Course detail

# Mathematics II

FSI-2MAcad. year: 2020/2021

Differential and integral calculus of functions of several variables including problems of finding maxima and minima and calculating limits, derivatives, differentials, double and triple integrals. Also dealt are the line and surface integrals both in a scalar and a vector field. At seminars, the MAPLE mathematical software is used.

Department

Learning outcomes of the course unit

Students will be made familiar with differential and integral calculus of more variables. They will be able to apply this knowledge in various engineering tasks. After completing the course students will be prepared for further study of physics, mechanics and other technical disciplines.

Prerequisites

Linear algebra, differential and integral calculus of functions of one variable.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Karásek J.: Matematika II (skriptum VUT)

Thomas G.B. - Finney R.L.: Calculus and Analytic Geometry, 7th edition

Sneall D.B. - Hosack J.M.: Calculus, An Integrated Approach

Mezník I. - Karásek J. - Miklíček J.: Matematika I pro strojní fakulty (SNTL 1992)

Rektorys K. a spol.: Přehled užité matematiky I,II (SNTL Praha, 1988)

Děmidovič B. P.: Sbírka úloh a cvičení z matematické analýzy

Eliáš J., Horváth J., Kajan J.: Zbierka úloh z vyššej matematiky I, II, III, IV (Alfa Bratislava, 1985)

Thomas G. B.: Calculus (Addison Wesley, 2003)

Rektorys K. a spol.: Přehled užité matematiky I,II (SNTL, 1988)

Satunino, L.S., Hille, E., Etgen, J.G.: Calculus: One and Several Variables, Wiley 2002

Thomas G.B., Finney R.L.: Calculus and Analytic Geometry (7th edition)

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

COURSE-UNIT CREDIT REQUIREMENTS: The course includes seminars and exercises in the computer lab. There are two written tests within the seminars. Students may achieve max 12 points in each of these two tests, i.e. 24 points altogether. The course-unit credit is conditional on obtaining at least 6 points in each written test. If the minimum number of points is not achieved, students may repeat the test during the first two weeks of the examination period.

FORM OF EXAMINATIONS:

The exam has a written part (at most 75 points) and an oral part (at most 25 points).

WRITTEN PART OF EXAMINATION (at most 75 points)

In a 120-minute written test, students have to solve the following four problems:

Problem 1: In basic properties of functions of several variables: domains, partial derivatives, gradient (at most 10 points)

Problem 2: In differential calculus of functions of several variables (at most 22 points)

Problem 3: In double and triple integral (at most 20 points)

Problem 4: In line and surface integral (at most 23 points)

The above problems can also contain a theoretical question.

ORAL PART OF EXAMINATION (max 25 points)

• Discussion based on the written test: students have to explain how they solved each problem. Should the student fail to explain it sufficiently, the test results will not be accepted and will be classified by 0 points.

• Possible theoretic question.

• Possible simple problem to be solved straight away.

• The results achieved in the written tests in seminars may be taken into account within the oral examination.

FINAL CLASSIFICATION:

0-49 points: F

50-59 points: E

60-69 points: D

70-79 points: C

80-89 points: B

90-100 points: A

Language of instruction

Czech

Work placements

Not applicable.

Aims

The course aims to acquaint the students with the basics of differential and integral calculus of functions of several variables. This will enable them to attend engineering courses and deal with engineering problems. Another goal of the course is to develop the students' logical thinking.

Specification of controlled education, way of implementation and compensation for absences

Attendance at lectures is recommended, attendance at seminars is required. The lessons are planned on the basis of a weekly schedule. The way of compensation for an absence is fully at the discretion of the teacher.

Classification of course in study plans

- Programme B-ZSI-P Bachelor's
specialization STI , 1. year of study, summer semester, 8 credits, compulsory

specialization MTI , 1. year of study, summer semester, 8 credits, compulsory - Programme B-PDS-P Bachelor's, 1. year of study, summer semester, 8 credits, compulsory
- Programme B-MET-P Bachelor's, 1. year of study, summer semester, 8 credits, compulsory

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Syllabus

Week 1: Functions in more variables: basic definitions, limit of a function, continuous functions, partial derivative.

Week 2: Higher-order partial derivatives, gradient of a function, derivative in a direction, first-order and higher-order differentials, tangent plane to the graph of a function in two variables.

Week 3: Taylor polynomial, local maxima and minima of functions in several variables.

Week 4: Relative maxima and minima, absolute maxima and minima.

Week 5: Functions defined implicitly.

Week 6: Double and triple integral, Fubini's theorem: calculation on normal sets.

Week 7: Substitution theorem, cylindrical a spherical co-ordinates.

Week 8: Applications of double and triple integrals.

Week 9: Curves and their orientations, first-type line integral and its applications.

Week 10: Second-type line integral and its applications, Green's theorem.

Week 11: Line integrals independent of the integration path, potential, the nabla and delta operators, divergence and curl of a vector field.

Week 12: Surfaces (parametric equations, orienting of a surface), first-type surface integral and its applications.

Week 13: Second-type surface integral and its applications, Gauss' theorem and Stokes' theorem.

Exercise

44 hours, compulsory

Teacher / Lecturer

prof. RNDr. Jan Čermák, CSc.

Ing. Lucie Fedorková

Mgr. Jana Hoderová, Ph.D.

Ing. Pavel Hrabec, Ph.D.

Mgr. Aleš Návrat, Ph.D.

Ing. Jiří Novák

Mgr. Jan Pavlík, Ph.D.

doc. Mgr. Pavel Řehák, Ph.D.

doc. Ing. Pavel Štarha, Ph.D.

Mgr. Viera Štoudková Růžičková, Ph.D.

doc. Mgr. Petr Vašík, Ph.D.

prof. RNDr. Miroslav Doupovec, CSc., dr. h. c.

Syllabus

The first week: calculating improper integrals, applications of the Riemann integral. Following weeks: seminars related to the lectures given in the previous week.

Computer-assisted exercise

8 hours, compulsory

Teacher / Lecturer

Syllabus

Seminars in a computer lab have the programme MAPLE as a computer support. Obligatory topics to go through: Plotting of the graph of a function of more variables (given by explicit, implicit or parametric equations), extrema of functions of more variables.

eLearning

**eLearning:** currently opened course