Here and are any two events and the last identification is referred to as parallel transport. The first identification is the canonical identification of vectors in the tangent space at any point with vectors in the space itself. The appearance of basis vectors in tangent spaces as first order differential operators is due to this identification. It is motivated by the observation that a geometrical tangent vector can be associated in a one-to-one manner with a directional derivative operator on the set of smooth functions. This is promoted to a "definition" of tangent vectors in manifolds "not" necessarily being embedded in . This definition of tangent vectors is not the only possible one as ordinary "n"-tuples can be used as well.