Differential Geometry and Relativity Theories tangent vectors derivatives paths 1 forms

In this lecture note we focus on some aspects of smooth manifolds which appear of fundamental importance for the developments of differential geometry and its applications to Theoretical Physics Special and General Relativity Economics and Finance In particular we touch basic topics for instance 1 definition of tangent vectors 2 change of coord inate system in the definition of tangent vectors 3 action of tangent vectors on coordinate systems 4 structure of tangent spaces 5 geometric interpretation of tangent vectors 6 canonical tangent vectors determined by local charts 7 tangent frames determined by local charts 8 change of local frames 9 tangent vectors and contravariant vectors 10 covariant vectors 11 the gradient of a real function 12 invariant scalars 13 tangent applications 14 local Jacobian matrices 15 basic properties of the tangent map 16 chain rule 17 diffeomorphisms and derivatives 18 transformation of tangent bases under derivatives 19 paths on a manifold 20 vector derivative of a path with respect to a re parametrization 21 tangent derivative versus calculus derivative 22 vector derivative of a path in local coordinates 23 existence of a path with a given initial tangent vector 24 tangent vectors as vector derivatives of paths 25 derivatives and paths 26 cotangent vectors 27 differential 1 forms 28 differential of a function 29 derivative versus differentials 30 critical points of real functions 31 differential of a vector function 32 change of covector frames 33 generalized contravariant vectors 34 generalized covariant vectors