Math 511 - Introduction to Analysis II

Time:	Tue & Thur 11 a.m. - 12:15 p.m.
Room:	GSM 230 (Graduate School of Management/ Parish Library) moved to SMLC 124
Instructor Info:	Anna Skripka, skripka [at] math [dot] unm [dot] edu
Office hours:	Tue and Thur 3:30 - 4:45 p.m., and by appointment, SMLC 212

Topics include differentiation of functions in Rⁿ, inverse and implicit function theorems, integration in Rⁿ, Fubini's theorem, change of variables, Stokes' theorem. Math 510 and Math 511 prepare graduate students for the Real Analysis Qualifying Exam.

Textbooks:
J.R. Munkres, Analysis on Manifolds, Westview Press, 1997, ISBN-10: 0201315963. (All differentiation and integration topics.)
W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Science/Engineering/Math, 3d edition, 1976, ISBN-10: 007054235X. (Differentiation.)
W.R. Wade, An Introduction to Analysis, Pearson, 4th edition, 2009, ISBN-10: 0132296381. (Advanced calculus differentiation and integration, undergraduate level examples, no proof of Stokes' theorem in the general case.)
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.)

Prerequisite: Math 510 or equivalent.
Grades: 40% homework, 30% each of two midterm exams.

Academic Honesty: You can work on the homework assignments in groups but you do need to write up your own solutions in your own words (identical or nearly identical solutions will receive zero credit). Submitting solutions obtained from third parties, in particular, from the internet forums is STRICTLY PROHIBITED. Here is a section on dishonesty in UNM Handbook.

American Disabilities Act: In accordance with University Policy 2310 and the American Disabilities Act (ADA), academic accommodations may be made for any student who notifies the instructor of the need for an accommodation. It is imperative that you take the initiative to bring such needs to the instructor's attention, as the instructor is not legally permitted to inquire. Students who may require assistance in emergency evacuations should contact the instructor as to the most appropriate procedures to follow. Contact Accessibility Services at 505-661-4692 for additional information.

TENTATIVE Schedule (will be updated at the end of each week)

Week	Important Dates	Lectures	Topics	Homework
1	Jan 13, 15	Munkres 1.3, 1.4. Lecture 1	Review of Math 510 analysis in Rⁿ.
2	Jan 20, 22	Munkres 1.1, 1.2, 2.5. Lecture 2	Review of linear algebra. Directional derivatives.	HW 1
3	Jan 27, 29; Drop with no grade: Jan 30	Munkres 2.5, 2.6, 2.7. Lecture 3	Differentiability of functions mapping R^m to R^n.	HW 2
4	Feb 3, 5	Wade 11.5, 11.7, 11.1, Rudin 9.42. Lecture 4	Taylor's formula. Local extremum. Differentiation of integrals.	HW 3
5	Feb 10, 12	Munkres 2.8, or Wade 11.6, or Rudin 9.24. L 5	Inverse function theorem. Prepare for Exam 1	HW 4
6	Feb 17, 19 (Exam 1)	Munkres 2.9. Lecture 6	Implicit function theorem.	HW 5
7	Feb 24, 26	Munkres 3.10, 3.11. Lecture 7	Integration over a rectangle.
8	Mar 3, 5	Munkres 3.11, 3.12. Lecture 8	Set of discontinuities of measure 0. Fubini's theorem.	HW 6
9	Mar 17, 19	Munkres 3.13. Lecture 9	Integration over a bounded set.	HW 7
10	Mar 24, 26	Munkres 3.14, 4.17. Lecture 10	Rectifiable sets. Fubini's theorem for simple regions. Change of variables and its applications. Prepare for Exam 2	HW 8
11	Mar 31, Apr 2 (Exam 2)	Munkres 4.17, 3.15. Lecture 11	Change of variables for improper integrals. Proof strategy.
12	Apr 7, 9; Withdrawal deadline: Apr 10	Munkres 4.16, 4.18, 4.19. Lecture 12	Partition of unity. Proof of the change of variables theorem.	HW 9
13	Apr 14, 16	Wade, Chapter 13. Lecture 13	Line and surface integrals. Fundamental theorems of vector calculus.	HW 10
14	Apr 21, 23	Wade, Chapter 13. Lecture 14	Applications of Green's, Stokes', and the Divergence theorems. Proofs in particular cases.
15	Apr 28, 30	Munkres, overview of Chapters 5,6,7. L 15	Differential forms. Stokes' theorem for manifolds.