| Spring 2022 MAT 319: Foundations of Analysis | Spring 2022 MAT 320: Introduction to Analysis | |
| Schedule | TuTh 11:30am-12:50pm S B Union 103-02 (through 3/1: joint lectures in S B Union 103-02) | TuTh 11:30am-12:50pm Earth & Space 131 (through 3/1: joint lectures in S B Union 103-02) |
| Instructor | Ljudmila Kamenova | Raanan Schul |
| Office hours | LK's web card | RS's web card |
| Recitation | MW 2:40-3:35pm Earth & Space 181, Physics P-117 | MW 2:40-3:35pm Earth & Space 183 |
| TA | Alexandra Viktorova, Mohamed El Alami | Owen Mireles Briones |
| Office hours | Viktorova's web card, El Alami's web card | Mireles Briones's web card |
| Description | A careful study of the theory underlying topics in one-variable calculus, with an emphasis on those topics arising in high school calculus. The real number system. Limits of functions and sequences. Differentiations, integration, and the fundamental theorem. Infinite series. | A careful study of the theory underlying calculus. The real number system. Basic properties of functions of one real variable. Differentiation, integration, and the inverse theorem. Infinite sequences of functions and uniform convergence. Infinite series. |
| Overview | The purpose of this course is to build rigorous mathematical theory for the fundamental calculus concepts, sequences and limits, continuous functions, and derivatives. We will rely on our intuition from calculus, but (unlike calculus) the emphasis will be not on calculations but on detailed understanding of concepts and on proofs of mathematical statements. | An introductory course in analysis, it provides a closer and more rigorous look at material which most students encountered on an informal level during their first two semesters of Calculus. Students learn how to write proofs. Students (especially those thinking of going to graduate school) should take this as early as possible. |
| Prerequisites |
C or higher in MAT 200 or permission of instructor; C or higher in one of the
following: MAT 203, 205, 211, 307, AMS 261, or A- or higher in MAT 127, 132, 142,
or AMS 161. Math majors are required to take either MAT 319 or MAT 320 | |
| Textbook | Bartle and Sherbert Introduction to Real Analysis, 4th edition | |
| Homework | Weekly problem sets will be assigned, and collected in the Wednesday recitation. The emphasis of the course is on writing proofs, so please
try to write legibly and explain your reasoning clearly and fully. You are encouraged to discuss the homework problems with others, but your write-up must be your own work.
Late homework will never be accepted, but under documented extenuating circumstances the grade may be dropped. Your lowest homework grade will be dropped at the end of the class. | |
| Grading | Homework: 20%, Midterm I: 20%, Midterm II: 20%, Final: 40%. | |
Syllabus/schedule (subject to change)
All joint lectures through 3/1 meet in S B Union 103-02.
The first recitation is on Wed 1/26.
Recommendations on choosing MAT 319 vs MAT 320 will be made based upon your performance on the first midterm and homework to that date.
Discussion Board: If you log on to
Blackboard
then you can use the
Discussion Board.
| Tue 1/25 | 1. | Joint class: Sets, induction (Schul) | Read Sections 1.1-1.3 and do Homework #1 on Blackboard | |
| Thu 1/27 | 2. | Joint class: Infinite sets. (Kamenova) | HW due 2/2: p.10 #5, 6, p.15 #2, 9, p.22 #4 | |
| Tue 2/1 | 3. | Joint class: Algebraic properties of the real numbers. (Schul) | Read Sections 2.1-2.3 | |
| Thu 2/3 | 4. | Joint class: Completeness of the real numbers. (Kamenova) | HW due 2/9: p.30 #8, 15, p.35 #3, 16, p.39 #6 | |
| Tue 2/8 | 5. | Joint class: Applications of the supremum property (Schul) | Read Sections 2.4-2.5 | |
| Thu 2/10 | 6. | Joint class: Intervals. (Kamenova) | HW due 2/16: p.45 #2, 4, 7, p.52 #3, 6 | |
| Tue 2/15 | 7. | Joint class: Sequences and limits. (Schul) | Read Sections 3.1-3.2 | |
| Thu 2/17 | 8. | Joint class: Limit theorems. (Schul) | HW due 3/2: p.61 #6, 8, 9, p. 69 #6, 16 | |
| Tue 2/22 | 9. | Joint class: Monotone sequences. (Kamenova) | Read Sections 3.3-3.4 | |
| Thu 2/24 | Joint Midterm I in SB Union 103-02. | |||
| Tue 3/1 | 10. | Joint class: Subsequences and the Bolzano-Weierstrass Theorem (Schul) | Read Section 3.4 | |
| Thu 3/3 | 11. | Joint class: Cauchy's criterion (Kamenova) | Read Section 3.5. HW due 3/9: p.77 #1, 7, p.84 #1, p.91 #5, 8 | |
| Tue 3/8 | 12. | Divergent sequences | Read Sections 3.6, 3.7. |
| Thu 3/10 | 13. | Infinite series | HW due 3/23: p.93 #5, 8, p.100 #3, 9, 11 |
| Tue 3/15 | Spring Break | ||
| Thu 3/17 | Spring Break | ||
| Tue 3/22 | 14. | Limits of functions | Read Sections 4.1,4.2. |
| Thu 3/24 | 15. | Limit theorems | HW due 3/30: p.110 #1, 6, p.116 #2, 3, 4 |
| Tue 3/29 | 16. | Continuous functions and combinations of continuous functions | Read Sections 5.1-5.3. |
| Thu 3/31 | 17. | Continuous functions on intervals | HW due 4/13: p.129 #7, 12, p.133 #2, 7, p.140 #6 |
| Tue 4/5 | 18. | Uniform continuity | Read Section 5.4. |
| Thu 4/7 | Midterm II in class | ||
| Tue 4/12 | 19. | Monotone and inverse functions | Read Sections 5.6, 6.1. |
| Thu 4/14 | 20. | Derivative | HW due 4/20: p.148 #7, p. 160 #1, 5, p.171 #7 |
| Tue 4/19 | 21. | Mean Value Theorem | Read Sections 6.2, 6.3. |
| Thu 4/21 | 22. | L'Hospital's rule | HW due 4/27: p.179 #6, 13, p.187 #5, 6, 7 |
| Tue 4/26 | 23. | Taylor's Theorem | Read Sections 6.4, 7.1. |
| Thu 4/28 | 24. | Riemann integral | HW due 5/4: p.196 #1, 4, 5, p.207 #8, 15 |
| Tue 5/3 | 25. | Riemann integrable functions | Read Sections 7.2, 7.3. |
| Thu 5/5 | 26. | Fundamental theorem of calculus |
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