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# Mathematics

2019/2020
ENG
Instruction in English
8
ECTS credits
Course type:
Elective course
When:
1 year, 1, 2 module

### Course Syllabus

#### Abstract

The goal of this course is to introduce the students to the basic notions of Linear Algebra, Analytic Geometry, Calculus with emphasis on applications. The first module introduces the basic techniques of linear algebra and the coordinate method. The second module deals with the fundamentals of differential and integral calculus. #### Learning Objectives

• The goal of this course is to introduce the students to the basic notions of Linear Algebra, Analytic Geometry, Calculus with emphasis on applications. #### Expected Learning Outcomes

• Performs operations with matrices, determines the rank of a matrix, calculates inverse matrices
• Is able to compute determinants by expanding it along a row or a column
• Solves systems of linear algebraic equations
• Knows the geometric interpretation of vectors, performs operations on vectors, calculates the scalar product of vectors
• Has the skill to work with equations of straight lines and planes, has an understanding of equations of curves of 2nd order
• Analyzes the sign-definiteness of quadratic forms
• Knows the concept of sets and basic numerical sets
• Knows the definition of a function as a mapping of sets, properties of functions
• Has the skill of calculating limits, applies limits when examining a function and plotting its graph
• Is able to calculate derivatives, uses first and second order derivatives in the study of a function and its graph
• Is able to determine the maximum and minimum of the function on the segment
• Is able to explore the function and build its graph
• Is able to calculate the simplest indefinite and definite integrals
• Is able to calculate partial derivatives of 1 and 2 orders, to investigate the function of 2 variables #### Course Contents

• Matrices
Matrices. Special types of matrices: diagonal matrices, triangular matrix. Matrix operations: addition and multiplication of matrices by a scalar. Matrix transposition. Matrix multiplication. Properties of operations. Inverse matrix.
• Determinants
The concept of the determinant of the "n-th" order. Calculation of determinants of the 2nd and 3rd order. Minors and cofactors of the matrix elements. Properties of determinants. The expansion of a matrix along a row (column). Simple matrix equations.
• Systems of linear equations
Systems of linear algebraic equations. System solution, joint and incompatible systems. Solving systems using the inverse matrix. Cramer's theorem. The rank of the matrix. Pivot rows and columns. Extended matrix of a system. Kronecker-Capelli theorem. Gauss method.
• Vector algebra
Vector space. Geometric interpretation of the vectors. Linear operations on vectors and their properties. Collinear vectors. Coplanar vectors. Linear combinations of vectors. Linearly dependent system of vectors. Linearly independent system of vectors and its properties. The basis of a vector space. The expansion of a vector in a basis. The scalar product of vectors and its properties, Euclidean space. The length (norm) of the vector. The angle between the vectors. . Cross product and its properties. Triple product of vectors and its properties.
• Elements of analytic geometry
Straight lines on a plane. Different forms of equations of a line. The distance from a point to a line. The mutual position of the lines. The angle between the lines. The main tasks associated with the construction of the equation of the line. The plane in space. Different forms of equations of the plane. The angle between the planes. The conditions of parallelism and perpendicularity of the planes. The distance from a point in space to a plane. A straight line in space. Equations of a straight line in space. The angle between the lines. The conditions of parallelism and perpendicularity of lines. The angle between the line and the plane. The conditions of parallelism and perpendicularity of the line and plane. Equations of curves of the second order (circle, ellipse, hyperbola, parabola). The construction of second-order curves according to given equations. 2nd order algebraic surfaces and their equations (sphere, ellipsoid, paraboloid, hyperboloid, cylindrical and conical surfaces).
Quadratic forms. The matrix of a quadratic form. Positive (negative)-definite and indefinite quadratic form. Sylvester criterion. Reduction of a quadratic form to canonical form.
• Sets and functions
The concept of a set. Algebra of sets. Basic number sets.
• Elementary functions
Defining a function as a mapping of sets. Numerical functions and their properties. Overview of elementary functions
• Limits and continuity
Numerical sequence. A neighborhood and -neighborhood of a point. The limit of a numerical sequence. Limit of a function. One-sided limits. Infinitesimals. The relationship between a function, its limit, and an infinitesimal. Basic theorems on limits. Types of uncertainties. Some useful limits. Equivalent infinitesimals, basic equivalents. Continuity of a function at a point; properties of functions continuous at a point. Discontinuities. Properties of functions continuous on a segment.
• Derivative
The definition of derivative and its mechanical meaning. The geometrical meaning of the derivative. Relations between continuity and differentiability of functions. Derivative of sum, difference, product and quotient. Derivative of composite and inverse functions. Derivatives of basic elementary functions. Derivative and differential. Derivatives and differentials of higher orders.
• Applications of derivative
Some theorems on differentiable functions. L’Hospital rule. Increasing and decreasing functions. Local extrema. The largest and smallest values of the function on the segment. Convexity of the function graph, inflection points. Study of a function and plotting it’s graph.
• Integral
Indefinite integral and its properties. Table of basic indefinite integrals. The main methods of integration. Definite integral as the limit of the integral sum. The Newton-Leibniz formula. Properties of definite integrals. Calculation of definite integrals. Improper integrals. Geometric and physical applications of certain integrals.
• Functions of several variables
Functions of two variables: basic concepts and properties. Partial derivatives of the first order and their geometric interpretation. Partial derivatives of higher orders. Differentiability and the differential of a function. Differentials of higher orders. #### Assessment Elements

• Test 1
• Test 2
• Test 3
• Test 4
• Exam #### Interim Assessment

• Interim assessment (2 module)
0.4 * Exam + 0.12 * Test 1 + 0.12 * Test 2 + 0.18 * Test 3 + 0.18 * Test 4 #### Recommended Core Bibliography

• Williams, G. (2019). Linear Algebra with Applications (Vol. Ninth edition). Burlington, MA: Jones & Bartlett Learning. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1708709