Abstract

The work is devoted to the operations of differentiation in the space of vector fields and smooth functions. In mechanics, it is widely used derivative of a scalar function of the vector. To some extent, like it is determined by the derivative of the vector to another vector. However, formally interpreting the derivative as division differentials are entered in consideration of scalar and vector derived vector on another vector, which may have application to the solu-tion of problems of mechanics. The definition of a derivative of a scalar vector field on another vector field. We prove a theorem on the representation of the scalar derivative in the form of a combination of partial derivatives. As a typical particular case is considered a scalar derivative in the radius vector, generating formalism linking it with the operator nabla. It is noted that in solving some problems in the mechanics to simplify the calculation coordinate system is chosen so that at least some vectors direction coincides with one of the coordinate axes. If it concerns the vector for derivation to be performed, in such cases, the formula for the three-dimensional case can not be used because some of this vector differentials are equal to zero. This circumstance makes it necessary to prove two theorems for the two-dimensional and one-dimensional case. The definition of a vector derivative of a vector field on another vector field. We prove a theorem on the representation of the derivative vector as a combination of partial derivatives. As a typical particular case consid-ered vector derivative of the radius vector, generating formalism linking it with the operator nabla. We prove similar theorems for two-dimensional and one-dimensional case. We give examples of applications of these results to problems of mechanics.

Keywords

Vector field, the scalar derivative, derivative vector, pointing vector, acceleration, speed.