For further development of the theoretical framework of the higher school an aspect of the empowerment of the author's mathematical model [1] becomes more important regarding traceability of the impact of musical and computer technologies (MCT) components on the formation of future music teachers’ (FMT) professional competence in the form of the complete differential dF in partial derivative ∂F/∂Xi due to Xi – influence functions of the use of instruments of  musical and computer technologies, which takes the general form:

dF = Σi [(∂F/∂Xi )dХі ],  where dХі – finite increment functions Xi .

Results and Interpretation

Traceability of the impact on the pedagogical process of formation of pedagogical competence of FMT  by MCT instruments  starts from influential factors ( XI ) instruments from MCT – functions X0, covering factors: X01 - functionality of the MCT instruments; X02 – quality indicators of software instruments; X0n – other factors of instruments. X1 comes the next - the impact of student mastery of MCT instruments, factors: X11 – individual capacity; X12 – the student’s master degree of the instruments; X1m – other factors of student’s MCT knowledge. X2 is traced then - the practical impact of student’s work on formation of competence, factors: X21  – achieving of the artistic mastery of FMT games, singing, conducting; etc., X22 – achievement of technical skill of FMT games, singing, conducting; etc., X2k – other factors of the student’s practical work of forming elements of professional competence through MCT. The whole process of traceability of the formation of each individual component of competence F are shown in this table

With the influence functions which are shown in the table, the model has the form

dF =  (∂F/∂Х0 )dХ0 + (∂F/∂Х1 )dХ1  + (∂F/∂Х2 )dХ2

The finite increment of functions: X0 , X1, Х2 is also possible to express it as complete differential in partial derivative:

dХ0 = (∂Х0 /∂Х01 )dХ01  + (∂Х0 /∂Х02 )dХ02  + … + (∂Х0 /∂Х0n )dХ0n ;      

dХ1 = (∂Х1 /∂Х11 )dХ11  + (∂Х1 /∂Х12 )dХ12  + … + (∂Х1 /∂Х1m )dХ1m ;

dХ2 = (∂Х2 /∂Х21 )dХ21  + (∂Х2 /∂Х22 )dХ22  + … + (∂Х2 /∂Х2k )dХ2k .

Taking into account the relative increment of the following functions in time dτ, after obvious transformations the mathematical model represents the form of the partial differential equation.

Сonclusion

The mathematical model allows to investigate the influence of musical and computer technologies for each component of professional competence separately, and also to investigate the traceability of an unlimited number of influential factors in the pedagogical process of formation of future music teachers’ professional competence.