Course detail

# Mathematics II

Primitive function, indefinite integrals, properties of indefinite integrals, overview of basic indefinite integrals, methods of integration. Integrating rational functions, trigonometric functions, selected types of irrational functions.
Newton integral, its properties and calculation. Defining the Riemann integral. Applications of the definite integral in geometry and physics.
Real two- and more-functions, composite functions. Limit of a function, continuous two- and more functions. Theorems on continuous functions. Partial derivatives of composite functions, higher-order partial derivatives. Transformations of differential expressions. Total differential of a function. Higher-order total differentials. Taylor polynomials of two-functions. Local maxima and minima of two-functions. One-functions defined implicitly. A two-function defined implicitly. Global maxima and minima. Finding global maxima and minima using realtive maxima and minima. Scalar field and its levels. Directional derivative of a scalar function, gradient. Tangent and normal plane to a 3D Curve. Tangent plane and normal to a surface defined implicitly.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

Students will known methods of solving undefinite and definite integrals and will be able to use methods successfully to important applied problems. Except this students will understand basic calculus of functions of several variables and its application to analysis of behavior of functions in three-dimenesional space.

Prerequisites

Basics of the 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. Notion of a primitive function. Properties of an indefinite integral. Integration methods for indefinite integral.
2. Integrating a rational function. Integrating a trigonometric function.
3. Integrating selected types of irrational functions. Newton integral, its properties and calculation. Riemann integral.
4. Applying calculus in geomery and physics.
5. Real functions two and more variables, composite functions. Limit and continuity of functions two and more variables. Theorems on continuous functions.
6. Partial derivatives, partial derivatives of a composite function, higher-order partial derivatives. Transformations of differential expressions.
7. The total differential of a function. Higher-order total differentials. Taylor polynomial of a two-function. Local maxima and minima of two-functions.
8. Functions defined implicitly. Two-functions defined implicitly.
9. Global maxima and minima. Simple problems in global maxima and minima using relative maxima and minima. Scalar field and its levels. Directional derivative of a scalar function, gradient.
10. 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

TRYHUK, V., DLOUHÝ, O.: Matematika I, Diferenciální počet funkcí více reálných proměnných. CERM - studijní opora v intranetu i tištěný text, 2004. (CS)
Larson R., Hostetler R.P., Edwards B.H.: Calculus (with Analytic Geometry). Brooks Cole, 2005. (EN)
Daněček, J., Dlouhý, O., Přibyl. O.: Matematika I, Modul 8, Určitý Integrál. CERM - studijní opora v intranetu i tištěný text, 2007. (CS)
Daněček, J., Dlouhý, O., Přibyl, O.: Matematika I, Modul 7, Neurčitý Integrál. CERM - studijní opora v intranetu i tištěný text, 2007. (CS)
HŘEBÍČKOVÁ, J., SLABĚŇÁKOVÁ, J., ŠAFÁŘOVÁ, H.: Sbírka příkladů z matematiky II. CERM, 2008. (CS)

K. F. Riley, M. P. Hobson, S. J. Bence. Mathematical Methods for Physics and Engineering: A Comprehensive Guide, ‎ Cambridge University Press, 3rd edition, 2006 (EN)

Klaus Weltner, S. T. John, Wolfgang J. Weber, Peter Schuster, Jean Grosjean. Mathematics for Physicists and Engineers: Fundamentals and Interactive Study Guide, Springer, 2023.

(EN)

Classification of course in study plans

• Programme B-P-C-GK Bachelor's

branch GI , 1. year of study, summer semester, compulsory

• Programme B-K-C-GK Bachelor's

branch GI , 1. year of study, summer semester, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

1. Notion of a primitive function. Properties of an indefinite integral. Integration methods for indefinite integral. 2. Integrating a rational function. Integrating a trigonometric function. 3. Integrating selected types of irrational functions. Newton integral, its properties and calculation. Riemann integral. 4. Applying calculus in geomery and physics. 5. Real functions two and more variables, composite functions. Limit and continuity of functions two and more variables. Theorems on continuous functions. 6. Partial derivatives, partial derivatives of a composite function, higher-order partial derivatives. Transformations of differential expressions. 7. The total differential of a function. Higher-order total differentials. Taylor polynomial of a two-function. Local maxima and minima of two-functions. 8. Functions defined implicitly. Two-functions defined implicitly. 9. Global maxima and minima. Simple problems in global maxima and minima using relative maxima and minima. Scalar field and its levels. Directional derivative of a scalar function, gradient. 10. Tangent and normal plane to a 3D curve. Tanget plane and normal to a surface defined explicitly.

Exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus

1. Integrating a rational function. 2. Integrating a trigonometric function. 3. Integrating selected types of irrational functions. Newton integral, its properties and calculation. Riemann integral. 4. Geometric and physical applications of calculus. 5. Real functions of two and more variables, composite function. Limit and continuity. 6. Seminar test I. 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. Taylor polynomial of functions of two variables. Local extreme of functions of two variables. 8. Functions defined implicitly. 9. Seminar test II. Global extreme. Scalar field and its levels. Directional derivative of a scalar function, gradient. 10. Tangent and normal plane to a 3D curve. Tangent plane and normal to a surface defined explicitly.