MAT 319 and 320
MAT 319 (Foundations of Analysis) and MAT 320 are taught as a single lecture until
the first midterm. Based on the results of the first midterm, students may switch
into MAT 320, which will proceed at a faster pace than MAT 320. Professor Martens
will teach MAT 319, and the two of us will share lectures up until the first midterm.
TENTATIVE SYLLABUS AS OF AUG 7, 2021
MAT 319 Instructor
Prof. Marco Martens, marco@math.sunysb.edu
Dept. Phone: (631)-632-4893
Dept. FAX: (631)-632-7631
my homepage
MAT 320 Instructor
Prof. Christopher Bishop, bishop@math.stonybrook.edu
Dept. Phone: (631)-632-8290
Dept. FAX: (631)-632-7631
my homepage
Course Summary
This is a course in real analysis or roughly "calculus with proofs".
We will start by recalling set notation, the principle of induction,
and the basic properties of finite and infinite sets. We then
discuss the axioms of the real numbers, and learn to rigorously
derive various familiar properties. When then study sequences
and limits. Around this point MAT 319 and 320 will separate. In
MAT 320 will will then discuss continuity, differentiation,
Riemann integration, sequences of functions and infinite series.
If time permits, we will discuss basic point set topology,
metric spaces, the generalized Riemann integral and
the Lebesgue integral.
Time and Place
For the first few weeks, MAT 319 and MAT 320 both meet Tuesdays and
Thursdays 9:45am-11:05am in Stony Brook Union room 103-02.
After the classes split MAT will continue in that room and MAT 320 will
meet in Mathematics P-131 (this room is directly behind the elevators on the
PL level of the Math Tower). Recitations meet twice a week as follows:
MAT 319 R01 82270 MW 11:45am-12:40pm Earth-Space Science 181
MAT 319 R02 88825 MW 11:45am-12:40pm Frey Hall 309
MAT 319 R03 94829 MW 11:45am-12:40pm Socbehav Sci N310
MAT 320 R01 81315 MW 11:45am-12:40pm Physics P116
Note that all recitations meet at the same time.
Office Hours
Office Hours MAT319: TuTh 11:30-12:30 or by appointment
Textbook
The textbook is "Introduction to Real Analysis, 4th Edition" by
Robert Bartle and Donald Sherbert.
Grades
Grades will be based on weekly problem sets, two midterms and a final exam.
Each component will count for 25% of the total grade.
Final Exam
The final exam is Tuesday, December 14, 8:00am-10:45am. Location will be
be announced later. Possibly all final exams will be online.
Blackboard
Blackboard
is the Stony Brook University class management system. Your
homework, quiz and exam grades will be posted here. Letter grades
for the course are posted in the Solar System. I will occasionally
post announcements in Blackboard; you should receive email notifications
whenever this occurs.
Solar System
Solar System
is the Stony Brook University administrative management system
(registration, bills,...). It is not used for classes, except
to post letter grades at the end of the semester..
Stony Brook Gmail
Check your firstname.lastname@stonybrook.edu email
here.
Stony Brook Virtual SINC Site
The
Virtual Sinc Site
gives you access to various software
packages on a university license, such as Mathematica and Matlab.
I don't plan to use any of these in this class, but this resource
may be helpful in other classes.
Using the virtual Sinc Site requires downloading the
Citrix receiver software (you will be prompted).
Important University Dates
Link to university academic calendars, including final exam calendars.
First day of classes: Monday August 23, 2021.
Labor Day, no classes: Monday Sept 6, 2021.
Fall break, No classes Mon Oct 11 and Tue Oct 12
Thanksgiving break:
Wed November 24 to Sunday November 28, 2021.
Last day of classes: Monday December 6, 2021.
Reading day: Tuesday December 7, 2021.
Finals:
Wednesday December 8 to Thursday December 16, 2021.
MAT 320 Final Exam: 8:00am-10:45am Tuesday, Dec 14, 2021
Commencement: Friday December 17, 2021
University deadlines
See the following page for deadlines for things like withdrawing from classes without
penalty, applying for P/NC, changing major, ...
University deadlines
Pass/No Credit
This policy allows you to set a threshold so that if you
score above the threshold in a class you get a that grade
on your transcript, and otherwise you get a P (for pass) or
NC (no credit), neither of which affects your GPA.
For example, if you set the threshold at C and if you get a
C- or a D you will get a P in the course (which means it won't
count towards major requirements, but also won't affect your GPA).
A grade of F gives an NC (also won't affect your GPA), and any
grade equal to or higher than your threshold will count as usual.
Generally, only one course per semester may be designated P/NC.
Check the Bulletin for precise dates and requirements; your major may
also have rules about which classes these may be used for.
SBU G/P/NC page.
Problem sets
There will be a problem set due each week (except the first week
of class and midterm weeks; sections covered the week before a
midterm are due the week after the midterm).
These should be handed in at the first recitation of the week.
Problems will be taken from the textbook and are listed in
the lecture schedule below. Problems from a section
are due in recitation the week after that section is covered
in lecture.
Tentative Lecture Schedule
Week 1, Aug 23 - Aug 27
Topics covered:
1.1 Sets and functions (problems 6, 14, 18, 21)
1.2 Induction (problems 7, 11, 17)
1.3 Finite and infinite sets (problems 4, 12, 13)
Week 2, Aug 30 - Sept 3
Topics covered:
2.1 Algebraic and order axioms (problems 4, 8, 17, 19)
2.2 Absolute value and the real line (problems 2, 17, 18)
2.3 The completeness property (problems 10, 12,13)
Week 3, Sept 6 - Sept 10
Topics covered:
2.4 Applications of the Supremum property
(problems 5, 7, 11, 19)
2.5 Intervals (problems 8, 12)
3.1 Sequences and their limits (problems 4, 5, 11, 17)
Week 4, Sept 13 - Sept 17
Topics covered:
3.2 Limit theorems (problems 2, 7, 15, 20)
3.3 Monotone sequences (problems 2, 7, 9)
3.4 Bolzano-Weierstrass theorem (problems 6, 9, 18)
Week 5, Sept 20- Sept 24:
Topics covered:
3.5 Monotone sequences (problems 4, 6, 11)
3.6 Properly divergence sequences
(problems 2, 5, 10)
3.7 Introduction to infinite series
(problems 5, 11, 15, 17)
Week 6, Sept 27 - Oct 1
Topics covered:
Review of Chapters 1-3
Midterm 1
Week 7, Oct 4 - Oct 8
Topics covered:
4.1 Limits of functions
(problems 6, 13, 16)
4.2 Limit Theorems
(problems 8, 12, 14)
4.3 Extensions of the limit concept
(problems 7, 11)
5.1 Continuous functions
(problems 3, 6, 12, 14, 15)
5.2 Combinations of continuous functions
(problems 3, 8, 11, 14)
Week 8, Oct 11 - Oct 15, Fall break, no classes Oct 11-12
Topics covered:
5.3 Continuous functions on intervals
(problems 6, 13, 18)
5.4 Uniform continuity
(problems 2, 6, 15, 16)
Week 9, Oct 18 - Oct 22
Topics covered:
5.5 Continuity and gauges
(no problems)
5.6 Monotone and inverse functions
(problems 2, 9, 10, 12, 13)
6.1 The deriviative
(problems 7, 9, 13, 17)
6.2 The mean value theorem
(problems 8, 11, 12, 13, 15)
6.4 Taylor's theorem
(problems 8, 12, 16)
Week 10, Oct 25 - Oct 29
Topics covered:
7.1 The Riemann integral
(problems 7, 8)
7.2 Riemann integrable functions
(problems 6 (give an example), 8, 9, 15)
Appendix C, The Riemann and Lebesgue criteria
The fundamental theorem
(problems 8, 14, 16, 21)
Week 11, Nov 1 - Nov 5
Topics covered:
8.1 Pointwise and uniform convergence
(problems 12, 19, 21,24)
8.2 Interchange of limits
(problems 3, 14, 17, 18)
8.3 Exponential and logarithmic functions
(problems 4, 9)
8.4 The trigonometric functions
(no problems)
Week 12, Nov 8 - Now 12
Topics covered:
9.1 Absolute conevergence
(problems 2, 7, 8, 15, 16)
9.2 Tests for absolute convergence
(problems 15, 17, 19)
9.3 Tests for non-absolute convegence
(problems 6, 9, 10, 12)
9.4 Series of functions
(problems 2, 3, 11, 14, 15)
Week 13, Nov 15 - Nov 19
Topics covered:
Review of Chapters 4-9
Midterm 2
Week 14, Nov 22 - Nov 26
Thanksgiving break, no class Thursday.
Week 15, Nov 29 - Dec 3
Last week of classes. Some recitations meet Mon Dec 6.
Topics covered:
11.1 Open and closed sets
(problems 10, 11, 16}
11.2 Compact sets
(problems 4, 9, 11)
11.3 Continuous functions
(problems 2, 7, 10)
11.4 Metric spaces
(problems 8, 9, 10)
Final Exam on Tuesday Dec 14, 8:00am-10:45pm.
Topics in the history of Calculus
Below are some reading about the history of calculus that
may be of interest.
Wikipedia page on
calculus.
Wikipedia page on
Issac Newton.
Wikipedia page on
Newton-Leibniz controversy.
Wikipedia page on
the discovery of the planet Neptune (using only mathematics).
Wikipedia page on Gauss.
Wikipedia page on Pappus.
Rankings of math departments - 2020
The 2021
Shanghai Ranking of mathematics departments around the world.
Stony Brook was placed 19th in the world and 10th in the United States.
Technology Support:
Student Technology Services.
TLT provides academic technology support to all students.
If you require assistance with Blackboard or other academic
technologies, please contact TLT at:
helpme@stonybrook.edu; Phone: 631.632.9602; Chat;
http://www.stonybrook.edu/helpme or visit a SINC Site.
Students who need assistance with their personal devices
can contact DoIT's service desk at: 631.632.9800,
submit an online request, or visit the Walk In Center on
the 5th floor of the Melville Library (West Campus),
Room S-5410. For more information, visit: https://it.stonybrook.edu/students
Required Syllabus Statements
The University Senate Undergraduate and Graduate Councils
have authorized that the following required statements appear
in all teaching syllabi (graduate and undergraduate courses)
on the Stony Brook Campus.
Student Accessibility Support Center Statement
If you have a physical, psychological, medical, or learning
disability that may impact your course work, please contact
the Student Accessibility Support Center, 128 ECC Building,
(631) 632-6748, or at sasc@stonybrook.edu. They will determine
with you what accommodations are necessary and appropriate.
All information and documentation is confidential.
Students who require assistance during emergency evacuation
are encouraged to discuss their needs with their professors
and the Student Accessibility Support Center. For procedures
and information go to the following website:
https://ehs.stonybrook.edu/programs/fire-safety/emergency-evacuation/evacuation-guide-people-physical-disabilities
and search Fire Safety and Evacuation and Disabilities.
Academic Integrity Statement
Each student must pursue his or her academic goals honestly
and be personally accountable for all submitted work.
Representing another person's work as your own is always wrong.
Faculty is required to report any suspected instances of academic
dishonesty to the Academic Judiciary. Faculty in the Health Sciences
Center (School of Health Technology & Management, Nursing,
Social Welfare, Dental Medicine) and School of Medicine are required
to follow their school-specific procedures. For more comprehensive
information on academic integrity, including categories of
academic dishonesty please refer to the academic judiciary website
at http://www.stonybrook.edu/commcms/academic_integrity/index.html
Critical Incident Management
Stony Brook University expects students to respect the rights, privileges,
and property of other people. Faculty are required to report to the
Office of Student Conduct and Community Standards any disruptive
behavior that interrupts their ability to teach, compromises the
safety of the learning environment, or inhibits students'
ability to learn. Until/unless the latest COVID guidance is
explicitly amended by SBU, during Fall 2021 "disruptive behavior”
will include refusal to wear a mask during classes.
For the latest COVID guidance, please refer to:
https://www.stonybrook.edu/commcms/strongertogether/latest.php