On the use of covariant and contravariant vector spaces in mechanics

The expediency of the use of contravariant and covariant vector spaces in mechanics is proved. To describe geometric and kinematic vector quantities, contravariant vector spaces are used. To describe dynamic quantities, covariant vector spaces are used. The algebraic operation of scalar multiplication of vectors of different spaces is replaced by the operation of tensor convolution. The operation of convolution of basic vectors of spaces of different nature induces the construction of mutual bases of these spaces. Using the operation convolution implements affine formalism and does not imply the presence of Euclidean structure. The operation of the outer product of vectors of different spaces is determined and the space of bivectors of mixed structure is constructed. The outer product of the basis vectors determines the mutual orientation of the bases. The correct definition of vector multiplication of vectors of spaces of different nature is given. The use of the algebra of external multiplication of covariant and contravariant vectors allows us to construct polyvectors of mixed structure. The vector flow is considered as a convolution of a covariant 2-vector of a physical quantity with a counter-invariant 2-vector of an area.