Path-dependent functions

Various path-dependent functions are described in a uniform manner by means of a series expansion of Taylor type. For this, “path integrals” and “path tensors” are introduced. They are systems of multicomponent quantities whose values are defined for an arbitrary path in a coordinated region of space in such a way that they carry sufficient information about the shape of the path. These constructions are regarded as elementary path-dependent functions and are used instead Of the power monomials of an ordinary Taylor series. The coefficients of such expansions are interpreted as partial derivatives, which depend on the order of differentiation, or as nonstandard covariant derivatives, called two-point derivatives. Examples of path-dependent functions are given. We consider the curvature tensor of a space whose geometrical properties are specified by a translator of parallel transport of general type (nontransitive). A covariant operation leading to “extension” of tensor fields is described.