  A self-contained introduction is presented of the notion of the (abstract)
differentiable manifold and its tangent vector fields. The way in which
elementary topological ideas stimulated the passage from Euclidean (vector)
spaces and linear maps to abstract spaces (manifolds) and diffeomorphisms is
emphasized. Necessary topological ideas are introduced at the beginning in
order to keep the text as self-contained as possible. Connectedness is
presupposed in the definition of the manifold. Definitions and statements are
laid rigorously, lots of examples and figures are scattered to develop the
intuitive understanding and exercises of various degree of difficulty are given
in order to stimulate the pedagogical character of the manuscript. The text can
be used for self-study or as part of the lecture notes of an advanced
undergraduate or beginning graduate course, for students of mathematics,
physics or engineering.
