0366.1102.01 DIFFERENTIAL AND INTEGRAL CALCULUS II

Time and place:

Tuesday     8.00-10.00 in Kaplun 108
Thursdays 10.00-12.00 in Physics 007

Grade: An examination at the end of the semester

The syllabus of the course

1. THE RIEMANN INTEGRAL
   Antiderivatives; the technique of integration. Partitions and Darboux Sums. The Riemann Integral. Alternate definitions of Riemann's Integral. Classes of integrable functions. Zero sets and the Lebesgue's criterion for Riemann integrability. Properties of integrable functions and of the integral. The indefinite Riemann integral. The fundamental theorem. Mean value theorems. Functions of bounded variation. Curves and length of a rectifiable curve. Improper Riemann integrals; conditional and absolute convergence, some simple tests for convergence.
2. SERIES OF CONSTANT TERMS
   Convergence and divergence. Associativity and commutativity, absolute and conditional convergence. Multiplication of series. Tests for convergence of a series: the root test, the ratio test, Kummer's test, Cauchy's condensation test, Dirichlet' test and Abel's test.
3. SEQUENCES OF FUNCTIONS
   Uniform convergence, Cauchy's criterion. Continuity and uniform convergence. Dini's theorem. The Weierstrass approximation theorem. The lemma of Ascoli-Arzela. Differentiability and uniform convergence. Integrability and uniform convergence.
4. SERIES OF FUNCTIONS
   Uniform convergence of series of functions. Test for uniform convergence. Consequences of uniform convergence: continuity, termwise differentiation, termwise integration. Power series. The radius of convergence: Abel's first and second theorems, the theorem of Cauchy-Hadamard. The properties of the sum of a power series. The arithmetic of power series. Taylor series. Real analytic functions.
5. FUNCTIONS OF SEVERAL VARIABLES
   Topology in: neighborhoods of a point, open and closed sets, the boundary of a set, accumulation points, bounded sets, compact sets, connected sets. Sequences of points in Rⁿ, convergence. The theorem of Weierstrass-Bolzano. The theorem of Heine-Borel. Vector functions of a vector, operations with vector functions. Bounded functions. Limits of vector functions. Continuity of vector functions, uniform continuity. Continuous vector functions of a compact set, of a connected set. Partial derivatives. Differentiability and the total differentials of functions of several variables. Partial derivatives of higher orders. Composite functions, differentials and partial derivatives. The mean value theorem and Taylor's theorem. Extreme values of functions of several variables.

1. Meizler, D. - Differential and Integral Calculus. (in Hebrew)
2. Friedman, A. - Advanced Calculus .
3. Cohn, B.Z. and Zaffrany, S. - Differential and Integral Calculus 1,2. (in Hebrew)
4. Hille, E. - Analysis, vol. 1, 2 .
5. Fichtengolz, G.M. - Course in Differential and Integral Calculus,1,2. (also in Russian)
6. White, A.J. - Real Analysis, an Introduction.