M 383C/CSE 386C Methods of Applied Mathematics I.

M 383C, Unique #54000     and     CSE 386C, Unique #64380
M 383C-WB, Unique #53999     and     CSE 386C-WB, Unique #64379
Fall 2020     


Prof. Todd Arbogast
Office: PMA (RLM) 11.162, Phone: 512-471-0166
Office: POB 5.334, Phone: 512-475-8628
E-Mail: arbogast@oden.utexas.edu
Office hours via Zoom: TTh 10:30-11:30.

Teaching Assistants

Mr. Frimpong A. Baidoo
E-Mail: fabaidoo@utexas.edu
Office hours via Zoom: TBD
Duties: Hold informal review and homework discussion sessions (twice per week by zoom). Hold office hours and monitor the Piazza class discussion page.

Mr. Andrew Ma
E-Mail: andygma567@gmail.com
Duties: Grade homework papers. Please use e-mail to discuss questions regarding the grading of specific assignments.


A bound set of lecturer-prepared notes, Functional Analysis for the Applied Mathematician, by T. Arbogast and J. L. Bona (2020 version) is available for purchase from the UT Copy Center (Document Solutions). It may be available directly from the GSB location.

Class Web Site

We will use the University's Canvas web site. You can access Canvas from my.utexas.edu. The Longhorns Online page has helpful general information, and Protect Texas Together has helpful COVID-19 information.

Hybrid/Blended Meetings

Official class times are MWF 11:00 a.m. - 12:00 noon in PMA (RLM) 6.104. Students may participate in person or online (sychronously and/or asychronously) as health and safety dictate. You are strongly encouraged to participate synchronously if possible.

In person. The classroom is designed to accommodate all enrolled students with appropriate social distancing. We will meet on select class days throughout the semester and work online otherwise (for example, there will be no in person meetings starting November 30 after the Thanksgiving break). Meeting attendees must follow the Rules for Safe Class Participation (including wearing masks), detailed below.

Online sychronous. Class lectures will be live-streamed via Zoom (available in Canvas). While on Zoom, please mute your microphone unless you want to speak. If you have a question, you can switch your audio on and ask it, and then mute your audio again afterwards. (You can also use the chat feature of Zoom, but your Instructor might not see your comment in a timely manner.) If your bandwidth is limited, you can turn your video off.

Online asynchronous. Class lectures will be recorded (possibly using the Lectures Online recording system) and accessible later in Canvas (or possibly on UT-box) for a reasonable period of time and as disk space allows.

Class recordings are reserved only for students in this class for educational purposes and are protected under FERPA. The recordings should not be shared outside the class in any form. Violation of this restriction by a student could lead to Student Misconduct proceedings.

Class Discussion Sessions and Piazza

A class discussion will be held twice per week via Zoom to discuss the material and homework problems.

In addition to the regularly scheduled office hours via Zoom, we will use the asynchronous class discussion tool (or message board) Piazza, available in Canvas. It is intended for discussion on the homework problems, but it can be used in many other ways (such as to set up virtual discussion groups). You can post your question there, and anyone in the class can offer a solution (which can be edited if someone else thinks they can be of additional help). The Instructors also have space for an answer.

Homework, Exams, and Grades

Homework will be assigned weekly in Canvas. Students are encouraged to work in virtual groups; however, each student must write up his or her own work, which will be uploaded to Canvas.

Exams will use the open book, open notes format (i.e., you may use the lecturer prepared notes, i.e., the textbook, and your own notes on the exams). Exams will be administered online using Canvas. Two mid-term exams will be given in approximately weeks six and eleven (the weeks of September 28 and November 2), and these exams will be scheduled outside of the class meeting time (to allow more time). The final exam will be comprehensive and given Friday, December 11, 2:00-5:00 p.m.

Scores will be recorded in Canvas. The final grade (not recorded in canvas) will use the plus/minus system and be based on the homework and the three exams, with somewhat greater emphasis on the final exam.

Course Description

This is the first semester of a course on methods of applied mathematics. It is open to mathematics, science, engineering, and finance students (among others). It is suitable to prepare graduate students for the Applied Mathematics Preliminary Exam in mathematics and the Area A Preliminary Exam in CSEM. The first semester is an introduction to functional analysis.

Semester I.

  1. Preliminaries (0 weeks)
    1. Elementary Topology
    2. Lebesgue Measure and Integration
    3. Complex Contour Integration
  2. Normed Linear Spaces and Banach Spaces (6 weeks)
    1. Basic Concepts and Definitions
    2. Some Important Examples
    3. Hahn-Banach Theorems
    4. Applications of Hahn-Banach
    5. The Open Mapping Theorem
    6. Uniform Boundedness Principle
    7. The Embedding of X into its Double Dual X**
    8. Compactness and Weak Convergence in a NLS
    9. The Dual of an Operator
  3. Hilbert Spaces (2 weeks)
    1. Basic Properties of Inner-Products
    2. Best Approximation and Orthogonal Projections
    3. The Dual Space
    4. Orthonormal Subsets
    5. Weak Convergence in a Hilbert Space
  4. Spectral Theory and Compact Operators (4 weeks)
    1. Definitions of the Resolvent and Spectrum
    2. Basic Spectral Theory in Banach Spaces
    3. Compact Operators on a Banach Space
    4. Bounded Self-Adjoint Linear Operators on a Hilbert Space
    5. Compact Self-Adjoint Operators on a Hilbert Space
    6. The Ascoli-Arzela Theorem
    7. Sturm-Liouville Theory
  5. Distributions (2 weeks)
    1. The Notion of Generalized Functions
    2. Test Functions
    3. Distributions
    4. Operations with Distributions
    5. Convergence and Approximations to the Identity
    6. Some Applications to Linear Differential Equations
    7. Local Structure of D'

Semester II. (Generally, the following topics are covered.)

  1. The Fourier Transform
  2. Sobolev Spaces
  3. Boundary Value Problems
  4. Differential Calculus in Banach Spaces
  5. The Calculus of Variations

Rules for Safe Class Participation

We will all need to make some adjustments in order to benefit from in-person classroom interactions in a safe and healthy manner. Our best protections against spreading COVID-19 on campus are masks (defined as cloth face coverings) and staying home if you are showing symptoms. Therefore, for the benefit of everyone, this means that all students are required to follow these important rules.

Those who are returning to campus are asked to go to this site and sign The Commitment to Protect Texas Together. This is a voluntary code of conduct and a statement of shared purpose.

If a student is not wearing a cloth face-covering properly in the classroom (or any UT building), that student must leave the classroom (and building). If the student refuses to wear a cloth face covering, class will be dismissed for the remainder of the period, and the student will be subject to disciplinary action as set forth in the university's Institutional Rules/General Conduct 11-404(a)(3). Students who have a condition that precludes the wearing of a cloth face covering must follow the procedures for obtaining an accommodation working with Services for Students with Disabilities.

COVID-19 Monitoring and Reporting

University Health Services is available for student health services. Click for COVID-19 specific information.

To help keep everyone at UT and in our community safe, it is critical that students report COVID-19 symptoms and testing, regardless of test results, as soon as possible to University Health Services (faculty and staff report to the HealthPoint Occupational Health Program (OHP)).

Sharing of Course Materials is Prohibited

No materials used in this class, including, but not limited to, lecture hand-outs, videos, assessments (quizzes, exams, papers, projects, homework assignments), in-class materials, review sheets, and additional problem sets, may be shared online or with anyone outside of the class unless you have my explicit, written permission. Unauthorized sharing of materials promotes cheating. It is a violation of the University's Student Honor Code and an act of academic dishonesty. I am well aware of the sites used for sharing materials, and any materials found online that are associated with you, or any suspected unauthorized sharing of materials, will be reported to Student Conduct and Academic Integrity in the Office of the Dean of Students. These reports can result in sanctions, including failure in the course.

The University of Texas at Austin Student Honor Code

"As a student of The University of Texas at Austin, I shall abide by the core values of the University and uphold academic integrity."

The University of Texas at Austin Code of Conduct

The core values of The University of Texas at Austin are learning, discovery, freedom, leadership, individual opportunity, and responsibility. Each member of the university is expected to uphold these values through integrity, honesty, trust, fairness, and respect toward peers and community.

Students with Disabilities

Students with disabilities may request appropriate academic accommodations from the Division of Diversity and Community Engagement (DDCE), Services for Students with Disabilities (SSD) at http://ddce.utexas.edu/disability.

Religious Holidays

Appropriate academic accommodation for major religious holidays is provided upon request.

Emergency Classroom Evacuation

Occupants of University of Texas buildings are required to evacuate when a fire alarm is activated. Alarm activation or announcement requires exiting and assembling outside. Familiarize yourself with all exit doors of each classroom and building you may occupy. Remember that the nearest exit door may not be the one you used when entering the building. Do not re-enter a building unless given instructions by the Austin Fire Department, the University Police Department, or the Fire Prevention Services office.

Counselling and Mental Health Services

Available at the Counseling and Mental Health Center, Student Services Building (SSB), 5th floor, M-F 8:00 a.m. to 5:00 p.m., phone 512-471-3515, web site www.cmhc.utexas.edu. Crisis Line is always available 24/7 at 512-471-2255.

If you have concerns about a student, faculty, or staff member in the UT-Austin community, contact the 24-hour Behavior Concerns Advice Line at 512-232-5050.