Course detail

Mathematics II (G)

FAST-0A7Acad. year: 2020/2021

Indefinite integral and its properties, basic integration formulae, methods of integration. Integration of rational functions. Integration of goniometric functions. Integration of some special irrational functions.
Definition of Riemann integral and some properties, calculation by Newton-Leibnitz formula. Methods of integration for definite integral. Improper integrals and their calculation. Geometric applications of definite integral.
Real function in two and more variables, composed function. Limit and continuity of functions of two and more variables. Theorems on continuous functions. Partial derivatives, partial derivatives of a composed functions, higher-order partial derivatives of a function of two and more variables. Transformations of differential expressions. Total differential of a function. Total differentials of higher orders. Taylor s Polynomial of a function of two variables. Local extremes of a function of two variables. Function in one variable given implicitly. Function in two variables given implicitly. Global extremes. Simple problems for global extremes by means of extremes subject to the constraints. Scalar field, level surfaces. Directional derivative, gradient. Tangent line and normal plane to the curve in space. Tangent plane and normal line to the surface given implicitly. Vector field, curl and divergence of a vector field.

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 he theory of one-functions(limit, continuous functions, graphs of functions, derivative, sketching the graph of a function).
Formulas used to calculate indefinite and definite integrals, and the basic integration methods.

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. Integrating a rational function.
2. Integrating a trigonometric function.
3. Integrating selected types of irrational functions.
4. Geometric and physical applicetions of calculus.
5. Real two- and more-function, composite function. Limit and continuous two- and more-functions.
6. Partial derivative, partial derivative of a composite function, higher-order partial derivatives. Transformations of differential expressions.
7. The total differential of a function. higher-order total differentials.
8. Taylor polynomial of a two-function. Local maxima and minima of two-functions.
9. Function defined implicitly. Two-function defined implicitly.
10. Global maxima and minima. Simple problems in global maxima and minima on teh basis of relative maxima and minima.
11. Scalar field and its levels. Directional derivative of a scalar function, gradient.
12. Tangent and normal plane to a 3D curve. Tanget plane and normal to a surface defined explicitly.

Work placements

Not applicable.

Aims

After the course, the students should understand the principles of integration of some more sophisticated elementary functions, some of the applications of teh definite integral.
They should acquaint themselves with the basics of calculus of two- and more-functions, including partial derivatives, implicit functions, understand the geometric interpretation of the total differential. Learn how to find local and glogal minima and maxima of two-functions, calculate directional derivatives.

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
BUDÍNSKÝ, B., CHARVÁT, J.: Matematika II. SNTL, Praha 1990
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. Integrating a rational function. 2. Integrating a trigonometric function. 3. Integrating selected types of irrational functions. 4. Geometric and physical applicetions of calculus. 5. Real two- and more-function, composite function. Limit and continuous two- and more-functions. 6. Partial derivative, partial derivative of a composite function, higher-order partial derivatives. Transformations of differential expressions. 7. The total differential of a function. higher-order total differentials. 8. Taylor polynomial of a two-function. Local maxima and minima of two-functions. 9. Function defined implicitly. Two-function defined implicitly. 10. Global maxima and minima. Simple problems in global maxima and minima on teh basis of relative maxima and minima. 11. Scalar field and its levels. Directional derivative of a scalar function, gradient. 12. Tangent and normal plane to a 3D curve. Tanget plane and normal to a surface defined explicitly.

Exercise

39 hours, compulsory

Teacher / Lecturer

prof. RNDr. Josef Diblík, DrSc.