Some vector operations

The work is devoted to a series of operations on the space of smooth functions and vector fields in three-dimensional Euclidean space. Zero values are divided into two categories: the first category includes the value of the content of which "empty", the second - consisting of values whose sum is equal to zero. The latter category includes the cross product of a vector by itself. We introduce the notion of the terms of the vector products, which is the first or ortopositive part and the second part or ortoonegative; application of this approach to the vector product of the Hamiltonian operator (nabla) on itself gives rise to a mixed vector differential operator of second order, is a key element in defining the concept of surface vector analysis - mixed gradient mixed derivative in the direction of mixed divergence and rotor. It is shown that the above mentioned operations are superficial differentiation, which can be regarded as the inverse problem to surface integration, swarms that these operations can be used to produce a series of expansions vector representations of the second order, part of which has an equivalent first order. Define the operation of the conjugate of the vector product of vector fields. It is shown that the function can be restored in its mixed gradient. Presented some of the physical interpretation of the concepts introduced, including the definition of Umov mixed gradient as a function of power, volumetric energy density of the force field as a mixed divergence of the function of the spatial distribution of forces, etc.