APPLICATION OF TENSOR ALGEBRA IN THE DIFFERENTIAL CALCULUS OF MULTIDIMENSIONAL MAPPINGS - Bokhonov, Yuriy (Physical and Mathematical Education, 2024)
Formulation of the problem. There are known formulas that can be used to find the derivative of each element of a multidimensional mapping. At the same time, the Jacobi matrix - its first derivative, the Hessian matrix - the second derivative of a scalar function of several variables, etc., are rarely used in practice. At the same time, using matrices as a technical device for solving similar problems is convenient and practical. Difficulties still arise on this path, for example, when writing the derivative of a matrix as a matrix. For an adequate description of such constructions, it is worth using tensor products of matrices, where together with ordinary matrices and vectors work with a formal vector - a linear operator, the elements of which are partial derivative operators. At the same time, the formulas for the derivative of arbitrary and differential order from the vector function become clear and transparent.Materials and methods. To study high-order derivatives of multidimensional mappings, the method of tensor (Kroneker) matrix products is widely used. At the same time, the derivative of an arbitrary order of the vector function is defined as the tensor degree of the formal differential operator of the first order - the transposed gradient vector. The action of such tensor expressions on a vector function gives its derivative of the appropriate order. This makes it possible to describe the construction of derivatives in the language of matrices, which is qualitatively different from finding partial derivatives of each component of a multidimensional mapping.The results. Formulas for the first and second derivatives of vector functions are proved and written out in detail by using tensor products of matrices, and it is also indicated how the derivative of arbitrary order is found. In classical courses of mathematical analysis, as a rule, the Jacobian matrix of the multidimensional mapping and the Hessian matrix (second derivative) of the scalar-valued function of the multidimensional argument are written out. The proposed article shows the algorithm for finding an arbitrary derivative as an operator acting in the tensor product of linear spaces, which allows a better understanding of this important construction of mathematical analysis.Conclusions. Wide application of tensor operations, in which the formal vector operator of the first-order derivative also works, turns out to be very effective. Moreover, in this way, it is possible to show the structure and find out which elements of linear spaces are the derivatives. In this way, it is possible to obtain all derivatives of the desired order at once instead of each partial derivative separately.
Keywords: derivative; differential; linear space; linear operator; vector function; matrix; tensor product; bilinear mapping; quadratic form