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Cartan's chief tool was the calculus of exterior differential forms, which he helped to create and develop in the ten years following his thesis and then proceeded to apply to a variety of problems in differential geometry, Lie groups, analytical dynamics, and general relativity. He discussed a large number of examples, treating them in an ...
Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to Élie Cartan with reference to his 1899 paper. [1]
one in differential geometry: = +, where ,, and are Lie derivative, exterior derivative, and interior product, respectively, acting on differential forms. See interior product for the detail. It is also called the Cartan homotopy formula or Cartan magic formula .
Henri Paul Cartan (French:; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. [1] [2] [3]He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of composer Jean Cartan [fr; de], physicist Louis Cartan [] and mathematician Hélène Cartan [], and the son-in-law of physicist Pierre ...
That is, df is the unique 1-form such that for every smooth vector field X, df (X) = d X f , where d X f is the directional derivative of f in the direction of X. The exterior product of differential forms (denoted with the same symbol ∧) is defined as their pointwise exterior product.
In mathematics, Cartan's equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up to a diffeomorphism. For example, if M and N are two Riemannian manifolds with metrics g and h , respectively, when is there a diffeomorphism
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principal motivations for, his ...
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, [1] is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.