Continuation of theory as a tool for activation of students of higher mathematics in technical university

The article analyzes the conceptual ideas of theorems in the course of mathematical analysis as a means of activating the training of students in technical university. The use of mathematical methods in production leads to the need for engineers who have a well-developed logical and algorithmic thinking, the formation of which does not in any way affect the very proof of the theorems on the course of higher mathematics. It is determined that the proof of the fundamental theorems of the theory of boundaries and some other sections causes considerable difficulties for students, which is due to the considerable abstraction of reasoning and not always clearly visible geometric meaning. Some ways to overcome these difficulties and increase the students' activity in mastering the theoretical material are given. First of all, according to the author, the main source of difficulty in assimilating abstract considerations is not the content, but the form of these considerations. The specificity of this form leads to the fact that students do not always see the essence of the concept behind symbols and other similar attributes, which creates the danger of the separation of the form of proof of the theorem from its content. The author proposed in the proof of each theorem conditionally to distinguish two components: 1) the logical component (LС), which contains the basic idea of proof; 2) a technical component (TC), which implements this idea by means of mathematical symbols and relationships. The presence of these two components corresponds to the existence in each subject or phenomenon of two categories such as content and form. The author suggests examples of LС proof of three important theorems of mathematical analysis. It is established that the selection of drugs in the proofs of theorems can be used as a means of direct control of the learning process. Along with the reference in the lectures of complete proofs of theorems it seems appropriate in some theorems to confine the statement only to the proof of the LС, offering students to implement the technique of proof independently. This technique is one of the ways of creating a problem situation and its use helps to stimulate the thinking of students.