Contents Notes |
Linear algebra -- Differential calculus -- Integral calculus -- Differential equations -- Theory of curves and surfaces -- Riemannian geometry. In this work an attempt is made to present a systematic basis for a general, absolute, coordinate and dimension free infinitesimal vector calculus. The beginnings for such a calculus appear in the literature quite early. Above all, we should mention the works of M. Frechet, in which the notion of a differential was introduced in a function space. This same trend, the translation of differential calculus to functional analysis, is pursued in a number of later investigations (Gateaux, Hildebrandt, Fischer, Graves, Keller, Kerner, Michal, Elconin, Taylor, Rothe, Sebastiao e Silva, Laugwitz, Bartle, Whitney, Fischer and others). In all of these less attention was paid to classical analysis, the theory of finite dimensional spaces. And yet it seems that already here the absolute point of view offers essential advantages. The elimination of coordinates signifies a gain not only in a formal sense. It leads to a greater unity and simplicity in the theory of functions of arbitrarily many variables, the algebraic structure of analysis is clarified, and at the same time the geometric aspects of linear algebra become more prominent, which simplifies one's ability to comprehend the overall structures and promotes the formation of new ideas and methods. |