Analysis II

Jacobs University, Spring 2018

News

• There is an extra tutorial/question session on Wed, May 23, 19:15-20:30 in West Hall 8.
• The tutorial on Wed, May 16, starts a bit later, it runs from 19:45-20:30 (in West Hall 8).
• Homework sheet 10 can be handed in until Wed (May 2) before the tutorial.
• No office hour on Tuesday, May 1 (public holiday).
• No office hour on Tuesday, April 17!
• The usual day for handing in the homework sheets is now Tuesday (before class)!
• During the tutorial on Wed, Mar. 22, 2018, the midterm problems are discussed!
• The midterm is on Mon, Mar. 19, 2018, Lecture Hall Research I, from 9:45 - 11:00 (new time!)
• No office hour on Tuesday, March 6!
• The regular tutorial starts on Wednesdays, 19:15 - 20:30 in West Hall 8.
• The midterm date was shifted to Mon, Mar. 19, 2018, Lecture Hall Research I, 9:00 - 11:00.
• The tutorial on Wed Feb 14, from 19:15 - 20:30 is in West Hall 8. The TA is there to give you some hints and help for solving the first homework sheet. Starting on Feb 21, the solutions to the previous homework sheets will be presented and explained.
• Note: The Tuesday classes will be in East Hall 4 from now on (i.e., starting February 13)!
• The first homework sheet is online, see below. There are 40 points and 8 bonus points. Please hand it in before class on Mondays.

Contact Information

Time and Place

Class: Mon 09:45 - 11:00, Lecture Hall Research I; Tue 14:15 - 15:30, East Hall 4

Tutorial Session: Wed 19:15 - 20:30, West Hall 8

Syllabus

Syllabus (as of Feb. 1, 2018) available here. (Note that the Syllabus will not be updated, the most recent information can be found on this website.)

Textbooks

The class mainly follows

• W. Rudin - Principles of Mathematical Analysis (McGraw-Hill, third edition)
• T. Tao - Analysis I/II (Springer, third edition)

Lecture Notes

Abhik Pal wrote very nice class notes in LaTex (thank you very much!):

• Lecture Notes by Abhik Pal

These will be continuously updated and extended.

Grading

The final grade is weighted as follows: Homework: 20%, Midterm: 30%, Final: 50%

Exams

There will be two exams, a midterm and a final. More details will be announced. Note that this class uses gradescope for grading exams, see here for more information.

Midterm Exam: Mon, Mar. 19, 2018, Lecture Hall Research I, 9:45 - 11:00

The exam begins at 9:45 sharp, so please come a few minutes earlier to be seated at 9:45. The midterm is about all material up to the Mar. 5 session. You only need to bring a pen, no calculators or notes! Please be aware of the University policy regarding missing exams: Policies for Undergraduate Studies. In particular, in case you are missing an exam due to illness (only with doctor's note) or an emergency, you have to notify me (by email) before the beginning of the exam!

Final Exam: Tue, May 29, 2018, IRC conference hall, 16:00-18:00

The exam begins at 16:00 sharp, so please come a few minutes earlier to be seated at 16:00. The final exam is about all material of the class, with an emphasis on the second half. You only need to bring a pen, no calculators or notes! Please be aware of the University policy regarding missing exams: Policies for Undergraduate Studies. In particular, in case you are missing an exam due to illness (only with doctor's note) or an emergency, you have to notify me (by email) before the beginning of the exam!

Homework Sheets

Each week on Tuesday (with exceptions) there will be a homework assignment. These are an integral part of the coursework and working on the exercise sheets consistently is the best preparation for the exams. Note that

• The two worst homework sheets are not considered for grading.
• It is encouraged to discuss the homework sheets with your classmates (e.g., discuss how to come up with the solution or what the right way of approaching the problem is). On the other hand, the solutions must be written down and handed in individually! Copying the solutions from somebody else is a violation of Academic Integrity.

Homework Sheets are to be handed in individually at the beginning of class. Alternatively, they can be deposited in a mailbox at the main entrance of Research I at some point before class on Tuesday (right next to the entrance there is a mailbox labeled "Analysis II"). No late hand ins except with a doctor's note that proves you were sick several days during the time you had working on the homework sheet!

Date	Sheet Number	Due Date
Feb. 12, 2018	Sheet 1	Feb. 19, 2018
Feb. 19, 2018	Sheet 2	Feb. 26, 2018
Feb. 26, 2018	Sheet 3	Mar. 05, 2018
Mar. 05, 2018	Sheet 4	Mar. 12, 2018
Mar. 12, 2018	Sheets 5 and 6	Apr. 03, 2018
Apr. 03, 2018	Sheet 7	Apr. 10, 2018
Apr. 10, 2018	Sheet 8	Apr. 17, 2018
Apr. 17, 2018	Sheet 9	Apr. 24, 2018
Apr. 24, 2018	Sheet 10	May 01, 2018
May 01, 2018	Sheet 11	May 08, 2018
May 08, 2018	Sheet 12	May 15, 2018

Class Schedule

Will be updated while class is progressing.

Date	Topics
Feb. 05, 2018	Riemann-Stieltjes Integral (definition)
Feb. 06, 2018	Riemann-Stieltjes Integral (refinement of partitions, existence criterion)
Feb. 12, 2018	Riemann-Stieltjes Integral (continuous functions, finitely many discontinuities, basic properties)
Feb. 13, 2018	Riemann-Stieltjes Integral (compositions, change of variables)
Feb. 19, 2018	Riemann-Stieltjes Integral (change of variables, fundamental theorem)
Feb. 20, 2018	Riemann-Stieltjes Integral (integration by parts, improper integrals, uniform convergence and integration and differentiation)
Feb. 26, 2018	Convergence, Series, Sequences (review, convergence tests)
Feb. 27, 2018	Convergence, Series, Sequences (more convergence tests, rearrangements)
Mar. 05, 2018	Power Series (Cauchy product, radius of convergence; derivative, integral and Cauchy product of power series)
Mar. 06, 2018	Curves and Differential Equations (definition of curves in R^n, differentiability, speed limit)
Mar. 12, 2018	Curves and Differential Equations (integrability, rectifiability and curve length)
Mar. 13, 2018	Curves and Differential Equations (rectifiability, reparametrization; intro to differential equations, vector fields, initial value problem)
Mar. 19, 2018	Midterm Exam
Mar. 20, 2018	Curves and Differential Equations (Lipschitz condition, Picard-Lindelöf, separation of variables)
Mar. 26, 2018	no class (Spring Break)
Mar. 27, 2018	no class (Spring Break)
Apr. 02, 2018	no class (Spring Break)
Apr. 03, 2018	Basic Topology (definition, continuity)
Apr. 09, 2018	Basic Topology (compactness)
Apr. 10, 2018	Basic Topology (Heine-Borel, connectedness)
Apr. 16, 2018	Basic Topology (connectedness, path connectedness)
Apr. 17, 2018	Differentiation in Rn (definitions of total derivative, directional derivative, partial derivative)
Apr. 23, 2018	Differentiation in Rn (connections between total derivative, directional derivative, partial derivative)
Apr. 24, 2018	Differentiation in Rn (equivalence of total continuous differentiability and continuous partial differentiability, the gradient and extrema)
Apr. 30, 2018	Differentiation in Rn (higher order derivatives and second order Taylor expansion)
May 01, 2018	no class (Labor Day)
May 07, 2018	Differentiation in Rn (Inverse Function Theorem)
May 08, 2018	Differentiation in Rn (Inverse Function Theorem)
May 14, 2018	Differentiation in Rn (Implicit Function Theorem)
May 15, 2018	Differentiation in Rn (Implicit Function Theorem)
May 29, 2018	Final Exam