Manifolds - Historically, it was some time before mathematicians realised the importance of the transition maps in defining manifolds. The origins of manifolds date back to Gauss and Riemann in the 1800's. It wasn't until 1913 that Weyl gave a modern definition of surfaces, and then not until 1936 that Whitney gave the current definition of manifolds in arbitrary dimensions.

Smooth Functions - Now we define smooth functions between manifolds. This obviously an necessary definition in order to do any calculus. The approach is to define differentiability in terms of differentiability via charts, which reduces the problem to smoothness in Euclidean space which we already understand. We will also introduce tangent vectors and differentials of smooth maps.

Tangent Vectors - For embedded sub-manifolds in Euclidean space, a tangent vector is the velocity vector (as a vector in the ambient Euclidean space) of a smooth curve lying on the sub-manifold. In order to make this definition, we need to use the ambient Euclidean structure. For a general manifold such a structure is not available, hence we need a way to generalise tangent vectors. The approach taken here is to observe that many curves may represent the same tangent vector, and our definition is to essentially define a tangent vector to be all curves with the same velocity vector in a chart! Next we can define the differential of a smooth function by mapping curves to curves by the function. In a chart, this notion recovers the usual notion of directional derivative.

Differentials of a Smooth Map - With a suitable definition of tangent vector, we can now define directional derivatives and the differential of a smooth function. By mapping curves to curves we define the differential applied to tangent vector as the equivalence class of curves mapped to by the function. When checking this is well defined independently of the choice of curve representing a tangent vector, we recover the Euclidean differential with respect to a chart, giving further justification to the definition of tangent vector.

Tangent Bundle - The tangent bundle is the set of all tangent vectors on a manifold. Contrast this with the tangent space at a point, which is just the tangent vectors based at that point. Thus the tangent bundle is the union of all tangent spaces. The tangent bundle inherits a manifold structure and this allows us to define the notion of vector field.