Czech

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Institute of Mathematics and Descriptive Geometry (MAT)

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.

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.

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Requirements for successful completion of the subject are specified by guarantor’s regulation updated for every academic year.

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. Notion of a primitive function, Newtons integral, its properties and computation. Definition of Riemann integral. -
Integration methods for indefinite and definite integrals.

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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.

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

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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)

Not applicable.