Scalar and vector differentiation of vectors

The work is devoted to the operations of differentiation in the space of vector fields and smooth functions. In mechanics the derivative of a scalar function of the vector is widely used. To some extent, it is like determined by the derivative of the vector to another vector. However, formally interpreting the derivative as division differentials is entered in consideration of scalar and vector derived on another vector, which may have application to the solution of problems of mechanics. 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 we consider 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 the derection of some vectors 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 cannot 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. We prove a theorem on the representation of the derivative vector as a combination of partial derivatives. As a typical particular case we consider the 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.