Math 320 - Real Variables I

Math 320 - Real Variables I - Fall 2018

Course information

This is the common page for the two sections of MATH 320 in Term 1 of the 2018W session (September to December 2018). Here you will find the course outline, suggested homework and practice problems, course policies, exam dates, common handouts and supplementary notes, other course information, and information on available resources. For section-specific information, please follow the individual section links below or contact your instructor.

Text

The required textbook for this course is Principles of Mathematical Analysis, third edition, by Walter Rudin. The textbook is available at the UBC Bookstore .

UBC course description

The real number system
Real Euclidean n-space
Open, closed, compact, and connected sets
Bolzano-Weierstrass theorem
Sequences and series
Continuity and uniform continuity
Differentiability and mean-value theorems

The material is spread over Chapters 1-5 of the textbook.

Pre-requisites

One of the following two requirements:

(a) a score of 68% or higher in MATH 226,
(b) one of MATH 200, MATH 217, MATH 226, MATH 253, MATH 263 and a score of 80% or higher in MATH 220.

Beginning-of-term registration information

At last report, all sections of Math 320 were almost full, with a few seats remaining.
If you are not registered in a section, please do not attend it without the instructor's approval.
Instructors do not have the authority to "fit you in". Such requests have to be processed by the math department office (Room 121 Mathematics Building) and approved by our undergraduate chair Mark MacLean (maclean at math dot ubc dot ca).

Grading Scheme

Your grade will be computed based on the following formula:

Final Exam 50%
2 midterms 15% + 15% = 30%
Course-common homework assignemts 20%

Please note that grades may be scaled to ensure fairness across sections.

Exam Dates and Policies

There will be two midterms in MATH 320. The duration of each midterm will be 50 minutes. The dates are:

Midterm 1:

Friday October 12, 9-9:50 AM in class.

Midterm 1 will cover all of Chapter 1 and part of Chapter 2 (pages 24-36, up until and not including the section on "Compact sets") of the textbook.

Practice problem set .

Midterm 1 solutions .

Midterm 2:

Friday November 9, 9-9:50 AM in class.

Midterm 2 will cover part of Chapters 2 and 3 (pages 36-58, starting from the section on ``Compact sets" up until and not including the section on "Series") of the textbook.

Practice problem set .

Midterm 2 solutions .
Midterms are non-cumulative, but the final exam is based on the entire syllabus for the course.

Final exam info:

The final examination will be held on December 18 at 7 pm in SWNG 221.
The final exam covers the entire course syllabus, as detailed in the week-by-week outline below.
Practice problem set .
Past final exams for this course may be found here .
The final exam will be common to all sections of MATH 320, and will account for 50% of a student's final grade. The remaining 50% will be based on term work. The final exam generally will not be weighted higher for students who perform better on this test than they did during the term, although some allowance may be made for students who perform significantly better (an increase of 30% or more over their term mark). Such decisions are based on the sole discretion of the instructor.

Exam aids: No unauthorized electronic devices will be allowed in the midterms or in the final exam. This includes calculators, cell phones, music players and all communication devices. Students should not bring their own formula sheets or other memory aids.

Please bring your student ID-s to both midterms and the final.

Policy on missed work

Missed homework: Late submissions will not be accepted, but the two worst homework grades will be dropped.
Missed midterms: If a student misses a midterm, a documented excuse must be provided or a mark of zero will be entered for that midterm. Examples of valid excuses are an illness which has been documented by a physician and Student Health Services, or an absence to play a varsity sport (your coach will provide you with a letter).
There will be no make-up midterms. If a valid excuse if provided, the weight of the missed midterm will be transferred to the final examination. To be eligible for this arrangement, you must notify your instructor of your failure to take the test within a week of the missed midterm, and come up with a timeline acceptable to both for producing appropriate documentation for your absence.

Please note that a student may NOT have 100% of their assessment based on the final examination. A student who has missed both midterms will fail the class, regardless of his or her performance in the final exam.

Missed final exam: You will need to present your situation to your faculty's Advising Office to be considered for a deferred exam. See the Calendar for detailed regulations . Your performance in a course up to the exam is taken into consideration in granting a deferred exam status (for instance, failing badly normally means you will not be granted a deferred exam). For deferred exams in mathematics, students generally sit the next available exam for the course they are taking, which could be several months after the original exam was scheduled.

Academic misconduct

UBC takes cheating incidents very seriously. After due investigation, students found guilty of cheating on tests and examinations are usually given a final grade of 0 in the course and suspended from UBC for one year. More information.
Note that academic misconduct includes misrepresenting a medical excuse or other personal situation for the purposes of postponing an examination or quiz or otherwise obtaining an academic concession.

Individual section links

Section 101 Mon Wed Fri 9-10AM Math 102 (Instructor: Joshua Zahl)
Section 102 Mon Wed Fri 9-10AM Geography 212 (Instructor: Malabika Pramanik, malabika at math)

Homework and practice problems

Homework assignments will be posted weekly and collected every Wednesday.

Homework set 1 , due September 12.
Homework set 2 , due September 19.
Homework set 3 , due September 26.
Homework set 4 , due October 3.
Homework set 5 , due October 10.
Homework set 6 , due October 24.
Homework set 7 , due October 31.
Homework set 8 , due November 7.
Homework set 9 , due November 21.

Help outside class

There will be a few office hours every week held by the instructor and the TA. Details will appear at the top of this website.
We will be using Piazza (www.piazza.com) as a class discussion platform. This is a question-answer forum designed for getting quick responses from classmates, instructors or TA-s. We encourage you to use this feature extensively and responsibly to pose questions, discuss solutions and ask for clarifications. A sign-up link and a class link will be emailed to you by your instructor. You will need to self-register for this course in Piazza using the link and a valid ubc email address.

Week-by-week course outline

This section will contain a summary of the material that will be covered in class, arranged by week.

Week 1: (Pages 1-9)
- The real number system
- Ordered sets and fields
- Supremum and infimum
- Examples

Week 2: (Pages 9-12, 17-21)
- Archimedean property of reals
- Integer roots of positive reals
- Real numbers as cuts and Cauchy sequences

Week 3: (Pages 12-17, 24-26)
- The complex field
- Cardinality
- Finite, countable and uncountable sets

Week 4: (Pages 24-30)
- Countability of rationals
- Uncountability of reals
- Cardinality of unions and intersections of sets

Week 5: (Pages 30-35)
- Metric space topology
- Open and closed sets
- Limit points

Week 6:
- Midterm review

Week 7: (Pages 36-40)
- Compact sets
- Examples
- Relations between closed sets and compact sets

Week 8: (Pages 36-40)
- Compact sets and the nested intersection property
- Heine-Borel theorem
- Sequences in compact sets

Week 9: (Pages 42-57)
- Connected sets
- Numerical sequences and series
- Cauchy sequences
- Lim sup and lim inf

Week 10: (Pages 58-75)
- Absolutely and conditionally convergent series
- Root and Ratio tests
- Power series
- Radius of convergence
- Rearrangement of series
- Riemann's theorem on rearrangements

Week 11: (Pages 83-93 )
- Continuous functions in metric spaces
- Continuity and compactness
- Continuity and connectedness

Week 12: (Pages 93-97 )
- Discontinuities
- Monotonic functions
- Review