Course detail

Mathematics 2

FSI-Z2M Acad. year: 2024/2025 Summer semester

The course provides an introduction to the differential and integral calculus of functions of more variables. It is also devoted to the fundamentals of the theory of ordinary differential equations and their systems. The main attention is paid to the use of the mathematical apparatus in solving some basic tasks in mathematical models of real problems. The course is the basis for successful completion of subsequent professional technical courses (machine design, technical mechanics, etc.).

Supervisor

doc. Ing. Jiří Šremr, Ph.D.

Department

Institute of Mathematics

Learning outcomes of the course unit

Prerequisites

Knowledge of linear algebra, differential and integral calculus of functions of one variable.

Planned learning activities and teaching methods

Assesment methods and criteria linked to learning outcomes

Conditions for awarding the course-unit credit (0-100 points, minimum 50 points):

  • submitting all the assigned homework,
  • written test (at least 50 of possible 100 points); students who fail to score 50 points will be allowed to resit the test during the first week of the examination period.

Conditions for passing the exam (0-100 points, minimum 50 points):

  • written test (maximum 80 points),
  • discussion about the test and the oral part of the exam (maximum 20 points),
  • maximum 100 points, the overall classification is given by ECTS grade scale.

Lecture: Attendance at lectures is obligatory and checked, only one unexpected absence is allowed, absence may be compensated for based on an agreement with the teacher.

Seminar: Attendance in seminars is obligatory and checked, only one unexpected absence is allowed, absence may be compensated for based on an agreement with the teacher.

Language of instruction

Czech

Aims

Students will be able to determine parameters needed in mathematical models of some real problems. They will acquire skills for analytical solution of some ordinary differential equations and their systems.

  • Knowledge of the fundamentals of selected mathematical theories, which are needed in mathematical modelling in physics, mechanics, and other technical disciplines.
  • The ability to think logically and systematically, to move from simpler to more complex and to express and argue accurately when solving problems.
  • The ability to apply a suitable mathematical apparatus in solving some basic tasks appearing in mathematical models of real problems.

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

The study programmes with the given course

Programme B-KSI-P: Mechanical Engineering Design, Bachelor's
branch ---: no specialisation, 5 credits, compulsory

Programme C-AKR-P: , Lifelong learning
branch CLS: , 5 credits, elective

Type of course unit

 

Lecture

26 hours, compulsory

Syllabus


  • Functions of more real variables (basic notions, graph, level curves, vector function, vector field).

  • Differential calculus of functions of more variables (partial derivatives, directional derivative, gradient, continuity, differential, tangent plane, linear and quadratic approximations, potential vector field, potential, differential operators).

  • Double and triple integrals (measure in 2D and 3D spaces, double integral, Fubini theorem, change to polar coordinates, triple integral, applications).

  • First-order ordinary differential equations (basic notions, direction field, initial value problem, solving of some first-order non-linear differential equations).

  • Higher-order ordinary differential equations (basic notions, linear differential equations, solving of higher-order non-homogeneous linear equations with constant coefficients, initial and boundary value problems).

  • Systems of first-order linear differential equations (solving of homogeneous systems of first-order linear equations with constant coefficients).

  • Trigonometric Fourier series (trigonometric system, Fourier series and its sum, Fourier series expansions).

Exercise

39 hours, compulsory

Syllabus


  • Basic properties of functions of more real variables, vector field, examples of a possible use in geometry and evaluation of line integrals.

  • Evaluation of partial derivatives, linear and quadratic approximations, potential vector field, potential function, local extremes, examples of a possible use in physics.

  • Evaluation of double and triple integrals, change of variables, examples of a possible use in geometry and physics.

  • Solving of selected types of first-order non-linear differential equations, examples of a possible use in geometry and physics.

  • Solving of higher-order non-homogeneous linear equations with constant coefficients, examples of a possible use in dynamics and problems of strength analysis.

  • solving of homogeneous systems of first-order linear equations with constant coefficients, illustration of solutions in the phase space.

  • Fourier series expansions, Dirichlet theorem.