FormsThis would be the first book in the series. It deals with differential forms following the Cartan-Kähler-Clifton approach, which consists simply in replacing the cardinal role of the vector field approach with the differential form approach. But these differentials are not the non-intuitive anti-symmetric multilinear functions of vector fields (or linear functions of antisymmetric tensor fields), but the integrands of multiple integrals, as in W. Rudin’s book “Principles of Mathematical Analysis”. Tangent vector and tensor fields should be used only where they are of the essence of the problem in question. Thus, for example, vector fields may be of the essence of the vector calculus, but not of the equivalent multivariable calculus. The essence of the calculus lies with the differential forms. For a diatribe against the use of tensors in those situations where forms are the right tool, see Section 1.2 of the Dover book “Differential Forms with Applications to the Physical Sciences” by Harley Flanders, titled “Comparison with Tensors” (of differential forms). The preeminence of forms over vector fields means in essence the replacement of Lie’s methods with Cartan’s methods. For a brief comparison of these methods see the introduction of “the Method of Equivalence and its Applications” by Robert B. Gardner. Finally, another characteristic of the Cartan-Kähler-Clifton approach is that tangent vectors are viewed in a passive role rather than in an active role, i.e. not viewed as operators on functions. In chapter 1, we summarize the multivariable calculus, removing any reference to vector equations and replacing them with component equations for the purpose of identifying the latter equations with the components of the differential form version of specific instances of Stokes generalized theorem. In addition there are collections of formulas of the “extended vector calculus” used in undergraduate and graduate physics books in electrodynamics, as well as differential operators in curvilinear coordinates, a uniqueness theorem, Helmholtz theorem, etc. Also the two versions of the concept of differential forms are discussed, since the differentials in Cartan, Kähler and Rudin are functions of hypersurfaces (lines, surfaces, volumes, etc). In chapter 2, we develop the exterior calculus of scalar-valued differential forms as a small jump from the first chapter. In chapter 3, we deal with Maxwell’s equations: the right form (according to Cartan) of writing them, performing integrations in four dimensions, extracting the information contained in those equations directly from their differential form version in four dimensions ir physical significance, solving them in accordance with Cartan’s view of how they shoulf be written. In chapter 4, we deal with integral invariants (which are evaluations of differential invariants, when viewed as functions of hypersurfaces), and with conservation laws. In chapter 5, we deal with exact and closed forms, finding local primitive of the last ones. We also give some fo the flavor of de Rham’s theorem and explain the significance of Aharonov-Bohm for the writing of Maxwell’s equations.. In chapter 6, we deal with a variety of topics having to do with manipulating differential forms: algebraic solving of equations with differential forms, the Frobenius theorem for differential forms, solving Riemann’s original problem of equivalence (both on a manifold and on its bundle of coframes, while viewing these as symbolic forms and without resort to vector fields), and deriving the equations of structure of Euclidean space as an effect related to the motion of a reference frame while leaving fixed the referred point (this will be at the basis of the “need” to resort to 5-dimensional Kaluza-Klein spaces. In chapter 7, we deal with the many different approaches to Lie derivatives, emphasizing the approaches of Cartan and Kähler, all of it from the perspective of transformations between integral curves of a differential system, rather than vector fields as differential operators. |
AuthorsJose Vargas and Doug Torr will make every effort to respond to questions on their papers and the subject of Cartan-Kähler Unification. |