Course detail

# Mathematics II

FAST-GA04Acad. year: 2022/2023

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

Number of ECTS credits

Mode of study

Guarantor

Department

Learning outcomes of the course unit

Prerequisites

Formulas used to calculate indefinite and definite integrals, and the basic integration methods.

Co-requisites

Planned learning activities and teaching methods

Assesment methods and criteria linked to learning outcomes

Course curriculum

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

Aims

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

Recommended optional programme components

Prerequisites and corequisites

Basic literature

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)

Recommended reading

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)#### Type of course unit

Lecture

Teacher / Lecturer

Syllabus

Exercise

Teacher / Lecturer

Syllabus