differential form

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]

differential form (plural differential forms)

1. (differential geometry, tensor calculus, sometimes as "differential p-form") 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.

   • 2001, Shigeyuki Morita, translated by Teruko Nagase and Katsumi Nomizu, Geometry of Differential Forms, American Mathematical Society, page 145:

     Now that the length of tangent vector is defined, the magnitude of a differential form is also determined. By making use of this fact we can give a more precise statement to the theorem of de Rham. That is, from the point of view of magnitude, we may prove that within the set of all closed forms representing a de Rham cohomology class, there is one and only one differential form that has the best shape. Such a form is called a harmonic form, and it can be characterized by using a differential operator called the Laplacian.

   • 2004, Ismo V. Lindell, Differential Forms in Electromagnetics, IEEE Press, page xiii,

     The present text attempts to serve as an introduction to the differential form formalism applicable to electromagnetic field theory. A glance at Figure 1.2 on page 18, presenting the Maxwell equations and the medium equation in terms of differential forms, gives the impression that there cannot exist a simpler way to express these equations, and so differential forms should serve as a natural language for electromagnetism.

   • 2009, Ravi P. Agarwal, Shusen Ding, Craig Nolder, Inequalities for Differential Forms, Springer, page 1:

     In this first chapter, we discuss various versions of the Hardy-Littlewood inequality for differential forms, including the local cases, the global cases, one weight cases, and two-weight cases. We know that differential forms are generalizations of the functions, which have been widely used in many fields, including potential theory, partial differential equations, quasiconformal mappings, nonlinear analysis, electromagnetism, and control theory; see [1-19], for example. During recent years new interest has developed in the study of the ${\displaystyle L^{p}}$ theory of differential forms on manifolds [20, 21]. […] The development of the ${\displaystyle L^{p}}$ theory of differential forms has made it possible to transport all notations of differential calculus in ${\displaystyle \mathbb {R} ^{n}}$ to the field of differential forms.

• The term often appears as differential ${\displaystyle p}$-form (or simply ${\displaystyle p}$-form), where ${\displaystyle p}$ is a variable representing a positive integer (the order of the tensor).

• Pfaffian form
• p-form

• differential p-form

• form
• harmonic form
• linear form
• multilinear form

• exterior algebra (= Grassmann algebra)
• exterior derivative
• exterior product (= wedge product)
• Riemannian manifold

• Differential form on Encyclopedia of Mathematics
• Differential k-Form on Wolfram MathWorld
• Antisymmetric tensor on Wikipedia.Wikipedia
• bilinear form on Wikipedia.Wikipedia
• Pfaffian form on Wikipedia.Wikipedia
• Riemannian manifold on Wikipedia.Wikipedia
• Tensor on Wikipedia.Wikipedia
• Tensor field on Wikipedia.Wikipedia
• Exterior algebra on Wikipedia.Wikipedia
• Hodge theory on Wikipedia.Wikipedia