Course detail
Mathematics II
FSI-2M-AAcad. year: 2016/2017
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.
Language of instruction
English
Number of ECTS credits
8
Mode of study
Not applicable.
Department
Offered to foreign students
Of all faculties
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.
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: There are two written tests (each at most 12 points) within the seminars and a semestral work from the computer support (at most 1 point).
The student can obtain at most 25 points alltogether within the seminars. Condition for the course-unit credit: to obtain at least 6 points from each written test. Students, who do not fulfil conditions for the course-unit credit, can repeat the written test during first two weeks of examination time.
FORM OF EXAMINATIONS:
The exam has an obligatory written part.
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 20 points)
Problem 3: In double and tripple integral (at most 20 points)
Problem 4: In line and surface integral (at most 20 points)
Above problems can also contain a theoretical question.
RULES FOR CLASSIFICATION
1. Results from seminars (at most 25 points)
2. Results from the written examination (at most 75 points)
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
The student can obtain at most 25 points alltogether within the seminars. Condition for the course-unit credit: to obtain at least 6 points from each written test. Students, who do not fulfil conditions for the course-unit credit, can repeat the written test during first two weeks of examination time.
FORM OF EXAMINATIONS:
The exam has an obligatory written part.
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 20 points)
Problem 3: In double and tripple integral (at most 20 points)
Problem 4: In line and surface integral (at most 20 points)
Above problems can also contain a theoretical question.
RULES FOR CLASSIFICATION
1. Results from seminars (at most 25 points)
2. Results from the written examination (at most 75 points)
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
Course curriculum
Not applicable.
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.
Recommended optional programme components
Not applicable.
Prerequisites and corequisites
Not applicable.
Basic literature
Rektorys K. a spol.: Přehled užité matematiky I,II (SNTL Praha, 1988)
Satunino, L.S., Hille, E., Etgen, J.G.: Calculus: One and Several Variables, Wiley 2002
Sneall D.B. - Hosack J.M.: Calculus, An Integrated Approach
Thomas G. B.: Calculus (Addison Wesley, 2003)
Thomas G.B. - Finney R.L.: Calculus and Analytic Geometry, 7th edition
Satunino, L.S., Hille, E., Etgen, J.G.: Calculus: One and Several Variables, Wiley 2002
Sneall D.B. - Hosack J.M.: Calculus, An Integrated Approach
Thomas G. B.: Calculus (Addison Wesley, 2003)
Thomas G.B. - Finney R.L.: Calculus and Analytic Geometry, 7th edition
Recommended reading
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)
Karásek J.: Matematika II (skriptum VUT)
Mezník I. - Karásek J. - Miklíček J.: Matematika I pro strojní fakulty (SNTL 1992)
Rektorys K. a spol.: Přehled užité matematiky I,II (SNTL, 1988)
Eliáš J., Horváth J., Kajan J.: Zbierka úloh z vyššej matematiky I, II, III, IV (Alfa Bratislava, 1985)
Karásek J.: Matematika II (skriptum VUT)
Mezník I. - Karásek J. - Miklíček J.: Matematika I pro strojní fakulty (SNTL 1992)
Rektorys K. a spol.: Přehled užité matematiky I,II (SNTL, 1988)
Classification of course in study plans
Type of course unit
Lecture
39 hod., 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.
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 hod., compulsory
Teacher / Lecturer
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 hod., 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.