| Fall 2018 MAT 319: Foundations of Analysis | Fall 2018 MAT 320: Introduction to Analysis | |
| Schedule | TuTh 10:00-11:20 Library E4320 | TuTh 10:00-11:20 Math P-131 ( through 10/2: joint lectures in Library E4320) |
| Instructor | Lisa Berger | Samuel Grushevsky |
| Office hours | Tu 1.30-2.30, Th 12.30-1.30 in Math 4-100A, Tu 11.30-12.30 in Math P-143 | TuTh 11.30-12.30, W 1.30-2.30 in Math 3-109 |
| Recitation | MW 11.00-11.53 Library E4320 | MW 11.00-11.53 Math P-131 |
| TA | Prithviraj Chowdhury | Jack Burkart |
| Office hours | W 2-3, Th 1-2 in Math 5-125A, W 3-4 in MLC | MW 2.30-3.30 in Math 2-105, Tu 2.30-3.30 in MLC |
| 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 | Kenneth Ross Elementary Analysis: The Theory of Calculus, 2nd edition | |
| Homework | Weekly problem sets will be assigned, and must be handed in in person in 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. Announced and unannounced quizzes may be given during the lectures or during the recitation sections, and additional in-class work may be completed and graded. Missed quizzes and in-class work may not be made up.Your lowest homework or quiz 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 10/2 meet in Library E4320.
First recitation on Wed 8/29, second recitation Wed 9/5.
During joint lectures through 10/2, students with last names starting A-L attend recitation in Library E4330, students with last names M-Z attend recitation in Math P-131
Recommendations on choosing MAT 319 versus MAT 320 will be made based upon your performance on the first midterm and homework to that date.
| Tue 8/28 | 1. | Joint class: Introduction, motivation: what are real numbers? (Grushevsky) | Read pages 1-19 | |
| Thu 8/30 | 2. | Joint class: Properties of numbers; induction; concept of a field. (Berger) | HW due 9/5: 1.3, 1.4, 1.10, 1.12, 2.2, 2.5, 3.1, 3.4, 3.6 | |
| Tue 9/4 | 3. | Joint class: Completeness axiom for real numbers; Archimedean property. (Berger) | Read pages 20-38 | |
| Thu 9/6 | 4. | Joint class: Infinity, unboundedness. Intro to sequences. (Berger) | HW due 9/12: parts eghimr of: 4.1,4.2,4.3,4.4; and 4.8,4.10,4.11,4.12,4.14 | |
| Tue 9/11 | 5. | Joint class: Limit of a sequence. (Grushevsky) | HW due 9/19: 5.2, 5.6, 7.3, 7.4, 8.1ac | |
| Thu 9/13 | 6. | Joint class: Limit laws for sequences. (Grushevsky) | Read pages 39-55 | |
| Tue 9/18 | 7. | Joint class: Divergence to infinity, more formal proofs. (Berger) | HW due 9/26: 8.3, 8.6, 8.8, 8.10, 9.1, 9.3, 9.5, 9.12, 9.14 | |
| Thu 9/20 | 8. | Joint class: Monotone and Cauchy sequences. (Berger) | Read pages 56-65 | |
| Tue 9/25 | 9. | Joint class: Subsequences. (Grushevsky) | No HW due 10/3 because of the midterm | |
| Thu 9/27 | Joint Midterm I in Library E4320. | Practice midterm 1, Practice midterm 2, Practice midterm 2 solutions | ||
| Tue 10/2 | 10. | Joint class: Subsequences. (Grushevsky) | Read pages 66-78 HW due 10/10: 10.1, 10.2, 10.5, 10.8, 10.9, 11.2, 11.4, 11.5, 11.8, 11.9 | |
| Thu 10/4 | 11. | Lim sup, lim inf, series | Read sections 12,14,15,16 | |
| Tue 10/9 | No class, Fall break | |||
| Thu 10/11 | 12. | Series and series convergence tests | HW due 10/17: PDF | |
| Tue 10/16 | 13. | Decimal expansions | Read sections 16,17,18 | |
| Thu 10/18 | 14. | Continuity | HW due 10/24: HW 7 | |
| Tue 10/23 | 15. | Properties of continuous functions | Read sections 18, 20 | |
| Thu 10/25 | 16. | Limits | HW due 10/31: HW 8 | |
| Tue 10/30 | 17. | Topology in a metric space | Read section 13 | |
| Thu 11/1 | 18. | Topology in a metric space | Read section 21 | |
| Tue 11/6 | 19. | Topology in R^n | HW due 11/7:HW 9 | |
| Tue 11/6 | Midterm review session by Jack Burkart, MAT P-131, 7pm-8pm | |||
| Thu 11/8 | Midterm 2 | Practice midterm, and Another practice midterm | ||
| Tue 11/13 | 20. | Continuity in a metric space | HW due 11/15 (change!):HW 10 | |
| Thu 11/15 | 21. | Connectedness and uniform continuity | Read sections 22, 19 | |
| Tue 11/20 | 22. | Uniform continuity | short HW due 11/20: HW 11(a) | |
| Thu 11/22 | No class - Happy Thanksgiving! | |||
| Tue 11/27 | 23. | Power series | Read sections 23,24,25 | |
| Thu 11/29 | 24. | Uniform convergence | short HW due 11/28: HW 11(b) | |
| Tue 12/4 | 25. | Derivative and mean value theorem | Read sections 28,29,31 | |
| Thu 12/6 | 26. | Taylor's theorem | HW due 12/5: HW 12 | |
Practice final for 320.
Jack's review session Mon December 17, 7.00PM-8.30PM in Math 5-127
Sam's extra office hours in Math 3-109: Mon Dec 17, 10.00-11.30, Tue Dec 18 12.30-1.30
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