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Fields of study the course is directly associated with
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The second part of the block of four courses in Mathematics in its extended version. In the whole course, the fundamentals of general algebra, linear algebra and mathematical analysis, including their applications in probability, statistics are presented. This semester is concerned with the basic concepts of Calculus including numerical and applied aspects. The students will be able to work both practically and theoretically with the derivative and integral (indefinite and definite intergral) and use them for solving various applied problems and for the analysis of behavior of functions of one real variable. Students will understand the theory and use of infinite number series and power series, they will also learn about applications of some integral transforms.
At the end of the course students will be able to:
work both practically and theoretically with the derivative and (indefinite and definite) integral ;
use calculus for solving various applied problems;
analyse the behavior of functions of one real variable;
understand the theory and use of infinite number series and power series;
use some integral transforms and Fourier series.
1. Creating the ZOO (4 weeks) – interpolation of data by polynomials and splines; axiomatics of real numbers; topology of real numbers; scalar sequences,limits of sequenses and functions; continuity and derivatives; introduction of elementary functions via continuity; power series and goniometric functions;
2. Differential and integral Calculus (5 weeks) – higher order derivatives and Taylor expansion; extremes of functions; Riemann and Newton integration (area, volumes, etc.); uniform convergence and their consequences; Laurant series in complex variable; numerical derivatives and integration; stronger integration concepts (Riemann-Stieltjes, Kurzweil)
3. Continuous models (3 week) – aproximation of functions via orthogonal systems; Fourier series (including the numerical aspects); integral transforms, discrete Fourier transform