Mathematical Analysis

Daily Schedule Term 4

Monday, Jan. 13 — Reading: Chapter 1, pp. 3-10, postulates (P1)-(P12) for the real numbers — How to read mathematics: (1) when the author “leaves something for the reader” stop and do it, and (2) keep a constant eye on what has so far been postulated (as an axiom), defined, or proven (in a theorem, a lemma, or a corollary) and be very careful not to use things that seem obvious but have not yet been postulated, defined, or proven — In-class: we got started on the end-of-chapter problems
Thursday, Jan. 16 — Problem Set 1: Write up Problems 1-3 (which have many subparts) — For all of the first three problems, it would be best to work in a two-column format where you do the work in the left column, and enumerate which postulates you used in the right column — To keep the verbosity manageable, let’s agree that we don’t have to repeatedly note that a-b means a+(-b), that a/b means a·b^-1, and that -(a·b)=(-a)·b=a·(-b) — Second Reading: Finish Chapter 1, and read the first three pages of Chapter 2 — In-class: scrutinizing what is meant by equality, proving (-(-a))=a, and proving that the additive inverse is unique

Monday, Jan. 20 — Problem Set 2: Let’s do Problems 5-7 (still in Chapter 1) — NB: to keep the verbosity of your proofs manageable, you may start using everything you have previously proven (but be sure you aren’t using things we haven’t proven!) — As an example, you don’t have to keep re-proving (-(-a))=a every time you use it, or that the multiplicative inverse (when it exists) and additive inverse are unique, because you know how that goes now — Reading: Finish Chapter 2 — In-class: We did inequality proofs
Thursday, Jan. 23 — No new reading — Problem Set 3: Just Problems 1 and 2 from Chapter 2 — In-class: How about we do a selection of the parts from Problem 3 and 4? — Are there other end-of-chapter problems that particularly interest you? — I find lots of them interesting-looking, such as 13, 14, and 15 — Avoid problems marked with an asterisk unless you are finding the others to be easy

Monday, Jan. 27 — Reading: First half of Chapter 3 to p. 44 (ending with commutativity of addition for functions and of multiplication for functions “should also present no difficulty”) — Problem Set 4: Problems 1-3 of Chapter 3
Thursday, Jan. 30 — Problem Set 5: Problems 5 and 6 (still from Chapter 3) — Reading: Finish Chapter 3 (but skip the Chapter 3 Appendix) and then continue through to p. 60 of Chapter 4 — In-class: Examples of Lagrange Interpolation, open and closed intervals, even and odd functions

Monday, Feb. 3 — Reading: Finish Chapter 4 — Problem Set 6: Problems 3, 4, 5 and 9 from Chapter 4 — In-class: We did some more strange functions (stair step and saw tooth), and then started into into Chapter 4 Appendices 1 and 3 (pp. 84-89) on Vectors and Polar Coordinates
Thursday, Feb. 6 — Reading: Appendices 1 and 3 of Chapter 4 — Problem Set 7: Problems 18(v) and 21(a) of Chapter 4 (on pp. 72-73), Problems 1, 2, and 3 of Appendix 1 (on p. 77-78), and Problems 6 and 9(i) of Appendix 3 (on pp. 88-89) — In-class: As review before the exam, we will do more problems from Chapters 1 to 4

Monday, Feb. 10 — Exam 1 on Spivak Chapters 1 to 4, including Appendices 1 and 3 of Chapter 4
Thursday, Feb. 13 — Reading: Study Chapter 5 to the bottom of p. 100 — In-class: Mapping the square function and Cube root limits

Monday, Feb. 17 — Reading: Finish Chapter 5 — Problem Set 8: Chapter 5, Problems 1-4, 7, and 8 — In-class: Focus on the limit of the inverse function (Theorem 2, Part 3, p. 102) — Illustration of g Inverse
Thursday, Feb. 20 — Reading: Chapter 6, p. 113-115, including the statement of Theorem 2, but not the proof — Problem Set 9: Still from Chapter 5, do Problems 6, 9, 10, and 11 — In-class: lots of time going over problems 6 and 10, limits at infinity, including 5-39(v), and a look ahead at Problem 2 of Chapter 6

Monday, Feb. 24 — Reading: Finish Chapter 6 — Problem Set 10: Chapter 6, Problems 1-5, but for Problem 2 just do 4-17’s functions — In-class: Preview the first eight theorems of Chapter 7