Irina Akimova

The paper considers the method of bringing the equations of curves and surfaces of the second order to the canonical form. The stages of practical calculating the orthogonal transformation resulting in a quadratic form to canonical form are illustrated on the examples. The prospects of using the application package in the educational process (namely, MathCAD environment) are shown. The procedure of finding the eigenval-ues and eigenvectors with this tool is demonstrated on the examples. Their finding is usually time-consuming, so for the speed of counting, it is advisable to use MathCAD program. The paper will be useful to students and lecturers during seminars on the subject "Linear Algebra".

The paper is devoted to teaching theory of normal systems solutions in the course “Differential Equations” and to the problems that arise in the presentation of the material. In studying the discipline, students face difficulties in finding the general solution of the normal systems of ordinary differential equations (ODE). One of the methods for solving systems is the method of allocation of integrable combinations, that is getting out of the system such equations that can be integrated and thus obtain the first integrals. Their aggregate determines the general decision or the general integral of the system. In this regard, the article describes the normal system of ODE and their symmetrical forms of recording. The basic concepts are interpreted: Cauchy problem, theorem of existence and uniqueness of normal ODE systems solutions, first integrals for systems. It illustrates in detail the method of finding integrable combinations, first integrals and general solutions of various systems on a wide range of tasks. The paper will be useful for students of technical universities and lecturers during seminars on the subject “Differential equations”.

The paper is devoted to the historical development of geometry. The purpose of the research is to show that apart from the geometry, which is taught in schools and universities, there is another geometry, called Lobachevskian geometry. It significantly differs from the Euclidean geometry. Coming from delivered purposes, the following major objectives are identified in this work: the main provisions of Euclidean geometry and the foundations of the Lobachevskian geometry, the consistency of Lobachevskian geometry. The paper will be useful to students of physical and mathematical departments of universities and pedagogic institutions of higher education. It can be used by lecturers and students in classes with profound study of mathematics.