Course detail

Mathematics (1)

FAST-0A1Acad. year: 2020/2021

Linear algebra (fundaments of matrix calculus, rank of a matrix, Gauss method of elimination, determinants, inverse matrices, solutions of systems of linear algebraic equations).
Vector calculus. Eigenvalues and eigenvectors of matrices.
Analytic geometry (scalar, vector and triple-scalar products of vectors, affine and metric problems for linear bodies in E3).
Real function in one real variable, limit and continuity (basic definitions and properties), derivative of a function (geometric and physical meaning, techniques of differentiation, basic theorems on derivatives, higher-order derivatives, sketching of graphs of functions, differential of a function, Taylors expansion of a function).
The Indefinite integral (basic properties, integration methods, technique of integration).
The Definite Integral (definition of Riemann integral, basic properties and calculus). Applications of the Integral in Geometry and Physics (area of a plane region, length of a curve, volume and surface area of a solid of revolution, first moments and center of mass).

Language of instruction

Czech

Number of ECTS credits

8

Mode of study

Not applicable.

Guarantor

doc. RNDr. Václav Tryhuk, CSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

Not applicable.

Prerequisites

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.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Course curriculum

1. Basics of matrices, elementarz transactions, rank of a matrix. Solving szstems of linear algebraic equations bz Gauss elimination method. Second and third order determinants.
2. N-th order determinants, expanding a determinant along a row or column. Calculating with determoinants. Cramer's rule. Inverse to a matrix.
3. Matrix equations. Linear dependence and independence of arithmetic vectors. Geometric vectors. Real linear space, basis and dimension of a linear space. The coordinates of a vector. Dot product a nd cross product of vectors, calculating in coordinates.
4. Scalar tripple product of vectors, calculating in coordinates. Straight line a nd plane in E3. Positional and metric problems.
5. Real function of one real variable, explicit and parametric definition of a function. Basic properties of functions. Composite and inverse functions. Elementary functions.
6. Polynomial and the basic properties of its roots, decomposition of a polynomial in C and R. The sign of a polynomial. Eigenvalues and eigenvectors of a square matrix. Rational functions, sign of a rational function.
7. Decomposing a rational function intopartial fractions. Limit of a function, continuous functions. Basic Theorems.
8. Extended definition of a limit. Derivative of a function, its geometric and physical interpretation, differentiation rules. Derivative of a and inverse composite function. Theorems about functions continuous on an interval.
9. Basic theorems of calculus. teh diferential of a fucntion. Higher-order derivatives. Taylor's theorem. Geometric interpretation of the first and second derivative, sketching the graph of a function, l`Hospital's rule, asymptotes.
10. Derivative of a function defined parametrically. The indefinite integral and anti-derivative, basic properties of teh indefinite integral. Newton integral. Basic integration formulas. Method of integration for the indefinite integral.
11. Integrating a rational function. Integrating trigonometric functions. Integrating selected types of irrational functions.
12. Riemann integral, its basic properties, using Newton-Leibnitz formula. Integration methods for teh definite integral.
13. Geometric applications of teh definite integral. Physical and engineering applications of the definite integral.

Work placements

Not applicable.

Aims

After completing the course, the students should be able touse the basics necessarz to deal with linear problems.
They should understand the basics of calculus and interpret beometrically some of the concepts, know how to apply the definite integral.

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

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

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

BUDÍNSKÝ, B. , CHARVÁT, J.: Matematika 1. SNTL 1990
Dlouhý, O., Tryhuk, V.: Diferenciální počet I, Limita a spojitost funkce. FAST - studijní opora v intranetu 2005
Horňáková, D.: Matematika I5. ECON publishing, s.r.o. 2001
Novotný, J.: Matematika I Základy lineární algebry. Akademické nakladatelství CERM, s.r.o. 2004
STEIN, S. K.: Calculus and analytic geometry. New York 1989

Recommended reading

Not applicable.

Type of course unit

 

Lecture

52 hod., optionally

Teacher / Lecturer

doc. RNDr. Václav Tryhuk, CSc.

Syllabus

1. Basics of matrices, elementarz transactions, rank of a matrix. Solving szstems of linear algebraic equations bz Gauss elimination method. Second and third order determinants. 2. N-th order determinants, expanding a determinant along a row or column. Calculating with determoinants. Cramer's rule. Inverse to a matrix. 3. Matrix equations. Linear dependence and independence of arithmetic vectors. Geometric vectors. Real linear space, basis and dimension of a linear space. The coordinates of a vector. Dot product a nd cross product of vectors, calculating in coordinates. 4. Scalar tripple product of vectors, calculating in coordinates. Straight line a nd plane in E3. Positional and metric problems. 5. Real function of one real variable, explicit and parametric definition of a function. Basic properties of functions. Composite and inverse functions. Elementary functions. 6. Polynomial and the basic properties of its roots, decomposition of a polynomial in C and R. The sign of a polynomial. Eigenvalues and eigenvectors of a square matrix. Rational functions, sign of a rational function. 7. Decomposing a rational function intopartial fractions. Limit of a function, continuous functions. Basic Theorems. 8. Extended definition of a limit. Derivative of a function, its geometric and physical interpretation, differentiation rules. Derivative of a and inverse composite function. Theorems about functions continuous on an interval. 9. Basic theorems of calculus. teh diferential of a fucntion. Higher-order derivatives. Taylor's theorem. Geometric interpretation of the first and second derivative, sketching the graph of a function, l`Hospital's rule, asymptotes. 10. Derivative of a function defined parametrically. The indefinite integral and anti-derivative, basic properties of teh indefinite integral. Newton integral. Basic integration formulas. Method of integration for the indefinite integral. 11. Integrating a rational function. Integrating trigonometric functions. Integrating selected types of irrational functions. 12. Riemann integral, its basic properties, using Newton-Leibnitz formula. Integration methods for teh definite integral. 13. Geometric applications of teh definite integral. Physical and engineering applications of the definite integral.

Exercise

52 hod., compulsory

Teacher / Lecturer

doc. RNDr. Václav Tryhuk, CSc.