Mathematical analysis I.1

mathematics
physics

obligatory courses

The course introduces the fundamental notions of differential calculus in one variable. It covers properties of real and rational numbers, mathematical induction, limits of sequences (including the Bolzano-Weierstrass theorem), convergence of series (from basic criteria to the Cauchy multiplication of series), limits and continuity of a function of one variable, properties of continuous functions (intermediate value property, Weierstrass' theorem), convex functions and the notion of a derivative.

1. Real numbers, infima and suprema of sets, Dedekind's axiom. Natural, integer, rational and irrational numbers, mathematical induction and its applications.

2. Limits of sequences (also infinite), Cauchy's condition, existence of limits of monotone sequences. Existence of n-th roots of positive numbers. Basic techniques for calculating limits, the definition of e. Cesaro-Stolz theorem. Subsequences, Bolzano-Weierstrass theorem.

3. Series of real and complex numbers, convergence of a series. Geometric series, expanding real numbers in numeral systems with different bases. Cauchy's condition for series. Series of non-negative numbers, comparison tests, Cauchy's condensation test, d'Alembert's ratio test, Cauchy's root test. Series of arbitrary numbers - dependence of the series' sum on the order of summation. Alternating series and Leibniz' test. Absolutely convergent series. Abel's and Dirichlet's tests. Cauchy's product of series and its convergence. Irrationality of e.

4. Limit of a function at a point, continuity of a function (Heine's and Cauchy's conditions), intermediate value property. Continuity of the inverse function. Weierstrass' extreme value theorem. Uniform continuity of a continuous function defined on a closed interval. Exponential, logarythmic, trigonometric and cyclometric functions.

5. Convex functions and their geometric interpretation. Jensen's inequality and its consequences: the inequality between arythmetic and geometric means, Schwartz' inequality. Derivative of a function and its basic properties, tangent to the graph of a function. Characterizing convexity in terms of difference quotients and the derivative.

1. A. Birkholc, Analiza matematyczna dla nauczycieli. PWN, Warszawa 1977.

2. B. P. Demidowicz, Zbiór zadań z analizy matematycznej, Naukowa Książka, Lublin 1992 (t. I) i 1993 (t. II i III).

Baranenkov, G. S.; Demidovich, B.; Efimenko, V. A.; Kogan, S. M.; Lunts, G.; Porshneva, E.; Sycheva, E.; Frolov, S. V.; Shostak, R.; Yanpolsky, A., Problems in mathematical analysis. (Under the editorship of B. Demidovich). Mir Publishers, Moscow, 1976.

3. G. M. Fichtenholz, Rachunek różniczkowy i całkowy. Tom 1-2, PWN, Warszawa 2007.

4. W. Kaczor, M. Nowak, Zadania z Analizy Matematycznej 1. Liczby rzeczywiste, ciągi i szeregi liczbowe, PWN, Warszawa 2005.

W. Kaczor, M. Nowak, Problems in Mathematical Analysis I, AMS Student Mathematical Library (Book 4), 2000.

5. K. Kuratowski, Rachunek różniczkowy i całkowy, PWN, Warszawa 1979.

K. Kuratowski, Introduction to calculus. International Series of Monographs in Pure and Applied Mathematics, Vol. 17. Pergamon Press, Oxford-Edinburgh-New York; PWN—Polish Scientific Publishers, Warsaw, 1969

6. W. Pusz, A. Strasburger, Zbiór zadań z analizy matematycznej Wydział Fizyki UW, Warszawa 1982.

7. W. Rudin, Podstawy analizy matematycznej, PWN, Warszawa 2000.

W. Rudin, Principles of mathematical analysis. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976.

8. P. Strzelecki, Analiza Matematyczna I (lecture notes in Polish),

http://dydmat.mimuw.edu.pl/sites/default/files/wyklady/analiza-matematyczna-i.pdf

Addition to the lecture notes (auth. M. Jóźwikowski, S. Kolasiński),

http://dydmat.mimuw.edu.pl/sites/default/files/wyklady/analiza-matematyczna-i-zadania.pdf

The student

1. • knows examples of irrational numbers and is able to prove their irrationality.
   • can determine the suprema and infima of subsets of real numbers.
   • is able to use the techniques of mathematical induction in proofs and reasonings.
2. • knows the notion of a limit of a sequence of real and complex numbers and its arithmetic properties.
   • knows the Bolzano-Weierstrass theorem and the Cauchy condition for converegence of sequences.
   • recognizes and determines the key properties of sequences of numbers given by a direct or recursive formula (monotonicoty, boundedness, convergence of the sequence or of its subsequences).
3. can present a definition of the exponential and trigonometric functions on the set of real numbers and knows the key properties of these functions.
4. • knows the notion of a series of numbers and the definition of its sum.
   • knows the fundamental properties of absolutely and conditionally convergent series.
   • is able to determine the convergence (or lack of it) of a series using several convergence tests.
   • distinguishes between absolute and conditional convergence of a series.
5. • knows the notion of a limit of a function of one real variable and its equivalent definitions.
   • is able to analyze the existence of a limit of an elementary function and to calculate this limit.
6. • knows the key properties of continuous functions of one real variable: the intermediate value property, Weierstrass' extreme value theorem, uniform continuity on closed intervals.
   • is able to analyze continuity and uniform continuity of functions defined on intervals.
   • is able to use the contiunuity of functions in certain qualitative problems, e.g., the intermediate value property in proofs of existence of solutions to specific equations.
7. knows the notion of a convex function, Jensen's inequality and examples of its applications to proofs of other inequalities.
8. knows the definition of a derivative of a function of one real variable, its geometric and physical interpretations.

Credit based on the sum of points from exercises, and tests.