Math 4001-5001 Analysis II
Semester 1, 2021-2022
Course Lecturer:
Dr. Judith Packer, Dept. of Mathematics
Tel: (303) 492-6979
Office: Math 227
Email: packer@colorado.edu
URL: http://math.colorado.edu/~packer
Course Syllabus: For course syllabus, click here!
This syllabus contains all university rules and regulations. Please abide by the University Mask mandate in lectures.
Course Information:
This course is meant to continue to familiarize the student with the theory of
functions of one or more real variables, with an initial emphasis on sequences and series of functions and
different types of convergence, and with a continuation into the theory of vector-valued functions of vector variables,
including the inverse function theorem and the implicit function theorem.
Topics to be covered include:
a rigorous treatment of infinite series of real numbers, sequences and series of functions,
convergence and uniform convergence of sequences and series of functions, Taylor's Theorem,
a brief review of vector spaces, in particular, n-dimensional Euclidean space;
differentiation, including
the definition of the total derivative and the chain rule; the mean value theorem, the inverse mapping theorem,
and the implicit function theorem.
Prerequisite:
Math 3001 Analysis 1 and Math 2130 Linear Algebra (or Math 2135).
Course Text:
We will use the text "Undergraduate Analysis", by Serge Lang, Second Edition, Springer, 2005.
Assessment:
- Assessment of homework performance: Homework will be assigned every week.
Some, but not all, of the problems will be graded.
Please note that the assignments for Math 5001 will include extra problems.
The assessment of homework performance will count for 25% of the final grade.
- In-class mid-term exam - Wednesday September 29, 9:10 a.m. - 10:05 a.m.:
22.5 % of final grade. You will be allowed a 3" by 5" notecard on this exam.
- Take-home mid-term exam - given out Wednesday, November 3, 2021, 10 a.m., due Wednesday November 10, 2021
11:59 p.m.(please note that takehome exams will be different for Math 4001 and Math 5001): ,
: 22.5 % of final grade.
- Click here for the Math 4001 take-home exam.
- Click here for the Math 5001 take-home exam.
- Final exam - Wednesday, December 15, 2021, 7:30 p.m. - 10 p.m., ECCR 118 : 30 % of final grade. You will be allowed a 3" by 5" notecard on this exam.
Lecture Hours and Venue:
MWF 9:10 a.m.- 10:00 a.m. ECCR 118.
Office Hours:
10:30 a.m. - 11:30 a.m., by ZOOM or in Math 227 MWF, or by appointment on ZOOM.
Homework Assignments (subject to change):
- Assignment 1: Read Sections 1 and 2 of Chapter IX, do problems on p. 213,
1 (b), 3, 4, 5(a),(b), 8, 9 (a), (c), due Wednesday, Sept. 1.
- Assignment 2: Read Sections 3 and 4 of Chapter IX, do problems on p. 214,
7, 10; p. 224, 1, 3 (a), (c), (e), (h) , due Friday, September 10. Also, begin the following extra problem (click here)
- Assignment 3: Due Friday, September 17: complete the extra problem (click here)
assigned last week; also, read Chapter VII Section 3 pp. 179-181 through Example 3, do problems 1 (a), (b), (c),
2, and 3, p. 187.
- Assignment 4, due Friday, September 25: read Chapter VII Section 3 pp. 181-182, Theorems
3.1 and 3.2, and Chapter IX, Section 5, pp. 229-231; do problems 4(b), 4(c) and 5 on p. 187;
do problems 1(a), 2, 4 on pp. 232-233 of the textbook.
- Assignment 5: Due Friday, October 8: read Chapter IX Section 6 pp. 234-238, do problems 1 (b), (d), (e), (g),
2 (b), (d), (e), and 3, p. 238; start reading Chaper IX, pp. 239--243, do problem 1, p. 243.
- Assignment 6: Due Friday, October 15: do problems 2 and 7, pp. 244-245; read Chapter V, Section 3, pp. 109-115,
do problems 1, 2, 4, p. 115.
- Assignment 7: Due Friday, October 22: Read Chapter XV Section 1 pp. 370-375, and Section 2 through p. 383.
Do problems 1, 3, 5, 13 on p. 375-377, and do problems 1, 6 (b) (d), 7 (b), on pp. 387-388.
- Assignment 8: Due Friday, October 29: Read Chapter XV, end of Section 2, Section 5, Section 6; Chapter XVII Sections 2-4.
Do problems 12 on p. 377; do problems 2, 3 on p. 410; problem 12 on p. 415, problem 2 on p. 473, and problem 1 on p. 477.
- Assignment 9: Due Friday, November 12: Read Chapter XVII Sections 5-6 (should be review as we have had lectures on this material),
and Chapter XVIII, Section 1 of the Lang textbook. Do problem 5 on p. 487; do problems 3, 4 on p. 520;
do problems 9 and 14 on pp. 536-537.
- Assignment 10: Due Friday, November 19: Read the Rudin hand-out, p. 221- top of p. 228. Do problems to be assigned.
- Assignment 11: Due Friday, December 3: Read the Rudin hand-out, p. 234- p. 238. Do problems to be assigned.
- Click here for a two-and-a-half hour final exam from Math 4001, given several years ago.
Homework:
Homework will be assigned weekly, and is due uploaded to Canvas at 11:59 p.m., usually on Fridays.
Some, but not all of the problems will be graded. Show
all your work.
Some famous mathematicians who worked in this area:
Back to the home page of Judith A. Packer
Last modified August 23, 2021.