Manifolds

Abstract

The main goal of the theory of differentiable manifolds is to extend many of the concepts and results of the multivariate calculus to sets that do not possess the structure of a normed vector space. Recall that the differentiability of a function of $\mathbb {R}^{n}$ to $\mathbb {R}^{m}$ means that around each interior point of its domain the function can be approximated by a linear transformation, but this requires the notions of linearity and distance, which are not present in an arbitrary set.