Course information

Mathematical Analysis: Mathematical Induction. Real numbers, sequences of real numbers, sequential completeness. Limit of a sequence, convergence tests. Series of real numbers, convergence tests. Calculus of functions of one variable. Trigonometric and inverse trigonometric functions. The notions of limit and continuity of a function, fundamental theorems. Derivative of a function, basic theorems, Taylor-McLaurin formula. Power series (Taylor – Mac-Laurin). The indefinite integral, basic methods of integration: integration by parts, the substitution method, integration of rational functions, trigonometric integrals. The Riemann integral of a real function, definition, examples, properties and applications. Improper integrals of first and second type: definition, simple and absolute convergence. Calculation and convergence tests. The integral test for convergence of series. Linear Algebra: Vector calculus, vector products. The equations of the line and the plane in 3-space and applications. The sphere, cylindric and conic surfaces. Surfaces of 2nd degree, projection of a space curve on the coordinate planes. Matrices, determinants, rank of a matrix. Linear systems of equations, Gauss elimination method, the method of Cramer, invertible matrices. Vector spaces and subspaces. Linear span, linear dependence-independence, basis of a vector space, change of basis matrix. Linear functions (definition, kernel, image, matrix). Linear transformations, examples. Eigenvalues and eigenvectors of linear transformations and matrices Cayley-Hamilton theorem, matrix diagonalization. Orthogonal and symmetric matrices. Quadratic forms and applications.