Mathematics 0600-S1-ChK-MATbr

Lecture:

Indefinite integral:
- the concept of an antiderivative function
- elementary integrals
- fundamental properties of integrals
- integration by parts
- integration by substitution
- integration of rational and trigonometric functions

Definite integral:
- the concept of a whole sum
- relationship with indefinite integral - Newton-Leibnitz theorem
- application for calculating surface areas
- improper integrals

Elements of linear algebra:
- Cartesian product
- the concept of linear space
- vectors and matrices and operations on them
- permutations and the sign of the permutation
- determinants and their basic properties
- scalar and vector products in R ^ 3
- inverse matrix
- matrix eigenvalues and vectors

Systems of linear equations:
- matrix representation of a system of linear equations
- Kronecker-Capelli theorem
- Cramer's rule

Vector space:
- basis in vector space
- linear independence of vectors
- Gram-Schmidt orthogonalization
- dependence of the linear mapping matrix on the basis

Multivariable Functions:
- partial derivatives
- local extremes of functions of several variables
- a study of local extremes of functions of two variables

Basics of differential equations
- basic concepts: a general and a particular solutions, an initial-value problem
- separable first-order ordinary differential equations
- first-order linear differential equations
- second-order linear differential equation with constant coefficients

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Tutorials:

Indefinite integral:
- integrals of elementary functions
- usage of basic properties of integrals
- integration by parts, including multiple
- integration by substitution
- integrals of rational functions
- integrals of trigonometric functions

Definite integral:
- calculation of definite integrals using the Newton-Leibnitz theorem
- calculation of surface areas of flat areas
- calculation of surface areas and volumes of revolving solids
- calculation of improper integrals of various types

Elements of linear algebra:
- calculating the sum and product of matrices
- illustration of the concepts: transposed, unit, symmetric, orthogonal, and triangular matrix
- calculation of determinants of 2x2 and 3x3 matrices from the rules of Sarrus
- the Laplace expansion for computingh matrix determinants of higher degrees
- determining the order of the matrix
- calculation of scalar and vector products in R ^ 3, finding the length of a vector and finding orthogonality
- determination of the inverse matrix
- finding matrix eigenvalues and eigenvectors

Systems of linear equations:
- saving systems of linear equations in matrix form
- Kronecker-Capelli theorem
- finding solutions to systems of linear equations from Cramer's rule
- finding solutions to systems of linear equations

Vector spaces - examples:
- R^2, R^3, vector space of polynomials
- basis and orthonormalization of vectors

Multivariable Functions:
- first and second-order partial derivatives
- local extremes of functions of multiple variables
- a study of local extremes of functions of two variables

Differential equations:
- solving a number of examples illustrating the types of equations discussed during the lectur