Exams

Final Exam

Take-home exam. Comprehensive, will cover all material learned in class.

Download here [Final Exam]

Deadline: Wed, May 7, 5:30pm
Bring the exam to my office MATH 610 between 3:30pm and 5:30pm on Wed, May 7.

Final Score

Two schemes for calculating your final score will be used: with or without final exam.

Scheme I = (3/10)ME1 + (3/10)ME2 + (1/5)FE + (1/5)HW
Scheme II = (3/8)ME1 + (3/8)ME2 + (1/4)HW

where FE, MEi, HW are the scores (in %) for Final Exam, Midterm Exam i, Homework, respectively.

You will have to email me by 5:30pm on Thur, May 1 to tell whether you will be completing the Final Exam (Scheme I will be used) or not (Scheme II will be used). If you do not contact me by then, your score will be calculated according to Scheme I.

Midterm Exam 2

Take-home exam.

Download here: [Midterm Exam 2] (Updated 4/22: fixed various typos in Problem 2)

Deadline: Thur, Apr 24, 12:00 noon (beginning of class)

Midterm Exam 1

Take-home exam.

Download here: [Midterm Exam 1]
(Updated 3/5: Added missing $I$ in the definition of $D_\alpha$)

Note: You will need the password given in class to open this file.

Deadline: Thur, Mar 6, 12:00 noon (beginning of class)

Course Log

Planned

- Thur, May 1: Review for Final Exam
- Tue, 04/29:  [M] §25 Integration of functions on manifolds

Covered

- Thur, 04/24: Overview of Midterm 2, [M] §23-24 Manifolds with boundary
- Tue, 04/22: Review for Midterm 2
- Thur, 04/17: [M] §23 Manifolds in Rⁿ
- Tue,  04/15: [M] §16 Partitions of Unity, §23 Manifolds in Rⁿ (start)
- Thur, 04/10: [M] §22 Integration on a Parametrized Manifold
- Tue, 04/08: [M] §21 Volume of the Parallelopiped, §22 Volume of A Parametrized Manifold
- Thur, 04/03: §45 Change of Variables Strong Form
- Tue, 04/01: §45 Jacobian Theorem, Change of Variables
- Thur, 03/27: §45 Transformations by Linear and Nonlinear Mappings
- Tue, 03/25: §45 Transformations of Sets with Content
- Thur, 03/13: Overview of Midterm 1
- Tue, 03/11: §44 Further Properties of Integral, Integral as Iterated Integral
- Thur 03/06: §44 Characterization of Content Function
- Tue, 03/04: Review for Midterm 1
- Thur, 02/27: §44 Content and Integral, Sets with Content
- Tue, 02/25: §43 Properties of Integral, Existence of Integral
- Thur, 02/20: §43 Content zero, Definition of Integral
- Tue, 02/18: no class - Thur, 02/13: §42 Inequality Constraints
- Tue, 02/11: §42 Extremum Problems with Constraints, Lagrange's Theorem.
- Thur, 02/06: §42 Local Extrema, Second Derivative Test.
- Tue, 02/04: §41 Implicit Functions
- Thur,1/30: §41 Surjective Mapping Theorem, Open Mapping Theorem, Inversion Theorem.
- Tue, 1/28: §41 Class C1, Approximation Lemma, Injective Mapping Theorem
- Thur, 1/23: §40 Interchange of Order of Differentiation, Taylor's Theorem
- Tue, 1/21: §40 Chain Rule and Mean Value Theorems.
- Thur, 1/16: §39 Partial derivatives, differentiation (cont)
- Tue, 1/14: §21 Linear functions, §39 Partial derivatives, differentiation

Extra Material

Extra material from [M] Munkres, Analysis on Manifolds

[Manifolds] [partition of unity]

Homework Assignments

(All problems are from [B] unless stated otherwise)

• #10 Due on Tue Apr 29:  from Munkres [M], §21 #2, §22 #1,3, §23 #3,5
• #9 Due Thur, Apr 10: 45.K, 45.M, 45.N, 45.P
• [#9 Partial Solutions]
• #8 Due Thur, Apr 3: 45.C, 45.D, 45.E, 45.F, 45.G, 45.H
• [#8 Partial Solutions]
• #7 Due Thur, Mar 27:  44.G, 44.H (a), 44.J (Use 43.R), 44.K, 44.O, 44.R
  [#7 Partial Solutions]
• #6 Due Tue, Mar 11: 43.B, 43.D, 43.L, 43.Q, 43.R, 43.S
  [#6 Partial Solutions]
• #5 Due Tue, Feb 25: 42.O, 42.P, 42.Q, 42.R, 42.S(a,c), 42.U
  [#5 Partial Solutions]
• #4 Due Thur, Feb 13: 41.O, 41.R, 41.W, 42.A(a-c), 42.B, 42.E
  [#4 Partial Solutions]
• #3 Due Thur, Feb 6: 41.D, 41.J, 41.K, 41.L, 41.U, 41.V
  [#3 Partial Solutions]
• #2 Due Thur, 1/30: 40.E, 40.L, 40.R, 40.S, 40.T, 40.U
  [#2 Partial Solutions]
• #1 Extended Tue, 1/28 (originally due Thur, 1/23):
  21.M, 39.E, 39.J, 39.T, 39.V, 39.W
  [#1 Partial Solutions]

Announcements

- There will be no class on Tue, Feb 18

Course Information

Schedule: TTh 12:00-1:15pm in MATH 215

Instructor: Arshak Petrosyan
Office Hours: TTh 1:30-2:30pm, or by appointment, in MATH 610

Course Description: MA44200 covers the foundations of real analysis in several variables, assuming the single variable notions of these concepts.
Prerequisite: MA44000

Textbook:
[B] R. Bartle, The Elements of Real Analysis, Second Edition, John Wiley & Sons, New York, 1975.
Additional text:
[R] W. Rudin, Principles of mathematical analysis, Third edition, McGraw-Hill, New York, 1976

Homework will be collected weekly on Thursdays. The assignments will be posted on this website at least one week prior the due date.

Exams: There well be two midterm exams and a  final project. The exact time and place will be specified as the time approaches.