CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2023/2024

# Calculus B4

Code Completion Credits Range Language
01MAB4 Z,ZK 7 2+4 Czech
Garant předmětu:
Milan Krbálek, Jiří Mikyška
Lecturer:
Miroslav Kolář, Jiří Mikyška
Tutor:
Martin Jex, Miroslav Kolář, Martin Kovanda, Jan Kovář, Milan Krbálek, Jiří Mikyška
Supervisor:
Department of Mathematics
Synopsis:

The course is devoted properties of functions of several variables, differential and integral calculus. Furthermore, the measure theory and theory of Lebesgue integral is studied.

Requirements:

Basic course of Calculus a Linear Algebra (in the extent of the courses 01MA1, 01MAB2, 01MAB3, 01LA1, 01LAB2 held at the FNSPE CTU in Prague).

Syllabus of lectures:

Differential calculus of functions of several variables - limit, continuity, partial derivative, directional partial derivative, total derivative and tangent plane, Taylor?s theorem, elementary terms of vector analysis, Jacobi matrix, implicit functions, regular mappings, change of variables, non-cartesian coordinates, local and global extremes. Integral calculus of functions of several variables - Riemann?s construction of integral, Fubiny theorem, substitution of variables. Curve and surface integral - curve and curve integral of first and second kind, surface and surface integral of first and second kind, Green and Gauss and Stokes theorems. Fundamentals of measure theory - set domain, algebra, domain generated by the semi-domain, sigma-algebra, sets H_r, K_r and S_r, Jordan measure, Lebesgue measure. Abstract Lebesgue integral - measurable function, measurable space, fundamental system of functions, definition of integral, Levi and Lebesgue theorems, integral with parameter, Lebesgue integral and his connection to Riemann and Newton integral, theorem on substitution, Fubiny theorem for Lebesgue integral.

Syllabus of tutorials:

1. Function of several variables (properties). 2. Function of several variables (differential calculus). 3. Function of several variables (integral calculus) 4. Curve and surface integral. 5. Measure Theory 6. Theory of Lebesgue integral.

Study Objective:

Knowledge: Investigation of properties for function of severable variables. Multidimensional integrations. Curve and surface integration. Theoretical aspects of measure theory and theory of Lebesgue integral. Skills: Individual analysis of practical exercises.

Study materials:

Key references:

[1] M. Giaquinta, G. Modica, Mathematical analysis - an introduction to functions of several variables, Birkhauser, Boston, 2009

Recommended references:

[2] S.L. Salas, E. Hille, G.J. Etger, Calculus (one and more variables), Wiley, 9th edition, 2002

Media and tools: MATLAB

Note:
Time-table for winter semester 2023/2024:
Time-table is not available yet
Time-table for summer semester 2023/2024:
Time-table is not available yet
The course is a part of the following study plans:
Data valid to 2024-08-13
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