fm18

Further Mathematics for Economics, M1314.001500, Fall 2018

Practical message: it will not be possible to sign up for this course with the override / cho an ji form. Because of the large number of emails I have received about this issue, I have not been able to reply to everyone.

Course web page

https://sites.google.com/site/oyvindthomassen/fm18

Time and place

Monday 09:30~12:20, Building 83 Room 401

No classes on the following days

24 September (Chuseok), 15 October (SNU Anniversary)

Exams

Mid-term exam, solution

Final exam, solution

Content / purpose

Our topic is often called (undergraduate) real analysis. But many of the concepts will be familiar to students who know some calculus, and it would not be wrong to say that this is a course in (rigorous) calculus. The main difference from a typical calculus course is that our focus is on proving theorems, rather than on calculating with numbers. The course should help you with the following:

Learn calculus in depth. (Caveat: we will not talk about integration at all.)
Become familiar with abstract reasoning and mathematical proofs.
Prepare for taking more advanced courses in economics or mathematics.

(There is no economics in this course. The "for economics" in the course title simply means that it covers mathematics that is useful for economists.)

Prerequisites

Mathematics for Economists (course 212.214) or similar (basic calculus and linear algebra). In fact very little background knowledge is needed. Any experience with mathematical proofs is helpful (but not strictly necessary).

Grading

Mid-term exam 40%, final exam 60%.

Reading:

Rudin: Principles of Mathematical Analysis, 3rd ed.

The book uses the numbering system chapter.paragraph, where the paragraph could be a definition, theorem, example, etc. The plan is to cover the following paragraphs (main topics in parentheses):

Chapter 2: 2.1 - 2.2, 2.15 - 2.42 (Metric spaces, compact sets)

Chapter 3: until 3.14 (Convergent sequences)

Chapter 4: until 4.20 (Limits of functions, continuity)

Chapter 5: all (Differentiation)

Chapter 9: until 9.29 (Functions of several variables) [Depending on how long it takes to cover chapters 2-5, we might drop some or all of chapter 9].

The campus bookstore sells the book. You might be able to find a pdf version of it online. A solutions manual by Roger Cooke is available here: http://digital.library.wisc.edu/1793/67009. You can find alternative solutions, discussions of many theorems etc. with google.

Lecture plan

(See list below of paragraphs that have been covered so far.)

The plan will be updated during the semester.

3 September, Lecture 1

Preliminaries: upper and lower bounds, absolute value, inner product, norm. (Lecture notes for this material here.)

2.1 - 2.2

2.15 - 2.18

2.23

The Preliminaries note and paragraphs 2.1-2.2, 2.16-2.17 are left for self study. Definitions 1.4 and 1.5 in the note and the fact that the real numbers have the least-upper-bound property will be needed soon (e.g. in 2.28). Make sure you understand why Theorem 3.2 in the Preliminaries note implies that |x-y| satisfies the requirements for a metric, as set out in Rudin's Definition 2.15. The definitions in 2.1-2.2 are important, but we will not use them quite yet, so you can refer back to them later.

10 September, Lecture 2

2.19 - 2.22

2.24 - 2.28

2.21-2.22 are left for self study. See 2.9 if the set notation in 2.22 is unfamiliar.

Homework for Lecture 3: write an alternative proof of Theorem 2.27 (a). Show directly that the closure of E is closed, instead of showing that the complement of the closure is open. Draw an illustration.

17 September, Lecture 3

2.29 - 2.34

24 September No lecture

1 October, Lecture 4

2.35 - 2.40

8 October, Lecture 5

2.41 - 2.42

2.7

3.1 - 3.2a

3.2b,c,d - 3.4 self study

3.5 - 3.6

3.8

15 October No lecture

22 October Mid-term exam

29 October, Lecture 6

3.7

3.9 - 3.12

3.13 - 3.14 self study

4.1

4.2 - 4.4 self study

4.5 - 4.6

4.7 self study

5 November, Lecture 7

4.8

4.9 - 4.10 self study

4.12 read on your own

4.13 - 4.17

12 November, Lecture 8

4.17 - 4.19

5.1 - 5.2

5.3 - 5.5, 5.14, 5.16 (only first two paragraphs on p. 112) read on your own - no need to study carefully

5.7 - 5.10

5.15

19 November, Lecture 9

9.1 - 9.3

26 November, Lecture 10

9.4 - 9.8

3 December, Lecture 11

9.10 (read on your own)

9.11

9.13

9.14 (self study)

9.16 - 9.17

9.20 - 9.21 (we will skip the proof of 9.21)

9.22 - 9.23

9.26 - 9.27

10 December Final exam

List of material covered so far

(s) means that you should study this material on your own although I did not talk about it in the lectures.

preliminary notes (s)

2.1 - 2.2 (s)

2.15

2.16 - 2.17 (s)

2.18 - 2.20

2.21 - 2.22 (s)

2.23

2.24 a,c

2.24 b,d (s)

2.25 - 2.42

2.7

3.1 - 3.2a

3.2b,c,d - 3.4 (s)

3.5 - 3.12

3.13 - 3.14 (s)

4.1

4.2 - 4.4 (s)