Course detail
Mathematics II-B
FSI-BM Acad. year: 2027/2028 Summer semester
The course covers the fundamentals of differential and integral calculus of functions of several variables and the
basic theory of ordinary differential equations: partial derivatives, the differential, extrema of functions of
several variables, implicit functions, multiple integrals and their transformations, and methods of solving
ordinary differential equations and their systems. A significant part of the course is devoted to engineering
applications. In computer-assisted exercises students learn to use mathematical software. The knowledge
acquired is a prerequisite for subsequent engineering courses.
Language of instruction
Czech
Number of ECTS credits
7
Supervisor
Department
Entry knowledge
Differential and integral calculus of functions in one variable.
Rules for evaluation and completion of the course
Attendance at lectures is recommended; attendance at
exercises is compulsory and checked. The forms of compensation for missed classes are determined by the exercise tutor.
REQUIREMENTS FOR COURSE-UNIT CREDIT:
Two written credit tests are given during the exercises. To obtain the course-unit credit, the student must score
at least half of the points in each test. A student who scores less than half of the points in a credit test may
retake it (each test exactly once).
FORM OF EXAMINATION:
The examination consists of a written part, which may be followed by an oral part. The written part is evaluated
with at most 100 points. The examiner may take the results of the credit tests into account in the overall
evaluation.
COURSE OF THE EXAMINATION:
The written part takes 90–120 minutes, depending on the difficulty of the problems. The written part contains at least one computational problem from each of the following topics:
1. Differential calculus of functions of several variables
2. Multiple integrals
3. Differential equations
- The written part may also include theoretical questions on the above topics.
- The possible oral part usually consists of theoretical questions and a discussion of the written work. The
student must be able to justify the solution procedure of each problem; otherwise the written work is not
accepted and is awarded zero points. An additional (simple) problem to be solved immediately may be assigned.
GRADING RULES:
0–49 points: F
50–59 points: E
60–69 points: D
70–79 points: C
80–89 points: B
90–100 points: A
Aims
The course aim is to acquaint the students with the theoretical basics of the above mentioned mathematical disciplines necessary for further study of engineering courses and for solving engineering problems encountered. Another goal of the course is to develop the students' logical thinking.
Students will acquire basic knowledge of mathematical disciplines listed in the course annotation and will be made familiar with their logical structure. They will learn how to solve mathematical problems encountered when dealing with engineering tasks using the knowledge and skills acquired. Moreover, they improve their skills in mathematical software, which can be used to solving problems.
Study aids
Lecture notes, collections of solved problems, sample tests and instructions for computer-assisted exercises are
available in the VUT e-learning system.
The study programmes with the given course
Programme B-PRP-P: Professional Pilot, Bachelor's, compulsory
Programme B-PDS-P: Industrial Design, Bachelor's, compulsory
Programme B-ENE-P: Energy, Bachelor's, compulsory
Programme B-STR-P: Engineering, Bachelor's
specialization AIŘ: Applied Computer Science and Control, compulsory
Programme B-STR-P: Engineering, Bachelor's
specialization KSB: Quality, Reliability and Safety, compulsory
Programme B-STR-P: Engineering, Bachelor's
specialization SSZ: Machine and Equipment Construction, compulsory
Programme B-STR-P: Engineering, Bachelor's
specialization STG: Manufacturing Technology, compulsory
Type of course unit
Lecture
39 hours, optionally
Syllabus
1. Functions of several variables, basic properties. Limit and continuity. Partial derivatives.
2. Gradient, directional derivative, differential, tangent plane.
3. Taylor polynomial and Taylor's theorem. Local extrema.
4. Constrained extrema (Lagrange multipliers), global extrema.
5. Implicit functions. Multiple integrals: definition and basic properties.
6. Fubini's theorem, integration over elementary (normal) domains.
7. Transformation of multiple integrals to polar and cylindrical coordinates.
8. Transformation of triple integrals to spherical coordinates. Applications of multiple integrals.
9. Ordinary differential equations (ODEs): basic concepts, existence and uniqueness, analytical methods for
first-order ODEs.
10. Higher-order ODEs; properties and solution methods of linear higher-order ODEs. Systems of first-order
ODEs.
11. Properties and solution methods of linear systems of first-order ODEs.
12. Applications of ODEs.
13. Boundary value problem for second-order ODEs.
Exercise
33 hours, compulsory
Syllabus
Review of differential calculus of a function of one variable – especially derivatives, indefinite and definite integrals.
Functions of several variables, domain of definition, limits and continuity. Partial derivatives.
Gradient, directional derivative, differential, tangent plane.
Taylor polynomial. Local extrema.
Constrained extrema (Lagrange multiplier method), global extrema.
Implicit functions. Multiple integrals and their basic properties.
Fubini's theorem, evaluation of integrals over elementary (normal) regions.
Transformation of multiple integrals into polar and cylindrical coordinates.
Transformation into spherical coordinates. Applications of multiple integrals.
First-order ordinary differential equations (ODEs), analytical solution methods.
Higher-order linear ordinary differential equations.
Systems of first-order linear ordinary differential equations.
Applications of ordinary differential equations.
Computer-assisted exercise
6 hours, compulsory
Syllabus
Seminars in a computer lab have suitable mathematical software as computer support. Obligatory topics correspond to the course syllabus.