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Spring
2021 MAT 319: Foundations of Analysis |
Spring
2021 MAT 320: Introduction to Analysis |
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Lecture
Schedule |
TuTh
11:30-12:50 (through 2/25: joint lectures online) |
TuTh
11:30-12:50 online |
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Instructor |
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Office
hours |
Tu,
Thu 1:00-2:00; MLC hour Tu 3:00-4:00 |
Tu,
Th 10:00-11:30 |
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Recitation |
MW
2:40-3:35; R01 in Engineering 145, R02 and R03 online |
MW
2:40-3:35 online |
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TA |
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Office
hours |
Rabah: Fri 10:30-11:30; MLC hours Th 5:30-6:30, Th 6:30-7:30 Zhukova: TBA |
TBA
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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.
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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.
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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, required for math majors.
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 analysis proofs. Students (especially those thinking
of going to graduate school) should take this as early as
possible.
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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.
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Textbook |
Robert
G. Bartle and Donald R. Sherbert, Introduction
to Real Analysis, 4th edition |
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Homework |
Weekly
problem sets will be assigned, and should be uploaded to
Blackboard. 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. |
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Grading |
Homework:
20%, Midterm I: 20%, Midterm II: 20%, Final: 40%. |
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Exam
Policy |
The two Midterms and the Final exam will be administered
synchronously through Blackboard. You will have to write
clearly, take a clear photo or scan of your work, and upload
it to Blackboard. Note that credit cannot be given for
manuscripts that are not legible and it is your
responsibility to upload a readable copy of your work in
time. In order to eliminatethe impact of internet
disruptions, sufficient extra time will be given for
uploading your work to Blackboard. The exams will be
proctored through Zoom by the instructor and the teaching
assistants. Every student who is taking an exam will join
the Zoom session of the class with camera enabled. Minor
internet disruptions during proctoring are permissible |
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Syllabus/schedule (subject to change)
All joint lectures through 2/25 are
online.
First recitation on Wed 2/3, second recitation Wed
2/10.
Recommendations on choosing MAT 319 vs MAT 320 will be made based
upon your performance on the first midterm and homework to that date.
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1. |
Joint
class: Introduction, sets and functions (Ntalampekos) |
Read
Chapter 1 |
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2. |
Joint
class: Well ordering property and mathematical induction;
finite countable and uncountable sets, Cantor's theorem (Ntalampekos) |
HW
due 2/10:
page 10, problems 5, 10, 16, 23, 24, |
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3. |
Joint
class: Field properties and order properties of real numbers;
the completeness property. (Ntalampekos) |
Read
pages 23-39 |
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4. |
Joint
class: Applications of the completeness property and intervals
(Ntalampekos) |
Read
pages 40-53 |
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5. |
Joint
class: Limit of a sequence and theorems about limits (Ebin) |
Read pages 54-69 HW
due 2/24: page 44, problems 3, 5, 7, 9, 13, 17; page
52, problems 3, 7, 9, 14 |
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6. |
Joint
class: Monotone sequences. (Ebin) |
Read
pages 70-77 |
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7. |
Joint
class: Subsequences and the Bolzano-Weierstrass theorem (Ebin) |
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8. |
Joint
class: Cauchy sequences. (Ebin) |
Read
pages 78-93 |
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9. |
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No
HW: prepare for the midterm |
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Joint
Midterm I,
March 2, 11:30-12:50 via Blackboard |
Practice midterm 1,
Practice midterm 2,
Practice
midterm 2 solutions |
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10. |
Everything from here on is for MAT320 only |
HW due 3/10: page 91, problems 1, 2, 4, 5, 10, 14 |
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11. |
Properly
divergent sequences and infinite series
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Read sections 3.6 and 3.7 |
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12. |
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13. |
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14. |
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15. |
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16. |
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17. |
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18. |
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19. |
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Second
midterm Possible topics for the exam: equivalence relations and equivalence classes; natural numbers, integers, rational numbers, algebraic numbers, real numbers (a complete ordered field) and complex numbers; absolute value;max, min sup and inf for subsets of the real numbers; Archimedian property; positive numbers have square roots; sequences and series and their properties; Bolzano Weierstrass theorem; inner product and norm for R^n; Schwartz inequality; metric spaces; R^n as a metric space; completeness for metric spaces; Compactness; Heine-Borel theorem; Is a bounded complete metric space necessarily compact; open and closed sets in a metric space; ratio test for convergence of series; harmonic series; convergence of alternating series; exponential function of a complex variable called E(z); E(z+w) = E(z) E(w); sine and cosine from E(ix); Continuity of a function from one metric space to another; |
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20. |
Read
pages
126
-143 |
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21. |
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No HW due November 12. review for exam |
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22. |
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HW due 11/19 17.15, 18.3, 18.5a, 18.9, 18.12b |
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23. |
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Read
pages 145-154 We did not do all of this in classs because it
is rather routine, but you are responsible for it |
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24. |
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HW due 11/24 19.1acde, 19.4, 19.7, 20.14, 20.17 |
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25. |
Read
pages 205-220 and 243-265 |
Practice final for 319
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