ABSTRACT
The purpose of this chapter is to generalize the differential calculus in Euclidean spaces to a kind of topological spaces called manifolds which are only locally Euclidean. The idea is to use local coordinate systems around each point which overlap smoothly. We will investigate how tangent vectors at a point transform under coordinate changes. Using this we will make equivalence classes forming a tangent space at each point of the manifold. Together they constitute the tangent bundle which is our first example of a vector bundle.
