Analysis-I
This is a one-semester course (15 weeks), with two 90-minute lectures and two 90-minute tutorials per week. The course will cover the following subjects.
- Abstract foundations
- Real numbers
- Limit and continuity
- Differential calculus
- Applications
We shall use the following two textbooks:
- V. A. Zorich, Mathematical Analysis-I, Springer, Berlin–Heidelberg, 2015.
- W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Book, New York, 1976.
All (mathematical) questions should be asked on this blog, to make sure that the answers are freely available to everybody. Note that MathJax is active, so that you can use Latex for formulas:
f(x)=n∑k=0f(k)(a)k!(x−a)k+o(|x−a|k).
August 15
ReplyDeleteLecture: Elements of mathematical logic. Sets and operations on them
Exercises: 2 of Section 1.1.5 and 1, 2, 3, 5, 6, 7 of Section 1.2.4 in Zorich.
August 16
ReplyDeleteLecture: Functions and their elementary properties (Sections 1.3.1-1.3.3 in Zorich)
Exercises: 3, 4, 5, 6, 7 of Section 1.3.5 in Zorich.
August 18
ReplyDeleteLecture: Relations (Section 1.3.4)
Exercises: 1, 2, 8a, 8c of Section 1.3.5
August 19
ReplyDeleteLecture: Cardinals (Sections 1.4.1 and 1.4.3)
Exercises: 9 of Section 1.3.5 and 1, 5, 6 of Section 1.4.4
August 22
ReplyDeleteLecture: Axioms of real numbers and general properties (Section 2.1)
Exercises: 5, 6, 7, 8 of Section 2.2.5
August 23
ReplyDeleteLecture: Mathematical induction and Archimedes principle (Sections 2.2.1-2.2.3)
Exercises: 1a, 1b, 1e, 1f, 1g, 2, 3, 4, 13 of Section 2.2.5
August 25
ReplyDeleteLecture: Irrational numbers and Archimedes principle (Sections 2.2.2 and 2.2.3)
Exercises: 1c, 1d, 19, 20 of Section 2.2.5
August 26
ReplyDeleteLecture: Approximation of irrational numbers (Sections 2.2.4)
Exercises: 15, 16, 17, 18 of Section 2.2.5
Revision week (August 27-September 4)
ReplyDeleteRead Theorem 1.19 and its proof in the Appendix (in Rudin)
Solve Exercises: 11, 21, 23, 24 of Section 2.2.5 (in Zorich)
The first mid-term test will take place on August 30, from 10:00 to 11:30.
ReplyDeleteSeptember 5
ReplyDeleteLecture: Completeness and its consequences (Section 2.3)
Exercises: 1, 2, 3, 4 of Section 2.3.4 (in Zorich)
September 8
ReplyDeleteLecture: Countable and uncountable sets; Section 2.4
Exercises: 1, 2, 3, 5a, 5b, 6, 7, 8 of Section 2.4.3
September 12
ReplyDeleteLecture: Limit of sequences and their properties; Section 3.1.1, 3.1.2, and 3.1.3 (a-b) in Zorich
Exercises: 1-4 of the file I sent by email
September 15
ReplyDeleteLecture: Upper and lower limits, and the number e; Section 3.1.3 in Zorich
Exercises: 1, 3, 6, 8 of Section 3.1.5
September 19
ReplyDeleteLecture: Series; Section 3.1.4 (a-b) in Zorich
Exercises: 7, 8, 11 in Chapter 3 of Rudin (pages 78-79)
September 22
ReplyDeleteLecture: Series; Section 3.1.4 (b-c) in Zorich
Exercises: 5a, 5b in Section 3.1.5 of Zorich and 12, 14 in Chapter 3 of Rudin (pages 79-80)
September 26
ReplyDeleteLecture: Definition and elementary properties of the limit of a function; Section 3.2 (a-b-c)
Exercises: See the list of exercises sent by email.
September 29
ReplyDeleteLecture: Exponential and logarithmic functions; Section 3.2 (d)
Exercises: Continue solving the exercises of the list sent earlier
Revision week (October 3-7)
ReplyDeleteConstruct the exponential and logarithmic function with all the details; Section 3.2.2 (d)
Solve Exercise 1 in Section 3.2.5 of Zorich
The second mid-term test will take place on October 6, from 16:00 to 17:30
ReplyDeleteOctober 10
ReplyDeleteLecture: Limit over a base. Criteria for the existence of limit; Sections 3.2.3, 3.2.4 (a-b) Exercises: 2, 5 of Section 3.2.5; 602-609 from Demidovich
October 13
ReplyDeleteLecture: Limit of a monotonic function. Comparison of asymptotic behaviour of functions; Section 3.2.4 (c-d)
Exercises: 3, 6, 8, 9, 10, 11 of Section 3.2.5
October 17
ReplyDeleteLecture: Continuous functions and their basic properties I; Sections 4.1 and 4.2
Exercises: 1, 2, 3, 4, 5 of Section 4.2.3
October 20
ReplyDeleteContinuous functions and their basic properties II; Sections 4.1 and 4.2
Exercises: 6, 8, 9, 10 of Section 4.2.3
October 24
ReplyDeleteLecture: Differentiability of a function: interpretation and examples; Sections 5.1.2-5.1.5
Exercises: 1, 2 of Section 5.1.6
October 27
ReplyDeleteLecture: Elementary applications in physics; van der Waerden example of nowhere differentiable continuous function; Sections 5.1.1-5.1.5
Exercises: 3, 4, 5 of Section 5.1.6
October 31
ReplyDeleteLecture: The basic rules of differentiation; Section 5.2
Exercises: 1, 2, 3 of Section 5.2.7
November 3
ReplyDeleteLecture: Applications in physics: Galilean and Lorentzian transformations, motion of a rocket, atmospheric pressure; Sections 5.2.5, 5.6.1, 5.6.2
Exercises: 4, 5, 6, 7 of Section 5.2.7
Revision week (November 7-11): Read about radioactive decay and falling bodies (Sections 5.6.3 and 5.6.4)
ReplyDeleteSolve Exercises 11, 12, 13 in Section 4.2.3 of Zorich
The third mid-term test will take place on November 12, from 9:45 to 11:45
ReplyDeleteNovember 14
ReplyDeleteLecture: Fermat, Rolle, Lagrange and Cauchy theorems; Sections 5.3.1 and 5.3.2
Exercises: 1240, 1241, 1246.2, 1247, 1255, 1256, 1257, 1266, 1267 in Demidovich
November 17
ReplyDeleteLecture: Taylor formula; Section 5.3.3
Exercises: 1381, 1387, 1388, 1393, 1397, 1403, 1406, 1413 in Demidovich; 1, 2, 4 of Section 5.3.4 in Zorich
November 21
ReplyDeleteLecture: Taylor formula, development of elementary functions and applications; Section 5.3.3 and 5.3.4
Exercises: 3, 5, 6, 7b, 8b in Section 5.3.4 of Zorich
For today's class, read the following sections in Zorich:
ReplyDelete5.6.1: Motion of a body of variable mass
5.3.3: Proposition 4 and its proof
5.3.4: Exercises 9 and 11 (a-b)
November 24
ReplyDeleteLecture: Study of functions with the help of differential calculus; Sections 5.4.1 and 5.4.2
Exercises: 1, 2, 3, 4 of Section 5.4.6; the examples of Section 5.6.3
November 28
ReplyDeleteLecture: Convex functions; Section 5.4.3
Exercises: 5, 6, 8 of Section 5.4.6
December 1
ReplyDeleteLecture: Hôpital's rule; graphs of functions; Section 5.4.4 and 5.4.5
Exercises: 9 of Section 5.4.6, Examples 27 and 28 of Section 5.4.5