Abstract
A scalar – vector approach to the study of general properties of scalar functions and vector fields is formulated. This approach is based on taking into account the general properties of their scalar and vector characteristics, as well as taking into account their invariants. The scalar – vector form of writing layered vector fields and an alternative version of this form of writing are discussed. New invariant characteristics of multidimensional scalar functions are introduced. For the name of these characteristics, it is proposed to use the terms - the Hamilton derivative and the Laplace derivative. A local volume approximation of a scalar function is introduced. It is essential that the nonlinear part of this approximation consists of two qualitatively different components. For the names of these components, it is proposed to use the terms-parabolic component and harmonic component. A factorized form of writing the divergence functions of a stationary layered vector field is obtained.
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