Course detail

Mathematics I

FAST-BA01Acad. year: 2014/2015

Linear algebra (basics of matrix calculus, Gauss elimination method, inverse to a matrix, determinants and their applications). Eigenvalues and eigenvectors of a matrix.
Basics of vector calculus. Linear spaces.
Analytic geometry (scalar, vector and scalar triple products, affine and metric problems for linear objects in E3).
Real function of one real variable. Sequences, limit and continuity of a function. Derivative of a function, its geometric and physical interpretation, basic theorems on derivatives, higher-order derivatives, differentials of a function, Taylor expansion of a function, sketching the graph of a function.
Antiderivative, indefinite integral, its properties and methods of calculation. Newton integral, its properties and calculation. Definition of Riemann integral. Applications of integral calculus in geometry and physics - area of a plane figure, length of a curve, volume and surface of a rotational body, static moments and the centre of gravity.
Functions in two and more variables. Limit and continuity, partial derivatives, implicit function, total differential, Taylor expansion, local minima and maxima, relative maxima and minima, maximum and minimum values of a function; directional derivative, gradient.

Language of instruction

Czech

Number of ECTS credits

11

Mode of study

Not applicable.

Guarantor

doc. RNDr. Václav Tryhuk, CSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

Students will get the understanding of the basics of differential and integral calculus of functions of one variable and the geometric interpretations of some of the concepts. They will master differentiating and sketching the graph of a function.
They will be able to perform operations with matrices and elementary transactions, to calculate determinants and solve systems of algebraic equations. They will get acquainted with applications of the vector calculus to solving problems of 3D analytic geometry.
Students will be able to calculate partial derivatives of functions of several variables. They will also learn how to find local and global minima and maxima of two-functions. They will get acquainted with directional derivatives of functions of several variables and their calculation.

Prerequisites

Basics of secondary school mathematics. Graphs of elementary functions (powers, square roots, quadratic function, direct and indirect proportion, absolute value, trigonometric functions) and basic properties of such functions. Simplification of algebraic expressions.
Geometric form of a vector and basics of analytical geometry in E3 (parametric equations of a straiht line, general equation of a plane, scala product of vectors and its application to metric and positional problems). Basic types and basic elements of conics, sketching their graphs.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations - lectures, seminars.

Assesment methods and criteria linked to learning outcomes

Successful completion of the scheduled tests and submission of solutions to problems assigned by the teacher for home work. Unless properly excused, students must attend all the workshops.

Course curriculum

1. Basics of matrix calculus, simple matrix rearrangement, rank of a matrix. Solutions to systems of linear algebraic equations by Gauss elimination method. Inverse to a matrix. Jordan`s calculation method. Matrix equations.
2. Determinants of second, third, and n-th order, expansions of determinants. Determinant calculation rules. Cramer`s rule for solving systems of linear algebraic equations. Real linear space, basis and dimension of a linear space. Linear spaces of arithmetic and geometric vectors. Coordinates of a vector. Eigenvalues and eigenvectors of matrices. - Scalar and vector products of vectors, calculation using coordinates.
3. Scalar triple product of a vector, calculation using coordinates. Straight line and plane in E3. Positional and metric problems in E3.
4. Real function in one real variable, explicit and parametric definition of a function. Composite and inverse functions. Elementary functions, cyclometric and hyperbolic functions.
5. Polynomial and its basic root properties, polynomial decomposition in real domain. Rational function. Sequence and its limit. Limit and continuity of a function. Basic theorems.
6. Derivative of a function, its geometric and physical importance, rules for differentiation. Derivative of a composite and inverse function. Rolle`s and Lagrange`s theorem. Differential of a function. Higher order derivatives and differentials.
7. Taylor`s theorem. l`Hospital`s rule. Asymptotes to the graph of a function. Geometric aspect of the first and second in determining the graph of a function. Antiderivative, indefinite integral and their properties.
8. Integration methods for indefinite and definite integral. Integration by parts and using substitutions. Integrating rational functions (no recurrence formulas), formulas needed to integrate trigonometric functions. Integrating trigonometric functions and irrational functions
9. Newton and Riemann integral and their properties. Integration by parts for definite integrals. Geometric and engineering applications of the definite integral
10. Engineering applications of the definite integral. Real function of several variables. Basic notions. Composite function. Limits of sequences, limit and continuity of two-functions. Partial derivative, partial derivative of a composite function, higher-order partial derivatives.
11. Total differential of a function, higher-order total differentials. Taylor polynomial of a function of two variables. Local maxima and minima of functions of two variables. Function in one variable defined implicitly.
12. Function of two variables defined implicitly. Some theorems of continuous functions, relative and global maxima and minima. Tangent and normal plane to a 3-D curve. Tangent and normal plane to a surface.
13. Scalar field, directional derivative, gradient. Revision, preparation for the exam.

Work placements

Not applicable.

Aims

After the course, students should know how to perform operations with matrices, elementary transactions, calculate determinants, solve systems of algebraic equations using Gauss elimination method.
They should understand the basics of calculus of functions of one variable and the basic interpretations of some of the concepts. They should master differentiating and sketching the graph of a function, understand and know how to integrate elementary functions, understand some applications of the definite integral (length of a curve, volume and surface area of a rotational body, static momentums and centre of gravity). They should acquaint themselves with the basic concepts of calculus of two and more-functions. They should be able to calculate partial derivatives, acquaint themselves with the concept of an implicit function. To understand the geometric interpretation of the total differential of a function. Learn how to find local and global minima and maxima of two-functions. To learn about the directional derivative and its calculation.

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. Praha, SNTL, 1987. (CS)
LANG, S.: Calculus of several variables. Springer Verlag, New York, 1988. (EN)
STEIN, S. K.: Calculus and analytic geometry. New York, 1989. (EN)

Recommended reading

DANĚČEK, J., DLOUHÝ, O.: Integrální počet I. CERM Brno, 2003. (CS)
DLOUHÝ, O.- TRYHUK, V.: Diferenciální počet I. CERM, 2004. (CS)
H. Čermáková a kolektiv: Sbírka příkladů z matematiky II. Akademické nakladatelství CERM, 2003. (CS)
J. Daněček a kolektiv: Sbírka příkladů z matematiky I. Akademické nakladatelství CERM Brno, 2003. (CS)
Kolektiv: Studijní opory předmětu BA01. FAST VUT, Brno, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp] (CS)
NOVOTNÝ, J.: Základy lineární algebry. CERM, 2004. (CS)
TRYHUK, V. - DLOUHÝ, O.: Modul GA01_M01 studijních opor předmětu GA01. FAST VUT, Brno, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp] (CS)
TRYHUK,V.- DLOUHÝ, O.: Diferenciální počet II. CERM, 2004. (CS)

Classification of course in study plans

  • Programme B-P-C-SI Bachelor's

    branch VS , 1 year of study, winter semester, recommended course

Type of course unit

 

Lecture

52 hod., optionally

Teacher / Lecturer

doc. RNDr. Václav Tryhuk, CSc.

Syllabus

1. Basics of matrix calculus, simple matrix rearrangement, rank of a matrix. Solutions to systems of linear algebraic equations by Gauss elimination method. Inverse to a matrix. Jordan`s calculation method. Matrix equations.
2. Determinants of second, third, and n-th order, expansions of determinants. Determinant calculation rules. Cramer`s rule for solving systems of linear algebraic equations. Real linear space, basis and dimension of a linear space. Linear spaces of arithmetic and geometric vectors. Coordinates of a vector. Eigenvalues and eigenvectors of matrices. - Scalar and vector products of vectors, calculation using coordinates.
3. Scalar triple product of a vector, calculation using coordinates. Straight line and plane in E3. Positional and metric problems in E3.
4. Real function in one real variable, explicit and parametric definition of a function. Composite and inverse functions. Elementary functions, cyclometric and hyperbolic functions.
5. Polynomial and its basic root properties, polynomial decomposition in real domain. Rational function. Sequence and its limit. Limit and continuity of a function. Basic theorems.
6. Derivative of a function, its geometric and physical importance, rules for differentiation. Derivative of a composite and inverse function. Rolle`s and Lagrange`s theorem. Differential of a function. Higher order derivatives and differentials.
7. Taylor`s theorem. l`Hospital`s rule. Asymptotes to the graph of a function. Geometric aspect of the first and second in determining the graph of a function. Antiderivative, indefinite integral and their properties.
8. Integration methods for indefinite and definite integral. Integration by parts and using substitutions. Integrating rational functions (no recurrence formulas), formulas needed to integrate trigonometric functions. Integrating trigonometric functions and irrational functions
9. Newton and Riemann integral and their properties. Integration by parts for definite integrals. Geometric and engineering applications of the definite integral
10. Engineering applications of the definite integral. Real function of several variables. Basic notions. Composite function. Limits of sequences, limit and continuity of two-functions. Partial derivative, partial derivative of a composite function, higher-order partial derivatives.
11. Total differential of a function, higher-order total differentials. Taylor polynomial of a function of two variables. Local maxima and minima of functions of two variables. Function in one variable defined implicitly.
12. Function of two variables defined implicitly. Some theorems of continuous functions, relative and global maxima and minima. Tangent and normal plane to a 3-D curve. Tangent and normal plane to a surface.
13. Scalar field, directional derivative, gradient. Revision, preparation for the exam.

Exercise

52 hod., compulsory

Teacher / Lecturer

doc. RNDr. Václav Tryhuk, CSc.

Syllabus

1. Absolute value, solutions to equations. Solving the quadratic equation in the complex domain. Conics. Graphs of selected functions. Elementary operations of matrices, rank of a matrix. Solutions of linear algebraic equations by Gauss elimination method.
2. Inverse to a matrix. Jordan‘s method. Matrix equations. Determinants. Expanding a determinant with respect to a row or a column. Rules for calculating determinants. Cramer rule applied to the solution of a system of linear algebraic equations. The eigenvalues and eigenvectors of a matrix.
3. Linear dependence and independence of arithmetic vectors. Dot, cross, and tripple scalar product of vectors, calculating in coordinates. Using scalar, vector, and scalar triple products in problems involving straight lines and planes.
4. Domains and graphs of selected types of elementary functions. Basic properties of functions. Composite function. Functions defined parametrically. Inverse functions. Functions inverse to trigonometric functions (graphs, ranges, inversion).
5. Polynomial, decomposition of a polynomial in real domain. Sign of a polynomial. Rational functions, sign of a rational function. Decomposition of a rational function into partial fractions. Limit of a sequence. Limit and continuous functions.
6. Limit and continuous functions. Derivative, differentiation rules, derivatives of elementary functions, derivative of a composite function. Derivative of composite function. Geometric aspect of a derivative. Equations of a tangent and normal to the graph of a function.
7. Higher-order derivatives. Differential used to estimate errors. Calculating higher-order differentials. Taylor polynomial. L` Hospital’s rule. Sketching the graph of a function.
8. Sketching the graph of a function. Indefinite integral. Integrating by simplification. Integrating by substitution.
9. Integrating by parts. Integrating rational functions. Integrating irrational functions. Integrating trigonometric functions.
10. Definite integral and methods for the definite integral. Check test. Geometric applications of the definite integral.
11. Geometric applications of the definite integral. Engineering applications of the definite integral. Domains. Calculating partial derivatives. Differential of a function.
12. Taylor polynomial (including higher-order differentials). Local minima and maxima of two-functions. Simple problems involving the search for global maxima and minima using relative maxima and minima. Implicit function of one variable (tangent and normal to the graph of a function defined implicitly).
13. Tangent plane and normal to the graph of a two-function defined implicitly. Tangent and normal plane to a 3-D curve. Credits.