Math 511 - Introduction to Analysis II (with Lebesgue integration), Spring 2021

The course is a continuation of Math 510, with newly included Lebesgue integration.
Along with Math 510, it helps graduate students to prepare for the Real Analysis Qualifying Exam.
Topics: Differentiation of functions in Rn, inverse and implicit function theorems, Lebesgue integration in Rn, Fubini's theorem, change of variables, Stokes' theorem.
Zoom Discussions & Lectures: Tue & Thur, 2:00 - 3:15 pm, to be posted on UNMLearn
Office hours: by email and by appointment in Zoom
Professor Info: Anna Skripka, skripka [at] math [dot] unm [dot] edu
 
Textbooks:
There is no required textbook; lectures and homework assignments will be influenced by different sources. Several recommended textbooks are listed below; if you want to have a single book covering all major topics, that would be "baby Rudin".

W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Science/Engineering/Math, 3d edition, 1976, ISBN-10: 007054235X (all topics).
N. L. Carothers, Real Analysis, Cambridge University Press, 1st edition, 2000, ISBN-10: 0521497566 (Lebesgue integration on R).
E. M. Stein, R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, 2005, ISBN-10: 9780691113869 (Lebesgue integration).
W.R. Wade, An Introduction to Analysis, Pearson, 4th edition, 2009, ISBN-10: 0132296381 (advanced calculus differentiation and multivariable Riemann integration, undergraduate level examples).
J.R. Munkres, Analysis on Manifolds, Westview Press, 1997, ISBN-10: 0201315963 (all differentiation and multivariable Riemann integration topics).
M. Spivak, Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus, W. A. Benjamin, Inc., New York-Amsterdam 1965, ISBN-10: 0805390219 (differential forms, Stokes' theorem on manifolds).
B.B. Hubbard, J.H. Hubbard, Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach, Prentice Hall College Div; 1st edition, 1998, ISBN-10: 0136574467 (undergraduate examples, light introduction to differential forms).
 
Prerequisites: solid Math 510 knowledge and skills.
Grades: 10% answers to questions asked during the class meetings in zoom, 40% homework, 50% midterm exam + final exam.
Policies: the homework is to be submitted on UNMLearn as a single PDF file. Please make sure that the files you submit are legible. Your work will be graded on the clarity, completeness, correctness of your reasoning and presentation. On the homework assignments you are allowed to work in groups, but you must write up your own solutions in your own words. Submitting solutions obtained from third parties, including the internet, is strictly prohibited. No late homework will be accepted. A make-up exam will be given for a missed exam only in case of a documented absence prescribed by the university (family emergency, serious medical problem, official UNM function). All objections to grades should be made within one week since the assignments are graded.
 
Schedule
 
Week
Important Dates
Chapters
Topics
Homework
1
Jan 19, 21
Munkres 1.3, 1.4, Stein-Shakarchi 1.1.
Review of Math 510 analysis in Rn and of linear algebra.
HW 1
2
Jan 26, 28
Munkres 1.1, 1.2, 2.5.
Review of linear algebra.
3
Feb 2, 4 (HW1)
Drop with no grade: Feb 5
Munkres 2.6, 2.7 or Rudin Chapter 9, Wade 11.4.
Differentiability of functions mapping Rm to Rn. Chain rule.
HW 2
4
Feb 9, 11 (HW2)
Wade 11.5, 11.7, 11.1.
Mean value theorem. Taylor's formula. Local extremum.
HW 3,
5
Feb 16, 18 (HW3)
Munkres 2.8, or Wade 11.6, or Rudin 9.24
Inverse function theorem.
HW 4,
6
Feb 23, 25 (HW4)
Munkres 2.9 or Rudin 9.28.
Implicit function theorem.
HW 5
7
Mar 2, 4 (HW5)
Carothers Ch. 16, Stein & Shakarchi 1.2.
Summary of Riemann & intro to Lebesgue integration.
8
Mar 9, 11 (Exam on topics of weeks 1-6)
Carothers Ch. 16, Stein & Shakarchi 1.2.
Outer measure. Review.
9
Mar 23, 25
Carothers Ch. 16, 17, Stein & Shakarchi 1.3, 1.4.
Measurable sets and functions.
10
Mar 30, Apr 1
Carothers Ch. 18, Stein & Shakarchi 2.1.
Lebesgue integral: simple and nonnegative functions.
HW 6
11
Apr 6, 8 (HW6)
Carothers Ch. 18, Stein & Shakarchi 2.1, 2.3.
Lebesgue integral: general case. Convergence. Fubini's theorem.
HW 7
12
Apr 13, 15 (HW7)
Withdrawal deadline: Apr 16
Munkres 4.16 - 4.19, Rudin 10.9, Wade 12.4.
Change of variables and its applications.
HW 8
13
Apr 20, 22 (HW8)
Wade Ch. 13.
Line and surface integrals. Fundamental theorems of vector calculus and their applications.
HW 9
14
Apr 27, 29 (HW9)
Wade, Ch. 13.
Green's, Stokes', the Divergence theorems. Proofs in particular cases.
HW 10
15
May 4, 6 (HW10)
Munkres, overview of Ch. 5,6,7.
Differential forms. Stokes' theorem for manifolds. Review (see Learn).
16
Final Exam: May 11 (Tuesday), 10 am - noon
 
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Disclaimer: The instructor reserves the right to change this syllabus. An up-to-date syllabus is posted on this webpage. It is your responsibility to know and understand the course policies.