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Mathematics

2019/2020
Учебный год
ENG
Обучение ведется на английском языке
8
Кредиты
Статус:
Курс по выбору
Когда читается:
1-й курс, 1, 2 модуль

Преподаватель

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

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

Expected Learning Outcomes

  • Performs operations with matrices, determines the rank of a matrix, calculates inverse matrices
  • Is able to calculate the determinants of matrices 2 and 3 of the order, to lay out the matrix in a row (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 2 orders
  • 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, the largest and smallest value 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

Course Contents

  • Matrices
    Matrices and their main types. Diagonal and unit 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. Minor and algebraic complement of the matrix element. Properties of determinants. The expansion of the matrix in a row (column). Simple matrix equations.
  • Systems of linear equations
    Systems of linear algebraic equations. System solution, joint and incompatible systems. Systems research. Cramer's theorem. Solving systems using the inverse matrix. Minor “k” matrix order. The rank of the matrix. Base rows and columns. Extended system matrix. Kronecker-Capelli theorem. Gauss method.
  • Vector algebra
    Vector space. Geometric interpretation of the vector. Linear operations on vectors and their properties. Collinear vector. Coplanar vector. Single unit vectors. The scalar product of vectors and its properties. Euclidean space. The length (norm) of the vector. The angle between the vectors. . Linear combination of a vector system. Linearly dependent vector system. Linearly independent system of vectors and its properties. The basis of the vector space. The expansion of a vector in a basis. Vector product and its properties. Mixed product of vectors and its properties.
  • Elements of analytic geometry
    Straight on the plane. Different equations of the 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 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. Direct in space. Different 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
    Quadratic forms. The matrix is quadratic. Positive (negative) and alternating quadratic form. Sylvester criterion. Reduction of a quadratic form to canonical form.
  • Sets and functions
    The concept of multitude. 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  is a neighborhood of a point. The limit of the numerical sequence. Function limit. One-sided limits. Infinitely large and infinitely small functions. The relationship between a function, its limit, and an infinitely small function. Basic limit theorems. Types of uncertainties. Signs of the existence of pre-business. Wonderful limits. Equivalent infinitesimal functions, an equivalent table. Continuity of a function at a point; properties of functions continuous at a point. Break points function. Properties of functions continuous on a segment.
  • Derivative
    The definition of a derivative is its mechanical meaning. Relations between continuity and differentiability of functions. Derivative of sum, difference, product and quotient. Derivative of complex and inverse functions. Derivatives of basic elementary functions. The geometrical meaning of the derivative. Derivative and differential functions. Derivatives and differentials of higher orders.
  • Applications of derivative
    Some theorems on differentiable functions. The rule of Lital. Ascending and descending functions. Maximum and minimum functions. The largest and smallest values of the function on the segment. Convexity of the function graph, inflection point. Complete study of the function and plotting it.
  • Integral
    Indefinite integral and its properties. Table of basic indefinite integrals. The main methods of integration. Defined integral as the limit of the integral sum. The Newton-Leibniz formula. Properties of a definite integral. Calculation of a definite integral. Own integrals. Geometric and physical applications of a certain integral.
  • 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 full differential of a function. Differentials of higher orders.
Assessment Elements

Assessment Elements

  • non-blocking Test 1
  • non-blocking Test 2
  • non-blocking Test 3
  • non-blocking Test 4
  • non-blocking Exam
Interim Assessment

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
Bibliography

Bibliography

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

Recommended Additional Bibliography

  • Fuad Aleskerov, Hasan Ersel, & Dmitri Piontkovski. (2011). Linear Algebra for Economists. Springer. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsrep&AN=edsrep.b.spr.sptbec.978.3.642.20570.5