Meaning of differential form | Babel Free

Definitions

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.

sometimes

Equivalents

Examples

“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.”

“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.”

“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 Lᵖ theory of differential forms on manifolds [20, 21].[…]The development of the Lᵖ theory of differential forms has made it possible to transport⟳ all notations of differential calculus in #92;mathbbRⁿ to the field of differential forms.”

CEFR level