Differential Geometry and Relativity Theories: tangent vectors, derivatives, paths, 1-forms - CARFÌ, David (Journal of Mathematical Economics and Finance, 2016)
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.