The introduction to the tensor analysis in curvilinear coordinates, as presented in this textbook, is  developed in such a way, and by employing such a notation, that almost all of the most significant  formulae obtained below with respect to a three-dimensional Euclidean space, which makes up the  starting point and the base of our reasoning,—nearly all of the most important formulae carry over,  without requiring any essential modification, to the general case of curved spaces as studied in the  Riemannian geometry. Such are, for example, the expression (2.7) for the length element ds² in terms  of the metric tensor g_ik; the formulae for raising (3.29) or lowering (3.30) the indices; the relations  (5.51) describing the variations in vector components produced by the vector’s parallel translation to  an infinitely close spatial point; the expressions (5.57) and (5.58) for the Christoffel symbols in  terms of the metric tensor; the relation (9.13) for the covariant derivative; and, finally, the formula  (9.21) defining the curvature tensor R^m_ikl . The distinguishing feature shared by the just mentioned  relations, resides in the fact that all of them are built in terms of the metric tensor g_ik without  referring to the specific nature of the latter. For this reason, the move from Euclidean geometry to the  Riemannian one may be carried out, from a formal standpoint, by simply redefining the metric tensor  (2.8) while keeping intact the entire structure of concepts, notions, and representations as established  with respect to curvilinear coordinates in three-dimensional Euclidean space.   For all those reasons, some of the above-mentioned relations, expressions, and formulae, if adopted  in advance as postulates, might lay the basis for an axiomatic development of the general theory of  Riemannian spaces, with no reference to the initial Euclidean space. Sometimes in the literature, an  introduction to the tensor analysis builds right from the start on such an axiomatic approach, and the  formal nature of the approach is fraught with dangers of its being learned formally as well. Tracing  the origins of the basic concepts and ideas, while supporting them visually by means of geometrical  images relative to the familiar three-dimensional Euclidean space, should help the reader in his effort  to overcome assimilating the concepts formally.  Tensorial differential operations in curvilinear coordinates Preface Welcome                                                                                                                           Lev Chebotarev                                                                                                                                  Montreal, April 2016.