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 ℝⁿ to the field of differential forms.

Differential form

Meaning

Noun