Mathematics of Curved Spacetime
Differentiable Manifolds: Defining an abstract manifold as a space consisting of patches that look locally like the Euclidean space and are smoothly sewn together. Introducing functions and curves on manifolds.
Tangent Spaces: Defining tangent vectors as derivations, or directional derivative operators acting on functions along curves. The set of tangent vectors at a point \(p\) of a manifold has the structure of a vector space under addition and scalar multiplication.
Derivative (or Tangent Map): Introducing the derivative of a smooth map between manifolds in two alternative ways. Narrowing in on the special case where \(f\) is a real-valued function. Introducing the notions of dual space and covectors, and finally showing the connection to the differential known from basic calculus.