| Fall 2022 MAT 319: Foundations of Analysis | Fall 2022 MAT 320: Introduction to Analysis | |
| Schedule | TuTh 1:15-2:35 SB Union 103-02 (through 9/22: joint lectures in SB Union 103-02) | TuTh 1:15-2:35 Physics P-112 |
| Instructor | Dimitrios Ntalampekos | Robert Hough |
| Office hours | TBD in Math Tower 3-117 | F 9-11am in Math 4-118, F 3-4pm in MLC |
| Recitation | MW 11:45-12:40 Frey Hall 309, 313, 217 | MW 11:45-12:40 Physics P116 |
| TA | Matthew Huynh, Emily Schaal | Astra Kolomatskaia |
| Office hours | TBD | TBD |
| 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 |
B or higher in MAT 200 or MAT 250 or permission of instructor; C or higher in one of the following: MAT 203, 211, 220, 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 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 9/22 meet in SB Union 103-02.
First recitation on Wed 8/24.
Recommendations on choosing MAT 319 vs MAT 320 will be made based upon your performance on the first midterm and homework to that date.
| Tue 8/23 | 1. | Joint class: Sets, functions induction (Hough) | Read Sections 1.1-1.2 | |
| Thu 8/25 | 2. | Joint class: Infinite sets, algebraic and order properties of the reals. (Ntalampekos) | Read Sections 1.3, 2.1 HW due 8/31: p.10 #6, 15, p.15 #1, 9, 18, p.22 #4, 12, p.31 #9, 21, 26 | |
| Tue 8/30 | 3. | Joint class: Absolute value on the Reals, Completeness of the Reals. (Hough) | Read Sections 2.2-2.3 | |
| Thu 9/1 | 4. | Joint class: Applications of the supremum. (Ntalampekos) | Read Sections 2.4 HW due 9/7: p.35 #2,4,17, p.39 #4, 6, 11, p.44 #4, 12, 13 | |
| Tue 9/6 | 5. | Joint class: Intervals, sequences and limits (Hough) | Read Sections 2.5, 3.1 | |
| Thu 9/8 | 6. | Joint class: Limit theorems, monotone sequences. (Ntalampekos) | Read Sections 3.2-3.3 HW due 9/14: p.52 #3, 6, p.61 #5, 11, 12, p.69 #6, 13, 14, p.77 #9, 10 | |
| Tue 9/13 | 7. | Joint class: Subsequences and Bolzano-Weierstrass. (Hough) | Read Section 3.4 | |
| Thu 9/15 | 8. | Joint class: The Cauchy criterion, divergent sequences. (Ntalampekos) | Read Sections 3.5-3.6 HW due 9/21: p.84 #3, 18, 19, p.91 #4, 6, 10, 13, p.93 #1, 6, 10 | |
| Tue 9/20 | 9. | Joint class: Infinite series. (Hough) | Read Section 3.7 | |
| Thu 9/22 | Joint Midterm I in SB Union 103-02. Spring 2020 Midterm 1, Spring 2020 solution, Fall 2022 solutions | |||
| Tue 9/27 | 10. | Limit of functions, limit theorems | Read Sections 4.1-4.2. |
| Thu 9/29 | 11. | Extensions of limits, continuous functions | Read Sections 4.3, 5.1. HW due 10/5: p.100 #3, 15, p.110 #3, 6, p.116 #10, 15, p.123 #8, p.129 #7, 11, 12 |
| Tue 10/4 | 12. | Combinations of continuous functions, continuous functions on intervals | Read Sections 5.2-5.3 |
| Thu 10/6 | 13. | Uniform continuity, gauges | Read Sections 5.4-5.5 HW due 10/12: p.134 #8, 9, 14, p.140 #6, 13, 14, p.148 #15, 16, p.152 #5, 9 |
| Tue 10/11 | Fall Break | ||
| Thu 10/13 | 14. | Monotone and inverse functions, the derivative | Read Sections 5.6, 6.1 HW due 10/19: p.160 #5, 6, 12, p.170 #14, 15, 17 |
| Tue 10/18 | 15. | Mean value theorem, L'Hospital's Theorem | Read Sections 6.2-6.3 |
| Thu 10/20 | 16. | Taylor's theorem, Riemann integral | Read Sections 6.4, 7.1 HW due 10/26: p.179 #5, 7, 13, 14, 15, p.187 #1, p.196 #3, 11 |
| Tue 10/25 | 17. | Riemann integrable functions, Fundamental Theorem of Calculus | Read Sections 7.2-7.3 |
| Thu 10/27 | Midterm 2 in Physics P-112 Practice Midterm 2 | ||
| Tue 11/1 | 18. | Darboux integral, numerical integration | Read Sections 7.4-7.5 |
| Thu 11/3 | 19. | Pointwise and uniform convergence, interchange of limits | Read Sections 8.1-8.2 HW due 11/9: p.215 #10, 15, 16, p.223 #17, 20, p.233 #9, 15, p.247 #20, 24, p.252 #7 |
| Tue 11/8 | 20. | Exponential and logarithm functions, trigonometric functions | Read Sections 8.3-8.4 |
| Thu 11/10 | 21. | Absolute convergence | Read Sections 9.1-9.2 HW due 11/16: p.259 #4, 5, 9, p.265 #6, p.270 #6, 8, 13, 14, p.276 #3, 17 |
| Tue 11/15 | 22. | Tests for non-absolute convergence, series of functions | Read Sections 9.3-9.4 |
| Thu 11/17 | 23. | Definitions and properties of generalized Riemann integral, Lebesgue integral | Read Sections 10.1-10.2 |
| Tue 11/22 | 24. | Infinite integrals, convergence theorems | Read Sections 10.3-10.4 HW due 11/30: p.280 #7, 12, p.286 #2, 5, 16, 19, p.300 #6, 11, 23, 24 |
| Tue 11/29 | 25. | Open, closed sets, compact sets | Read Sections 11.1-11.2 |
| Thu 12/1 | 26. | Continuous functions, metric spaces | Read Sections 11.3-11.4 |
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