![]() ![]() This process is experimental and the keywords may be updated as the learning algorithm improves. These keywords were added by machine and not by the authors. 19, the language of differential forms shows that Maxwell’s theory of electromagnetism fits Einstein’s theory of special relativity, whereas the language of classical vector calculus conceals the relativistic invariance of the Maxwell equations. It turns out that Cartan’s differential calculus is the most important analytic tool in modern differential geometry and differential topology, and hence Cartan’s calculus plays a crucial role in modern physics (gauge theory, theory of general relativity, the Standard Model in particle physics). It emerged in the study of point mechanics, elasticity, fluid mechanics, heat conduction, and electromagnetism. ![]() Cartan’s calculus has its roots in physics. The key idea is to combine the notion of the Leibniz differential df with the alternating product a∧ b due to Grassmann (1809–1877). The History of Differential Forms from Clairaut to Poincaré by Victor J.Cartan’s calculus is the proper language of generalizing the classical calculus due to Newton (1643–1727) and Leibniz (1646–1716) to real and complex functions with n variables. ![]() This is what you were really asking and looking for. But he only made sparing use of Grassmann's anti-commuting rule, just one place in the treatise, as far as I know. Before Maxwell stripped down Hamilton's quaternions to a vector algebra and applied it in his treatise in the 1870's, he used differential forms in his papers in the 1850's and 1860's and also made much more use of them in the treatise than he did vectors. The exterior algebra (that is: where differentials anti-commute with each other) is from Grassmann in the 1840's. This requires only $C^2$-ness, which is what is already required for the theorem. It might be possible to use the 2nd order Taylor's Theorem with remainder to directly establish necessity, eliminating the need to use any infinite series expansion. ![]() Differential forms came first and the general integrability theorem actually preceded differential forms, going back to Clairaut, 1739-1740.įor an equation of the form $A dx + B dy = dC$, Clairaut used Taylor expansions to prove the necessity of $∂A/∂y = ∂B/∂x$ and indefinite integrals to prove sufficiency. Victor Katz remarks elsewhere that the connection between differential forms and the big three theorems of vector calculus, as expressed by the generalized Stokes theorem, did not appear in textbooks until the second half of the 20th century, the first occurrence probably being in a 1959 Advanced Calculus text. Volterra, Sulle funzioni coniugate, Rendiconti Accademia dei Licei (4) 5 (1889), 599-611. Goursat, Sur certaines systèmes d'équations aux différentielles totales et sur une généralisation du problème de Pfaff, Ann. Cartan, Sur certaines expressions différentielles et sur le problème de Pfaff, Ann. $$\int_M d\omega=\int_\omega,$$įirst stated in coordinate free form by Volterra in 1889. These three theorems were all special cases of a generalised Stokes (Gauss, Green, Stokes) could be easily stated using differentialįorms, it was Edouard Goursat (1858-1936) who in 1917 first noted that That is to say, d is an antiderivation of degree 1 on the. Although Cartan realized in 1899 that the three theorems of vector calculus The exterior derivative is defined to be the unique -linear mapping from k -forms to (k + 1) -forms that has the following properties: df is the differential of f for a 0 -form f. ![]()
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