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differential form

noun

Etymology

From early 20th century. The concept was clarified chiefly by French mathematician Élie Cartan (1869–1951). In his fundamental paper, 1899, Sur certaines expressions différentielles et le problème de Pfaff, Annales Scientifiques de l'École Normale Supérieure (3), tome 16, Cartan used the (French) term expression différentielle, and in his 1922, Leçons sur les invariants intégraux, Hermann, he used the terms exterior differential form and exterior derivative.

  1. derived from mathematician Élie Cartan

Definitions

  1. A completely antisymmetric tensor (of order p) that is defined on a Riemannian manifold

    A completely antisymmetric tensor (of order p) that is defined on a Riemannian manifold; an expression, derived by applying a formalism to said tensor, that represents an integrand over the manifold.

    • The notion of differential form combines the concepts of multilinear form (itself an extension of linear form) and smooth function.
    • The choice of a Riemannian manifold - roughly speaking, a differentiable manifold whose every point has a tangent space with a defined metric - means it is possible to define a differential form over it.
    • Differential forms provide a unified approach to defining integrands over curves, surfaces and higher-dimensional manifolds, as well as providing an approach to multivariable calculus that is independent of coordinates.

The neighborhood

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sense glosses and etymology drawn from English Wiktionary · source · CC-BY-SA