USING MATRIX CALCULUS IN PROBLEMS OF TRANSFORMING EXPRESSIONS WITH DERIVATIVES OF MULTIDIMENSIONAL MAPPINGS

Abstract

Formulation of the Problem. In problems of substitution of variables in differential expressions containing derivatives of multidimensional mappings, the use of global constructions is proposed, in particular, the Jacobi matrix of the mapping. This achieves formalization that helps to understand the conceptual side of the problem. The proposed technique works especially effectively in the most complex problems, when it is necessary to switch to new variables, both independent and dependent.

Materials and Methods. For the application of substitutions in differential expressions containing partial derivatives of functions of many variables, the Jacobi matrix of mappings, its behavior when switching to new variables, and the formula for differentiating a composite mapping are widely used. In this case, the most useful is the use of the derivative in Leibniz notation. The methodology differs significantly from that found in many manuals. It should be noted that this topic is typically included in the programs of higher educational institutions with advanced studies in mathematical analysis.

Results. Problems related to transformations of differential expressions in which old variables are expressed in terms of new ones, and new variables in terms of old ones, are considered. The examples cover the main problems on the specified topic, as well as those of increased complexity. It has been convincingly demonstrated that matrix calculus, specifically the Jacobi matrix of differential mapping, is effective in solving such problems.

Conclusion. The proposed work emphasizes the active use of matrix techniques in problems involving the transformation of variables in differential expressions, particularly first-order differential equations with multiple variables. Thus, the Jacobi matrix proves to be an effective construction, making the transformation process more understandable and transparent. The use of the Jacobi matrix helps students better understand the material, making the solution easier and clearer. The technique can also be introduced in courses on differential equations, including ordinary and partial derivatives.