsibility to conduct themselves in a mature and appropriate manner and will be held accountable for their behavior in accordance with Boise State University Policy #2050.