Mathematical Seminar for Engineers (FSI-0MF)

Aim of the course: The main aim of the seminar is to get students to:

• think when solving problems requiring the use of a basic mathematical apparatus,
• adopt a creative approach to solving problems,
• "not be afraid" to try different solutions if the first idea does not lead to the goal.

Another goal is to show the students that there is often a difference between the ability to solve basic problems in a common "mathematical" formulation and the ability to solve a problem in which a suitable mathematical apparatus needs to be found.

Acquired knowledge and skills: The students will try applying the mathematical apparatus discussed in courses Mathematics I, II, BM, and Mathematics 1, 2 to various problems that are solved in mechanics and other technical disciplines. After completing the course:

• the students learn to think through the assignment and select essential information from the known facts,
• the students will be able to create a "strategy" for solving the problems and choose the appropriate mathematical apparatus.

The students will also understand that even very simple tasks can be quite laborious and vice versa, so they could be better prepared to solve the problems in the follow-up courses, where there are usually many unknowns or parameters and the necessary mathematical apparatus is not part of the formulation of the studied problem.

The course is designed for students of the 1st year of study, it supplements courses Mathematics I, II, BM, and Mathematics 1, 2. Within the seminar, slightly different problems than typical "algorithmic" tasks will be solved; the problems will be formulated in such a way that students are forced to find the necessary mathematical apparatus on their own. However, there will be no difficult or tricky problems from mathematical competitions, the problems will be inspired by those solved in physics, mechanics and other technical disciplines, in which many unknowns or parameters usually occur and that can be solved in different ways.

The course is taught in the form of seminars with an emphasis on mutual cooperation and discussion.

Some of the discussed topics:

A possible use of complex numbers in geometry.
Vectors in solid mechanics.
Vector functions in dynamics of motion.
Positive /negative definiteness of matrices, criteria for local extremes of functions of several variables.
Method of Lagrange multipliers.
Optimization problems in mechanics.
Determinants in stability of equlibria of dynamical systems.
Linearization in models of population dynamics.
Mathematical modelling in evaluation of geometric characteristics in mechanics.
Applications of line and surface integrals.
Basic algebraic structures, vector space.

STEWART, James, Daniel CLEGG a Saleem WATSON. Calculus: early transcendentals. 9th Edition. Australia: Cengage, 2021, xxx, 1214 stran, A158 : ilustrace, grafy. ISBN 978-0-357-11351-6.

MUSILOVÁ, Jana a Pavla MUSILOVÁ. Matematika I: pro porozumění i praxi. 2., dopl. vyd. Brno: VUTIUM, 2009, xi, 339 s. : barev. il. ; 26 cm. ISBN 978-80-214-3631-2.

MUSILOVÁ, Jana a Pavla MUSILOVÁ. Matematika pro porozumění i praxi: netradiční výklad tradičních témat vysokoškolské matematiky. II/1-2. Brno: VUTIUM, 2012, xiv, 341 s. : barev. il. ISBN 978-80-214-4071-5.