Analysis I

Brief overview of the mathematical foundations (aimed at showing the extent of the subject and interesting threads rather than completeness), with a slightly broader discussion of numerical series.

Classical discussion of the differential calculus of functions of one variable, with an emphasis on investingation of functions and Taylor series. Detailed discussion of indefinite and definite integrals of functions of one variable.

1. Basic information about mathematical proofs, mathematical logic and set theory.

2. Sequences and numerical series. Geometric series. Convergence criteria: d'Alembert, Cauchy. Harmonic series divergence. Euler number e.

3 Functions of one variable. Limit of a function, continuity, differentiability. Properties of the derivative. Chain rule.

4. Local and global extremes. Convexity, asymptotes. investigation of functions of one real variable.

5. Inverse function theorem. Derivative of inverse function. Lagrange's mean value theorem. Taylor's theorem. L'Hospital's rule.

6. Power series. Overview of elementary functions. The exponential function. Logarithm. Trigonometric, hyperbolic and cyclometric functions.

7. Sequences and functional series, uniform convergence.

8. Definite integral (Riemann integral). Approximate methods of calculating integrals. Newton-Leibnitz theorem. Improper integrals.

9. Basic information on the generalization of the notion of the integral (Stieltjes integral, Lebesque integral), sets of zero measure. Integral criterion of series convergence.

10. Basic information about Fouriere series

W. Rudin: Principles of mathematical analysis

W.Krysicki, L.Włodarski: Analiza matematyczna w zadaniach,

M.Gewert, Z.Skoczylas, Analiza matematyczna I

Student:

1. Learns the basic mathematical apparatus of mathematical analysis and other branches of higher mathematics, necessary for the further study of physics.

2. Gains computational skills and the ability to use mathematical tools to formulate and solve problems in physics and related disciplines.

3. Can carry out basic mathematical reasoning.

4. Uses mathematical language to describe physical reality.

5. Possesses computational skills in the field of differential and integral calculus of functions of one variable.

6. Is familiar with the issues of higher mathematics which are important for the further study of physics.

7. Can apply the methods of higher mathematics to the problems of mathematical and natural sciences.

During classes, students solve computational problems and are given homework. The emphasis is on acquiring several skills, described as the main learning outcomes. The effects are checked by written tests (two during the semester). Activity in classes and creativity in the approach to solved problems are also assessed. After completing the education in the Mathematical Analysis, there is a written and oral exam to verify the acquired knowledge.