Course detail

Mathematics I (G)

FAST-0A6Acad. year: 2021/2022

Geometric vectors in E3, vector operations. Applications of vector calculus in spherical trigonometry. Vector space, basis, dimension, vector coordinates. Applications of vector calculus in analytic geometry.
Linear algebra (basics of matrix calculus, rank of a matrix, Gauss method of elimination for linear systems). Inverse matrices, determinants. Eigenvalues and eigenvectors of matrices.
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, sketching of graphs of functions, differential of a function, Taylor s expansion of a function).
Primitive function, indefinite integral, properties of indefinite integral, basic integration formulae, methods of integration.

Language of instruction

Czech

Number of ECTS credits

6

Guarantor

prof. RNDr. Josef Diblík, DrSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

Not applicable.

Prerequisites

Basics of mathematics as taught at secondary schools. Graphs of elementary functions (powers and roots, quadratic function, direct and indirect proportion, absolute value, trigonometric functions) and basic properties of such functions. Simplifications of algebraic expressions.
Znát pojem geometrického vektoru a základy analytické geometrie ve třírozměrném euklidovském prostoru (parametrické rovnice přímky, obecná rovnice roviny, skalární součin vektorů a jeho použití při řešení metrických a polohových úloh). Umět určovat typy a základní prvky kuželoseček, kreslit jejich grafy.

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. Geometrical vectors in three dimensional Euclidean space, operations with vectors.
2. Applications of vector calculus in spherical trigonometry.
3. Vector space, base, dimension, coordinates of a vector.
4. Application of vector calculus in analytic geometry.
5. Matrices, systems of linear algebraic equations, Gaussian elimination method.
6. Inverse matrix, determinants.
7. Eigenvalues and eigenvectors of a matrix.
8. Real function of a one real variable, explicit and parametric expression of a function. Basic properties of functions. Composite fuction and inverse function. Elementary functions (including inverse trigonometric functions and hyperbolic functions).
9. Polynomials and rational functions.
10. Sequences and theirs limits, limit and continuity of a function.
11. Derivative of a function, its geometrical and physical meaning, derivation rules. Derivative of a composite function and of an inverse function. Derivatives of elementary functions.
12. Derivatives of higher order, geometrical meaning of first order and second order derivatives for investigation of behavior of a function, l Hospitals rule, asymptotes.
13. Properties of function, continuous on an interval. Basic theorems of differential calculus (Rolles theorem, Lagranges theorem). Differential of a function. Taylors theorem. Derivative of a function given in a parametric form.
14. Notion of a primitive function, Newtons integral, its properties and computation. Definition of Riemann integral. -
Integration methods for indefinite and definite integrals.

Work placements

Not applicable.

Aims

After the course the students should be acquaited with the general properties of geometric vectors, know all about the dot, cross, and scalar triple products of geometric vectors, understanding their role in spherical trigonometry. Rhey should be able to apply such products when solving metric and positionalproblems in 3D analytic geometry.
They should be able to compute with matrices, perform elementary transactions, and calculate determinants, solve systems of linear algebraic equations by Gauss elimination method.
Pochopit základní pojmy diferenciálního počtu funkce jedné proměnné a geometrické interpretace některých pojmů. Zvládnout derivování a naučit se řešit úlohu průběhu funkce.
They should understand the principles of integration of elementary functions.

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 I. SNTL Praha 1987
STEIN, S. K.: Calculus and analytic geometry. New York 1989

Recommended reading

Not applicable.

Type of course unit

 

Lecture

39 hours, optionally

Teacher / Lecturer

prof. RNDr. Josef Diblík, DrSc.

Syllabus

1. Geometrical vectors in three dimensional Euclidean space, operations with vectors. 2. Applications of vector calculus in spherical trigonometry. 3. Vector space, base, dimension, coordinates of a vector. 4. Application of vector calculus in analytic geometry. 5. Matrices, systems of linear algebraic equations, Gaussian elimination method. 6. Inverse matrix, determinants. 7. Eigenvalues and eigenvectors of a matrix. 8. Real function of a one real variable, explicit and parametric expression of a function. Basic properties of functions. Composite fuction and inverse function. Elementary functions (including inverse trigonometric functions and hyperbolic functions). 9. Polynomials and rational functions. 10. Sequences and theirs limits, limit and continuity of a function. 11. Derivative of a function, its geometrical and physical meaning, derivation rules. Derivative of a composite function and of an inverse function. Derivatives of elementary functions. 12. Derivatives of higher order, geometrical meaning of first order and second order derivatives for investigation of behavior of a function, l Hospitals rule, asymptotes. 13. Properties of function, continuous on an interval. Basic theorems of differential calculus (Rolles theorem, Lagranges theorem). Differential of a function. Taylors theorem. Derivative of a function given in a parametric form. 14. Notion of a primitive function, Newtons integral, its properties and computation. Definition of Riemann integral. - Integration methods for indefinite and definite integrals.

Exercise

39 hours, compulsory

Teacher / Lecturer

prof. RNDr. Josef Diblík, DrSc.