SubjectsSubjects(version: 928)
Course, academic year 2022/2023
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Functions of two variables - OKBM3M047A
Title: Funkce dvou proměnných
Guaranteed by: Katedra matematiky a didaktiky matematiky (41-KMDM)
Faculty: Faculty of Education
Actual: from 2022
Semester: summer
E-Credits: 4
Examination process: summer s.:
Hours per week, examination: summer s.:0/0, Ex [HT]
Extent per academic year: 14 [hours]
Capacity: unknown / unknown (unknown)
Min. number of students: unlimited
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: combined
Note: course can be enrolled in outside the study plan
enabled for web enrollment
priority enrollment if the course is part of the study plan
Guarantor: prof. RNDr. Ladislav Kvasz, DSc., Dr.
Teacher(s): RNDr. František Mošna, Ph.D.
Pre-requisite : OKBM3M032A
Annotation -
Last update: RNDr. František Mošna, Ph.D. (30.01.2023)
Vector spaces, neighborhood of a point, convergence, functions of several variables, limits, continuity, derivative in direction, partial derivative, differential, tangent planes, normals, implicit function, curves, surfaces, coordinate transformation, multiple integral, substitution, Fubini's theorem, curvilinear and surface integral, use.
Aim of the course -
Last update: RNDr. František Mošna, Ph.D. (30.01.2023)

The primary goal of the course is to acquaint students with the basic concepts, knowledge and connections of an infinitesimal number of functions of two variables, following similar courses on functions of one variable. The secondary goal is to check, repeat and consolidate knowledge from previous courses, especially from mathematical analysis, but also geometry (curves, surfaces) or algebra (vector spaces, linear, quadratic forms).

Descriptors -
Last update: RNDr. František Mošna, Ph.D. (31.01.2023)

school teaching - a total of 14 h

preparation for school teaching - a total of 20 h

reading mathematical literature 36 h

homework - 10 h

expected total time load of students - 80 h

Literature -
Last update: RNDr. František Mošna, Ph.D. (30.01.2023)


● František Mošna: Inženýrská matematika (ČZU Praha)

● Zuzana Došlá, Ondřej Došlý: Diferenciální počet více proměnných (přírodovědecká fakulta MU Brno)

● Josef Kalas, Jaromír Kuben: Integrální počet funkcí více proměnných (přírodovědecká fakulta MU Brno)

● Serge Lang: Calculus of Several Variables, Springer N. York 1987 


●  Walter Rudin: Principles of Mathematical Analysis,McGraw-Hill 1976 
●  Bruno Budinský, Jura Charvát: Matematika II. (stavební fakulta ČVUT Praha) 
● Jaroslav Tišer, Jan Hamhalter: Diferenciální počet funkcí více proměnných (elektrotechnická fakulta ČVUT Praha) 
● Jaroslav Tišer, Jan Hamhalter: : Integrální počet funkcí více proměnných (elektrotechnická fakulta ČVUT Praha) 
● Eva Dontová: Matematika IV. (fakulta jaderné fyziky a inženýrství ČVUT Praha) 
● Štěpán Pelikán, Tomáš Zdráhal: Matematická analýza - funkce více proměnných (Universita J.E.Purkyně, Ústí n. L.) 
● Ondřej Zindulka: Vektorové pole (stavební fakulta ČVUT Praha) 
● Jiří Brabec: Matematická analýza II. (stavební fakulta ČVUT Praha) 

Requirements to the exam -
Last update: RNDr. František Mošna, Ph.D. (13.02.2023)
The exam consists of written and oral parts. The written part will be focused on students' numerical knowledge and will contain examples for calculating derivatives in the direction and by the vector, differentials, finding extrema, calculating double and curve integrals. It will be possible for students to complete the written part already during the semester in the form of tests. The oral part of the exam is aimed at understanding the discussed concepts, relationships and contexts and usually consists of three questions (the first question examines a concept, definition, statement, context, introduction..., in the second question the student has to decide on the validity of the presented statement and his justify or support a decision with a counterexample, the third question refers to some kind of inference, proof, problem solving, etc.).
Syllabus -
Last update: RNDr. František Mošna, Ph.D. (30.01.2023)

Introductory part

• repetition - linear vector spaces, scalar, vector and external product (geometric meaning, determinants), lines - equations, parametrization according to distance, planes, functions

• convergence, neighborhood, distance of points (metric, norm - Euclidean, summation, maximum), points - interior, exterior, boundary, limit, isolated, sets - open, closed, bounded, convex, continuous, compact, area.

Differential calculus

• real function of two variables (R2->R), domain, contours, sections, limit (on a set, on a domain), continuity

• derivative in direction (Gâte's differential and derivative), partial derivative, total differential (Fréchet's derivative), mutual relations, gradient - geometric meaning

• derivatives of higher orders (interchangeability of mixed second derivatives), second differential, Taylor's theorem

• extrema - local, absolute, bound extremes (substitution method and Lagrange multipliers)

• search for tangent planes, tangents in direction, derivatives of implicitly specified functions

• coordinate transformation - polar, (cylindrical), spherical

Integral calculus

• multiple (double, triple) integral, calculation of content (circle), volume (sphere, cone), center of gravity (triangle, tetrahedron), moments, Fubini's theorem, substitution theorem - connection between determinant and volume, content

• curves in R2 (explicit, implicit, parametric expression), tangent, normal, length of curve (circle), divergence, (3rd component of rotation), curve integral, Green's theorem

• surfaces in R3, divergence, rotation, surface integral, Stokes, Gauss-Ostrogradsky theorem.

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