Czech

Not applicable.

Institute of Mathematics and Descriptive Geometry (MAT)

Students will achieve the subject`s main objectives. They will get the understanding of the basics of differential and integral calculus of functions of one variable and the geometric interpretations of some of the concepts. They will master differentiating and sketching the graph of a function.
They will be able to perform operations with matrices and elementary transactions, to calculate determinants and solve systems of algebraic equations. They will get acquainted with applications of the vector calculus to solving problems of 3D analytic geometry.
Students will be able to calculate partial derivatives of functions of several variables. They will also learn how to find local and global minima and maxima of two-functions. They will get acquainted with directional derivatives of functions of several variables and their calculation.

Basics of secondary school mathematics. Graphs of elementary functions (powers, square roots, quadratic function, direct and indirect proportion, absolute value, trigonometric functions) and basic properties of such functions. Simplification of algebraic expressions.
Geometric form of a vector and basics of analytical geometry in E3 (parametric equations of a straiht line, general equation of a plane, scala product of vectors and its application to metric and positional problems). Basic types and basic elements of conics, sketching their graphs.

Not applicable.

Not applicable.

Not applicable.

1. Basics of matrix calculus, simple matrix rearrangement, rank of a matrix. Solutions to systems of linear algebraic equations by Gauss elimination method. Inverse to a matrix. Jordan`s calculation method. Matrix equations.
2. Determinants of second, third, and n-th order, expansions of determinants. Determinant calculation rules. Cramer`s rule for solving systems of linear algebraic equations. Real linear space, basis and dimension of a linear space. Linear spaces of arithmetic and geometric vectors. Coordinates of a vector. Eigenvalues and eigenvectors of matrices. - Scalar and vector products of vectors, calculation using coordinates.
3. Scalar triple product of a vector, calculation using coordinates. Straight line and plane in E3. Positional and metric problems in E3.
4. Real function in one real variable, explicit and parametric definition of a function. Composite and inverse functions. Elementary functions, cyclometric and hyperbolic functions.
5. Polynomial and its basic root properties, polynomial decomposition in real domain. Rational function. Sequence and its limit. Limit and continuity of a function. Basic theorems.
6. Derivative of a function, its geometric and physical importance, rules for differentiation. Derivative of a composite and inverse function. Rolle`s and Lagrange`s theorem. Differential of a function. Higher order derivatives and differentials.
7. Taylor`s theorem. l`Hospital`s rule. Asymptotes to the graph of a function. Geometric aspect of the first and second in determining the graph of a function. Antiderivative, indefinite integral and their properties.
8. Integration methods for indefinite and definite integral. Integration by parts and using substitutions. Integrating rational functions (no recurrence formulas), formulas needed to integrate trigonometric functions. Integrating trigonometric functions and irrational functions
9. Newton and Riemann integral and their properties. Integration by parts for definite integrals. Geometric and engineering applications of the definite integral
10. Engineering applications of the definite integral. Real function of several variables. Basic notions. Composite function. Limits of sequences, limit and continuity of two-functions. Partial derivative, partial derivative of a composite function, higher-order partial derivatives.
11. Total differential of a function, higher-order total differentials. Taylor polynomial of a function of two variables. Local maxima and minima of functions of two variables. Function in one variable defined implicitly.
12. Function of two variables defined implicitly. Some theorems of continuous functions, relative and global maxima and minima. Tangent and normal plane to a 3-D curve. Tangent and normal plane to a surface.
13. Scalar field, directional derivative, gradient. Revision, preparation for the exam.

Not applicable.

After the course, students should know how to perform operations with matrices, elementary transactions, calculate determinants, solve systems of algebraic equations using Gauss elimination method.
They should understand the basics of calculus of functions of one variable and the basic interpretations of some of the concepts. They should master differentiating and sketching the graph of a function, understand and know how to integrate elementary functions, understand some applications of the definite integral (length of a curve, volume and surface area of a rotational body, static momentums and centre of gravity). They should acquaint themselves with the basic concepts of calculus of two and more-functions. They should be able to calculate partial derivatives, acquaint themselves with the concept of an implicit function. To understand the geometric interpretation of the total differential of a function. Learn how to find local and global minima and maxima of two-functions. To learn about the directional derivative and its calculation.

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Not applicable.

Not applicable.

BUDÍNSKÝ, B. - CHARVÁT, J.: Matematika I. Praha, SNTL, 1987. (CS)
LANG, S.: Calculus of several variables. Springer Verlag, New York, 1988. (EN)
STEIN, S. K.: Calculus and analytic geometry. New York, 1989. (EN)

DANĚČEK, J., DLOUHÝ, O.: Integrální počet I. CERM Brno, 2003. (CS)
DLOUHÝ, O.- TRYHUK, V.: Diferenciální počet I. CERM, 2004. (CS)
H. Čermáková a kolektiv: Sbírka příkladů z matematiky II. Akademické nakladatelství CERM, 2003. (CS)
J. Daněček a kolektiv: Sbírka příkladů z matematiky I. Akademické nakladatelství CERM Brno, 2003. (CS)
Kolektiv: Studijní opory předmětu BA01. FAST VUT, Brno, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp] (CS)
NOVOTNÝ, J.: Základy lineární algebry. CERM, 2004. (CS)
TRYHUK, V. - DLOUHÝ, O.: Modul GA01_M01 studijních opor předmětu GA01. FAST VUT, Brno, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp] (CS)
TRYHUK,V.- DLOUHÝ, O.: Diferenciální počet II. CERM, 2004. (CS)