Mathematics 311 Homepage

311 INTRODUCTION TO ANALYSIS I (3+0) 3 credits

Continuation of MATH 310. Emphasizes proving theorems about series, uniform convergence, functions of several variables: limits, continuity, differentiation, extrema, integration, implicit and inverse function theorems. Prerequisite(s): MATH 310 with a "C" or better. Corequisite(s): MATH 330.

Instructor  Course Section                       Time
------------------------------------------------------------------------
Eric Olson  1001 Math 311 INTRO TO ANALYSIS II   MWF 10:00-10:50am AB206

Course Information

Instructor:

Eric Olson

email:

Please contact me through WebCampus

Office:

DMS 238 MWF from 11am to noon

Homepage:

https://fractal.math.unr.edu/~ejolson/311/

Live Stream:

If you can't come to class due to sickness, quarantine or other reasons, please join via the Zoom link in WebCampus.

Primary Texts:

Gerald Folland, Advanced Calculus, Second Edition, Pearson. (pdf)

Other Reading:

Eric's Lecture notes from Math 310, Spring 2024.

Frank Dangello, Michael Syfried, Introductory Real Analysis, Houghton Mifflin Company, 2000.

Joseph Taylor, Foundations of Analysis, Twelfth Edition, American Mathematical Society, 2012.

Jay Cummings, Real Analysis, A Long-Form Mathematics Textbook, 2019.

Walter Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill, 1976.

Robert C. Wrede, Murray Spiegel, Schaum's Outline of Advanced Calculus, Second Edition, McGraw-Hill.

Student Learning Outcomes

• Write cogent proofs using different methods like direct proof, indirect proof, proof by contradiction, proof by induction.
• Demonstrate an understanding of the algebraic structure and the topology of Euclidean space.
• Apply theorems about differentiability and integrability of vector functions of several variables.
• Test infinite series for convergence.

Class Handouts

Homework

HW1 due Sept 13 (solutions)

Turn in (page 33) 1.6#4

Practice (page 29) 1.5#5, 1.5#7, (page 33) 1.6#1ab, 1.6#6, (page 38) 1.7#6, 1.7#8

HW2 due Sept 27 (solutions)

Turn in (page 74) 2.3#2abc

Practice (page 41) 1.8#4, 1.8#5, (page 52) 2.1#2, 2.1#9, (page 61-62) 2.2#1abc, 2.2#8, (page 77) 2.5#5

HW3 due Oct 4 (solutions)

Turn in (page 84) 2.6#10

Practice (page 41) 1.8#1, (page 84) 2.6#2, 2.6#11, (page 94-95) 2.7#5, 2.7#10 (page 100) 2.8#2

HW4 due Nov 1 (solutions)

Turn in (page 133) 3.3#5ab

Practice (page 119) 3.1#2, (page 125) 3.2#4, (page 139) 3.4#4abcde

HW5 due Nov 15 (solutions)

Turn in (page 167) 4.2#7

Practice (page 158) 4.1#6, (page 167) 4.2#3,4,5 (page 176) 4.3#5abc

Quizzes

• Quiz 1: Change of Variables Theorem
• Quiz 2: Definition of Line Integral
• Quiz 3: Implicit Function Theorem
• Quiz 4: Green's Theorem
• Quiz 5: The Integral Test

Lecture Notes

• Lecture 1: Monotone Convergence Theorem
• Lecture 2: Bolzano Weierstrass in Rⁿ
• Lecture 3: Compact Sets

• Lecture 4: Connected Sets
• Lecture 5: More Connected Sets

• Lecture 6: Arcwise Connected
• Lecture 7: Derivatives
• Lecture 8: More Derivatives

• Lecture 9: The Chain Rule
• Lecture 10: Mean Value Theorem
• Lecture 11: Implicit Differentiation

• Lecture 12: Taylor's Theorem
• Lecture 13: Estimate the Remainer
• Lecture 14: The Hessian

• Lecture 15: Lagrange Multipliers
• Lecture 16: Implicit Functions
• Lecture 17: Curves in the Plane

• Lecture 18: More on Curves
• Lecture 19: Surfaces and Higher Dimensions

• Lecture 20: Inverse Function Theorem
• Lecture 21: Implicit Function for Systems
• Lecture 22: Partition of [a,b]

• Lecture 23: Integration on R
• Lecture 24: Fundamental Theorem of Calculus

• Lecture 25: Sets of Zero Content
• Lecture 26: Integration on R²
• Lecture 27: Mean Value Theorem for Integrals

• Lecture 28: Change of Variables in R
• Lecture 29: Linear Change of Variables
• Lecture 30: Matrix Transformations

• Lecture 31: Double and Iterated Integrals
• Lecture 32: Arc Length

• Lecture 33: Line Integrals
• Lecture 34: Green's Theorem
• Lecture 35: Heine-Borel Theorem

• Lecture 36: Partitions of Unity
• Lecture 37: Proof of Green's Theorem

• Lecture 38: Proof Green's Theorem
• Lecture 39: Infinite Series
• Lecture 40: Ratio and Root Tests

• Lecture 41: Alternating Series

Announcements

[12-Dec-2024] Sample Final Exam

[13-Dec-2024] Final Exam

• Everything on the review for the Midterm.
• Material from all homework assignments and quizzes.
• The following additional definitions:
  • A smooth curve (page 123)
  • A partition and a refinement of a partition (page 148)
  • Upper and lower Rieman sums (page 148)
  • Upper and lower integrals (page 149)
  • A set of zero content (page 154 and 161)
  • Characteristic or indicator function χ_S (page 160)
  • A set to be Jordan measurable (page 162)
  • 5.7 Line integral of a scalar function (page 215)
  • 5.9 Scalar-valued line integral of a vector field (page 217)
  • Rectifiable curve (page 219)
  • x-simple, y-simple and simple regions (page 223)
• Statement without poof of the additional theorems:
  • 3.9 The Implicit Function Theorem for Systems (page 118)
  • 3.18 The Inverse Mapping Theorem (page 137)
  • 4.26 Fubini's Theorem on Iterated Integrals (page 169)
  • 4.41 Change of Variables for Multiple Integrals (page 183)
  • 5.12 Green's Theorem General Statment (page 223)
  • 6.19 Rearrangment of Absolutely Convergent (page 298)
  • 6.19 Rearrangment of Conditionally Convergent (page 298)
• Proofs of the following additional theorems:
  • 3.1 Implicit Function Theorem for One Equation (page 114)
  • 4.10 Bounded Monotone Functions are Integrable (page 152)
  • 4.11 Continuous Functions are Integrable (page 152)
  • 4.15 The Fundamental Theorem of Calculus (page 155)
  • 4.24 The Mean Value Theorem for Integrals (page 166)
  • 5.11 C¹ Curves are Rectifiable (page 220)
  • 5.12 Green's proof for a Simple Region (page 223)
  • 6.3 Geometric Series Formula (page 281)
  • 6.8 The Integral Test (page 285)
  • 6.9 The p test for ∑₁^∞ n^−p (page 286)
  • 6.12 The Limit Comparison Test (page 288)
  • 6.13 The Ratio Test (page 289)
  • 6.14 The Root Test (page 290)
  • 6.17 Absolute Convergent implies Convergent (page 295)

[9-Dec-2024] Quiz 5

[6-Dec-2024] Quiz 4

Note that the way I did problem 5.2#3 was to apply Green's Theorem and then switch to polar coordinates. There may be other ways.

[4-Dec-2024] Quiz 3

[2-Dec-2024] Quiz 2

[27-Nov-2024] Quiz 1

[8-Oct-2024] Sample Midterm

[7-Oct-2024] Homework 3 Solutions

[11-Oct-2024] Midterm

• Material from HW1, HW2 and HW3.
• The following definitions:
  • The open ball B(r,a) of radius r about a.
  • An interior point and a boundary point.
  • A bounded set, a convex set and a compact set.
  • A connected set and a disconnected set.
  • A function f to be of class C^k.
  • Multi-index notation.
  • The definition of derivative for f:Rⁿ→R.
  • The definition of derivative for F:Rⁿ→R^m.
• Statement without proof of the following theorems:
  • Taylor's theorem in Rⁿ using multi-index notation.
  • Bolzano-Weierstrass theorem in Rⁿ.
  • The Heine-Borel Theorem in Rⁿ.
  • Spectral theorem for symmetric H in R^n×n.
  • The chain rule for f(g(t)) where f:Rⁿ→R and g:R→Rⁿ.
  • The conditions on ∂_jf that imply f is differentiable at a.
• Proofs of the following theorems:
  • Every bounded monotone sequence in R is convergent.
  • Bolzano-Weierstrass theorem in Rⁿ.
  • A continuous function on a compact set is uniformly continuous.
  • A continuous function on a compact set attains its maximum.
  • If S is connected and f continuous then f(S) is connected.
  • If S is open and connected then S is arcwise connected.

[6-Oct-2024] Homework 2 Solutions

[24-Sep-2024] Homework 1 Solutions

[19-Jan-2024] Welcome Fall 2024

This section of Math 311 is in person. I will maintain an online archive of course materials including lecture notes, assignments and other announcements in case you miss a class or want to compare your notes.

Tentative Course Schedule

#   Date      Chapter    Topic
------------------------------------------------------------------------
1   Aug 26    1.5        Completeness
2   Aug 28    1.5        Bolzano Weierstrass Theorems
3   Aug 30    1.6        Compactness

    Sep 02    ***        Labor Day
4   Sep 04    1.7        Connectedness
5   Sep 06    1.8        Uniform Continuity

6   Sep 09    2.1        Differentiability in One Variable
7   Sep 11    2.2        Differentiability in Several Variables
8   Sep 13    2.3        The Chain Rule

9   Sep 16    2.4        The MVT and Implicit Functions
10  Sep 18    2.5        Implicit Functions
11  Sep 20    2.6        Higher-Order Partial Derivatives

12  Sep 23    2.7        Taylor's Theorem
13  Sep 25    2.8        Critical Points
14  Sep 27    2.9        Extreme Values

15  Sep 30    2.10       Derivatives of Vector-Valued Functions
16  Oct 02    3.1        The Implicit Function Theorem
17  Oct 04    3.2        Curves in the Plane

18  Oct 07    3.2        Curves in the Plane (continued)
19  Oct 09    3.3        Surfaces and Curves in Space
20  Oct 11               Midterm

21  Oct 14    3.4        Inverse Mapping Theorem
22  Oct 16    B.2        The Implicit Function Theorem
23  Oct 18    B.2        The Implicit Function Theorem (continued)

24  Oct 21    4.1        Integration on the Line
25  Oct 23    4.1        Integration on the Line (continued)
    Oct 25    ***        Nevada Day

26  Oct 28    4.2        Integration in Higher Dimensions
27  Oct 30    4.2        Integration in Higher Dimensions (continued)
28  Nov 01    4.2        The Mean Value Theorem for Integrals

29  Nov 04    4.4        Change of Variables on the Line
30  Nov 06    4.4        Change of Variables for Multiple Integrals
31  Nov 08    4.4        Matrix Transformations

    Nov 11    ***        Veteran's Day
32  Nov 13    5.1        Arc Length
33  Nov 15    B.4        Double and Iterated Integrals

34  Nov 18    5.1        Line Integrals
35  Nov 20    5.2        Green's Theorem
36  Nov 22    B.1        The Heine-Borel Theorem

37  Nov 25    B.7        Partitions of Unity
38  Nov 27    B.7        Proof of Green's Theorem
    Nov 29    ***        Thanksgiving Family Day

39  Dec 02    6.1/6.2    Infinite Series and Nonnegative Terms
40  Dec 04    6.3        Absolute and Conditional Convergence
41  Dec 06    6.4        More Convergence Tests

42  Dec 09               Review

*** Final exam Monday, Friday 13 from 10:15am-12:15pm in AB206

Grading

     Midterm                  100 points
     5 Quizzes                  5 points each
     5 Homework Assignments     5 points each
     Final Exam               100 points
    ------------------------------------------
                              250 points total

    Grade        Minimum Percentage
      A                 90 %
      B                 80 %
      C                 70 %
      D                 60 %

Quiz and Exam Schedule

Course Policies

Communications Policy

Late Policy

Plagiarism

Diversity

Statement on Academic Success Services

Equal Opportunity Statement

Statement on Audio and Video Recording

Academic Conduct

Final Exam