ABSTRACT
In this chapter we introduce the foundational notions of differentiable manifolds, vectors at a point, vector fields, and tensor fields. We begin with a summary of basic facts about differentiation in ℝD , since we aim to transfer the known calculational concepts to the case of differentiable manifolds. We provide the general definition of a differentiable manifold, introduce coordinates, and discuss diffeomorphisms. The directional derivative leads us to the algebraic definition of a vector as an element of the tangent space at a point. In the next step, we move from vectors at a point to vector fields defined over the entire manifold. Finally, we generalize to the multilinear structure and introduce general tensor fields on manifolds.
