MATH 2XX3

(Winter 2020)

Syllabus & Policies

Lecture Schedule

As of March 17: Official Plans for our Course

Instructor: Dr. Madnick (madnickj [at] mcmaster [dot] ca)

Office: Hamilton Hall 319

Teaching Assistant: Uyen Le (leu [at] mcmaster [dot] ca)

Video Lectures & Tutorials:

https://www.youtube.com/channel/UC5MpCyyJhD2GIry3QF5g5cQ/videos

Virtual Office Hours: Platform is Zoom

Assignments

Assignment 1. (Due: Fri Jan 24) Solutions.
Assignment 2. (Due: Fri Feb 7) Solutions.
Assignment 3. (Due: Fri Feb 28) Solutions.
Assignment 4. (Due: Fri Mar 13) Solutions.
Assignment 5. (Due: Thu Apr 2) Solutions.

Assignment 5: Submit the assignment through Crowdmark. The deadline is: Thu Apr 2 at 9:00 pm (eastern time).

Tutorial Problems

Set 1 (Jan 13-15) Solutions.
Set 2 (Jan 20-22) Solutions.
Set 3 (Jan 27-29) Solutions.
Set 4 (Feb 3-5) Solutions.
Set 5 (Feb 10-12) Solutions.

Set 6 (Feb 24-26) Solutions.
Set 7 (Mar 2-4) Solutions.
Set 8 (Mar 9-11) Solutions.
Set 9 (Mar 16-18) Solutions.
Set 10 (Mar 23-25) Solutions.

Lecture Notes

Unit I: Foundations
- Chapter 1: Logic, Sets, and Functions
- Chapter 2: Subsets of Euclidean Space
- Interlude A: Review of Single-Variable Calculus
Unit II: Multivariable Functions
- Chapter 3: Limits and Continuity
- Interlude B: Vector Spaces & Linear Transformations
- Chapter 4: Differentiation
- Interlude C: Diagonalization & Quadratic Forms
- Chapter 5: Approximation and Optimization
- Chapter 6: The Inverse and Implicit Function Theorems
Unit III: Fourier Series
- Interlude D: Pointwise & Uniform Convergence
- Interlude E: Normed Spaces & Inner Product Spaces
- Chapter 7: Fourier Series

"Don't just read it; fight it! Ask your own question, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? ... Where does the proof use the hypothesis?" -- Paul Halmos

Resources

Basic Proof Strategies
Calculus 1 Exercises (These are problems that I assign to my students when I teach Calculus 1. You should be able to solve all of these.)

For Fun

A nice article about the Weierstrass function and its impact on mathematics
An animation of a Fourier series
A nice discussion of the Isoperimetric Problem using Parseval's Formula (Sec 7.5)

Midterm 1 Information.

Date: Thu Feb 13 (evening)
Seating 1: 6:30 pm - 7:55 pm (85 minutes) (last name: A - M)
Seating 2: 8:10 pm - 9:35 pm (85 minutes) (last name: N - Z)
Location: MDCL 1102

Midterm Conflicts / Alternate Writes: If you have an academic conflict with Midterm 1, you can sign up for an alternate write here: https://www.childsmath.ca/childsa/forms/main_login.php

Alternate Write 1: Thu Feb 13. 4:30 pm - 5:55 pm (85 minutes). Location: HH 207
Alternate Write 2: Fri Feb 14. 8:55 am - 10:20 am (85 minutes). Location: BSB B156

Content: Chapters 1, 2, 3, and Sections 4.1-4.2. This material corresponds to Assignments #1 and #2 as well as Tutorial Sets 1 -- 4. (Note: Although Interludes A and B are not the focus of the test, they will be assumed as background knowledge.)

Format: Midterm 1 will have four problems.

#1: Complete the definitions.
#2: Analyze a function. (i.e.: Given a function, prove that it is continuous / discontinuous / differentiable / not differentiable / class C1 / not of class C1.)
#3: Analyze a set. (i.e.: Given a set, prove that it is open / not open / closed / not closed.)
#4: Five True/False + One Multiple Choice.

Advice:

Know definitions by heart: this will help you complete #1 quickly. Problems #2 and #3 are quite long; probably most of the test will be spent on those two problems. (But save time for #4, which tests understanding of the definitions & theorems we have learned.)
Proving statements under pressure can be tricky. So: Make sure you can solve all the problems on the relevant assignments and tutorials. Review the proofs, examples, and theorems in the lecture notes. Make sure you know the triangle inequality and the trivial inequality.

Solutions to Version A

Solutions to Version B

Midterm 2 Information

Test Begins: Wed Mar 25 (1:00 pm) (eastern time)
Test Ends: Fri Mar 27 (1:00 pm) (eastern time)
Platform: The test will appear on Crowdmark. Students will submit their solutions on Crowdmark.

Format: Midterm 2 will have five problems.

Midterm 2 is an open-book test. This means that during the test, you are encouraged to consult the Lecture Notes, the Custom Courseware text, the Assignment solutions, the Tutorial solutions, and any notes you have written during the term.

During the test, you are not permitted to: Collaborate with anyone, ask anyone for hints, or discuss the test problems with anyone (other than Dr. Madnick). Your work must be completely your own. You may not use any resources other than the ones mentioned above. Calculators and computer algebra systems are not permitted. Violation of any of these rules is academic misconduct.

Questions during the test? E-mail Dr. Madnick.

Midterm 2 is designed to take approximately 4 hours to complete. The time frame is 48 hours to accommodate students' varying schedules (and varying internet connections). This is in keeping with Universal Design for Instruction (UDI) principles. Any student who is registered with SAS and is impacted by this UDI principle should contact their SAS coordinator directly.

Content:

Sections 4.3-4.5, Chapter 5, Chapter 6. That is:
Assignments #3 and #4. Tutorial Sets 5 -- 8.
Parts of Interludes B and C might be helpful. Parts of Section 4.2 might be helpful.

Advice: Midterm 2 is challenging. This is by necessity: Open-book take-home tests are more difficult than in-class tests. The focus can no longer be on memorization, but instead on conceptual understanding and problem-solving ability. Some advice:

Start the test as early as you are able. The test is designed to take approximately 4 hours. So, it is to your advantage to at least attempt all the problems on Wednesday. If you cannot solve them all on Wednesday, that’s okay: You can sleep on it and try again the next day.
Be very familiar with the Lecture Notes, preferably before the test begins. The Lecture Notes are there to help you. Don’t be shy about consulting them during the test.
A couple of problems on Midterm 2 do not resemble any of the assignment or tutorial problems. (That was a deliberate choice on my part.) When faced with an unfamiliar problem, it is natural to feel stuck or lost. Take a deep breath and remember that you have all the tools that you need to solve the problem.

Solutions to Midterm 2

Final Exam Information.

Test Begins: Thu Apr 16 (1:00 pm) (eastern time)
Test Ends: Sat Apr 18 (1:00 pm) (eastern time)
Platform: The test will appear on Crowdmark. Students will submit their solutions on Crowdmark.

General Format:

The final exam is an open-book test. This means that during the test, you are encouraged to consult the Lecture Notes, the Custom Courseware text, the Assignment solutions, the Tutorial solutions, and any notes you have written during the term.

During the test, you are not permitted to: Collaborate with anyone, ask anyone for hints, or discuss the test problems with anyone (other than Dr. Madnick). Your work must be completely your own. You may not use any resources other than the ones mentioned above. Calculators and computer algebra systems are not permitted. Violation of any of these rules is academic misconduct.

Questions during the test? E-mail Dr. Madnick.

The final exam is designed to take approximately 4 hours to complete. The time frame is 48 hours to accommodate students' varying schedules (and varying internet connections). This is in keeping with Universal Design for Instruction (UDI) principles. Any student who is registered with SAS and is impacted by this UDI principle should contact their SAS coordinator directly.

Content: The final exam will have eight problems.

#1, #2: Open sets, Closed sets, Bounded sets
#3: Analyze a function (i.e.: Given a function, prove that it is continuous / discontinuous / differentiable / not differentiable / class C1 / not of class C1.)
#5: Global Optimization
#6: Fourier Series
#4, #7, #8: Various topics

The final exam is cumulative, covering Chapters 1 - 7. All five assignments are relevant. Although the "Interlude" lectures are not the focus of the test, they will be assumed as background knowledge.

General Advice:

Start the test as early as you are able. The test is designed to take approximately 4 hours. So, it is to your advantage to at least attempt all the problems on Thursday. If you cannot solve them all on Thursday, that’s okay: You can sleep on it and try again the next day.
Be very familiar with the Lecture Notes, preferably before the test begins. The Lecture Notes are there to help you. Don’t be shy about consulting them during the test.

Specific Advice:

Know the difference between an elliptic paraboloid, a hyperbolic paraboloid (saddle), and a parabolic cylinder.
If you're stuck on a problem, ask yourself:
- Will drawing a picture help?
- Can I use the precise definitions of the terms mentioned in the problem?
- Are there theorems in the lecture notes that might help?
- Have I seen a similar problem in the assignments, tutorials, or lecture notes?

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