Course detail

Mathematics I

FAST-GA01Acad. year: 2022/2023

Linear algebra (basics of matrix calculus, rank of a matrix, solution of linear systems by Gauss elimination method). Inverse matrix, determinants. Eigenvalues and eigenvectors of a matrix.
Geometrical vectors in three dimensional Euclidean space, operations with vectors. Applications of vector calculus in spherical trigonometry. Vector space, base, dimension, coordinates of a vector. Application of vector calculus in analytic geometry.Real function of one real variable, limit and continuity of a function (basic notions and properties), derivative of a function (geometrical and physical meaning, techniques of differentiation, basic theorems on derivatives, higher order derivatives, sketching the graph of a function, differentials of a function, Taylor expansion of a function).

Language of instruction

Czech

Number of ECTS credits

8

Mode of study

Not applicable.

Guarantor

prof. RNDr. Josef Diblík, DrSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

After passing the course students will have necessary skills for performing operations with vectors defined either generally or by their coordinates. Except this application of vectors in the spherical trigonometry will be deeply explained. The next outpiuts are: application of vector calculus in metrix and positional problems in analytical geometry, operations with matrices and solving systems of linear algebraic equations.
Bringing off basic differential calculus will permit successfully analyse problems of behavior of analytical curves.

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.

Definition of a geometric vector and basics of 3D analytic geometry (parametric equations of a straight line, dot product of vectors and its applications to metric and positional problems). Identifying the the types and basic properties of conics, sketching graphs of conics)

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. Matrices, systems of linear algebraic equations, Gaussian elimination method.
2. Inverse matrix, determinants.
3. Geometrical vectors in three dimensional Euclidean space, operations with vectors.
4. Applications of vector calculus in spherical trigonometry.
5. Vector space, basis, dimension, coordinates of a vector.
6. Eigenvalues and eigenvectors of a matrix.
7. Application of vector calculus in analytic geometry.
8. Real function of 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 their 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 sketching the graph of a function, l Hospital's rule, asymptotes.
13. Properties of functions continuous on an interval. Basic theorems of differential calculus (the Rolle and Lagrange theorems). Differential of a function. Taylor's theorem. Derivative of a function given in a parametric form.

Work placements

Not applicable.

Aims

Being able to compute with matrices, performing elementary transactions, and calculating determinants, solving systems of linear algebraic equations by Gauss elimination method. Getting acquainted with the general properties of geometric vectors, without using coordinates. Knowing all about the dot, cross, and scalar triple products of geometric vectors, understanding their role in spherical trigonometry. Being able to apply such products when solving metric and positional problems in 3D analytic geometry. Understanding the basic ideas of the calculus of one- and two-functions including geometric interpretations of some concepts. Mastering differentiation, being able to sketch the graph of a function.

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

Dlouhý O., Tryhuk V.: Diferenciální počet I, Limita a spojitost funkce. FAST - studijní opora v intranetu, 2005. (CS)
Dlouhý, O., Tryhuk, V.: Diferenciální počet funkce jedné reálné proměnné. FAST, 2008. (CS)
Dlouhý O., Tryhuk V.: Diferenciální počet I, Derivace funkce. FAST - studijní opora v intranetu, 2005. (CS)
Larson R., Hostetler R.P., Edwards B.H.: Calculus (with Analytic Geometry). Brooks Cole, 2005. (EN)
Novotný, J.: Základy lineární algebry. FAST - studijní opora v intranetu i tištěné texty, 2005. (CS)
Tryhuk, V., Dlouhý, O.: Vektorový počet a jeho aplikace. FAST - studijní opora v intranetu, 2005. (CS)

Recommended reading

Chris McMullen. Calculus with Multiple Variables Essential Skills Workbook: Includes Vector Calculus and Full Solutions, ‎ Zishka Publishing, 2021 (EN)

Classification of course in study plans

  • Programme B-P-C-GK Bachelor's

    branch GI , 1 year of study, winter semester, compulsory

  • Programme B-K-C-GK Bachelor's

    branch GI , 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

39 hod., optionally

Teacher / Lecturer

prof. RNDr. Josef Diblík, DrSc.

Syllabus

1. Matrices, systems of linear algebraic equations, Gaussian elimination method. 2. Inverse matrix, determinants. 3. Geometrical vectors in three dimensional Euclidean space, operations with vectors. 4. Applications of vector calculus in spherical trigonometry. 5. Vector space, basis, dimension, coordinates of a vector. 6. Eigenvalues and eigenvectors of a matrix. 7. Application of vector calculus in analytic geometry. 8. Real function of 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 their 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 sketching the graph of a function, l Hospital's rule, asymptotes. 13. Properties of functions continuous on an interval. Basic theorems of differential calculus (the Rolle and Lagrange theorems). Differential of a function. Taylor's theorem. Derivative of a function given in a parametric form.

Exercise

39 hod., compulsory

Teacher / Lecturer

prof. RNDr. Josef Diblík, DrSc.

Syllabus

1. Geometrical vectors in E3, 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 and inverse functions. Elementary 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 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. Differential of a function. Taylor’s theorem. Derivative of a function given in a parametric form. Primitive function, Newtons integral, its properties and computation. Riemann’s integral. Integration methods for indefinite and definite integrals.