Spring 2020 MAT 319: Foundations of AnalysisSpring 2020 MAT 320: Introduction to Analysis
ScheduleTuTh 11:30-12:50 Earth&Space 131 (through 3/3: joint lectures in Earth&Space 131)TuTh 11:30-12:50 Math P-131
InstructorLjudmila KamenovaRobert Hough
Office hoursW 1-3pm in Math Tower 3-115, F 2:30-3:30pm in the MLCF 9-11am in Math 4-118, F 3-4pm in MLC
RecitationMW 10:00-10:53 Earth & Space 69, Frey Hall 309MW 10:00-10:53 Math P-131
TADaniel Brogan, Christina KarafylliaSaman Habibi Esfahani
Office hoursBrogan M 1-2pm, Tu 10-11am in Math Tower S-240A, F 10-11am in MLC,
Karafyllia MW 11am-12pm in Math Tower 3-102, W 1-2pm in MLC
M 8-10am in Math Tower 3-106, M 11am-12pm in the 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
TextbookBartle 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.
GradingHomework: 20%, Midterm I: 20%, Midterm II: 20%, Final: 40%.

Syllabus/schedule (subject to change)
All joint lectures through 3/3 meet in Earth & Space P-131.
First recitation on Wed 1/29.
During joint lectures through 3/3, students with last names starting A-H attend recitation in Earth & Space 69, students with last names starting I-R attend recitation in Frey Hall 309, students with last names S-Z attend recitation in Math P131.

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 1/281. Joint class: Sets, induction (Hough)Read Sections 1.1-1.3
Thu 1/302. Joint class: Infinite sets. (Kamenova)HW due 2/5: p.10 #5, 6, 9, p.15 #1, 2, 9, 18, p.22 #4, 9, 12
Tue 2/43. Joint class: Algebraic properties of the real numbers. (Hough)Read Sections 2.1-2.3
Thu 2/64. Joint class: Completeness of the real numbers. (Kamenova)HW due 2/12: p.30 #8, 9, 15, p.35 #3, 4, 16, p.39 #1, 6, 10, 11
Tue 2/11 5. Joint class: Applications of the supremum property (Hough)Read Sections 2.4-2.5
Thu 2/136. Joint class: Intervals. (Kamenova)HW due 2/19: p.45 #2, 4, 7, 11, 13, 14, 19, p.50 #3, 6, 14
Tue 2/187. Joint class: Sequences and limits. (Hough)Read Sections 3.1-3.2
Thu 2/208. Joint class: Limit theorems. (Hough)HW due 2/26: p.59 #6, 8, 9, 16, 17, 18, p. 68 #6, 13, 16, 18
Tue 2/259. Joint class: Monotone sequences. (Kamenova)Read Sections 3.3-3.4
Thu 2/27 Joint Midterm I in Earth & Space 131. Practice Midterm 1, Practice midterm 1 solutions
Tue 3/310.Subsequences and the Bolzano-Weierstrass Theorem (Kamenova) Read Section 3.4

The following syllabus below is only for MAT 320, in Math P131
Thu 3/511. Cauchy's criterion, divergent sequencesRead Sections 3.5-3.6. HW due 3/11: p.77 #1, 7, 9, 10, p.84 #1, 3, 9, p.91 #5,8, p.93 #8
Tue 3/1012. Infinite series, limits of functionsRead Sections 3.7, 4.1-4.3
Thu 3/1213. Limit theoremsHW due 3/25: p.100 #3, 9, 11, p.110 #1, 6, 16, p.116 #2, 4, 5, p.123 #11
Tue 3/17 Spring Break
Thu 3/19 Spring Break
Tue 3/2414. Continuous functions Read Sections 5.1-5.4
Thu 3/2615. Uniform continuity HW due 4/1: p.129 #7, 12, p.133 #2, 7, 12, p.140 #1, 6, 11, p.148 #7, 16
Tue 3/3116. Monotone functions, inverse functionsRead Sections 5.5-5.6, 6.1-6.2
Thu 4/217. Derivative, Mean Value TheoremHW due 4/8: p.152 #5, p.160 #1, 5, 8, p.170 #7, 10, p.179 #5, 6, 11, 13
Tue 4/718. L'Hospital's rule, Taylor's TheoremRead Sections 6.3-6.4
Thu 4/9 Midterm II in class
Tue 4/1419.Riemann integralRead Sections 7.1-7.4
Thu 4/1620.Fundamental theorem of calculusHW due 4/22: p.205 #3, 8, 15, p.215 #16, 17, 18, p.223 #13, 21, p.233 #8, 15
Tue 4/2121.Pointwise and uniform convergenceRead Sections 8.1-8.4, 9.1-9.2
Thu 4/2322.Exponential, log and trig functions, absolute convergenceHW due 4/29: p.247 #19, 23, p.252 #4, 9, 18, p.259 #5, 9, p.265 #8, p.269 #3, 14, p.262 #3
Tue 4/2823.Series of functionsRead Sections 9.3-9.4, 10.1-10.2, Appendices C, E
Thu 4/3024.Lebesgue integralHW due 5/6: p.280 #9, 12, 13, p.286 #6, 14, 17, p.300 #6, 8, 9, 15
Tue 5/525.Improper and infinite integralsRead Sections 10.3-10.4, 11.1-11.4
Thu 5/726.Metric spaces
Final Exam: Tuesday May 19, 11.15AM-1.45PM

Review session by Saman: TBD.

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