In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The...

In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section...

and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form, α...

vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values...

complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients. Complex forms have...

has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called harmonic. The theory...

shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry...

In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced...

. More generally, any covariant tensor field – in particular any differential form – on N {\displaystyle N} may be pulled back to M {\displaystyle M}...

field. Where no such symmetry exists, Gauss's law can be used in its differential form, which states that the divergence of the electric field is proportional...

structure, in terms of a system of differential forms. The idea is to take advantage of the way a differential form restricts to a submanifold, and the...

In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal...

to introduce the notion of the pullback of a differential form. Roughly speaking, when a differential form is integrated, applying the pullback transforms...

In mathematics, differential forms on a Riemann surface are an important special case of the general theory of differential forms on smooth manifolds...

In differential geometry, a Lie-algebra-valued form is a differential form with values in a Lie algebra. Such forms have important applications in the...

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions...

In differential geometry, an equivariant differential form on a manifold M acted upon by a Lie group G is a polynomial map α : g → Ω ∗ ( M ) {\displaystyle...

ideal Differential geometry, exterior differential, or exterior derivative, is a generalization to differential forms of the notion of differential of a...

calculus, a differential or differential form is said to be exact or perfect (exact differential), as contrasted with an inexact differential, if it is...

In algebra, the ring of polynomial differential forms on the standard n-simplex is the differential graded algebra: Ω poly ∗ ( [ n ] ) = Q [ t 0 , . ....

geometry and the theory of complex manifolds, a logarithmic differential form is a differential form with poles of a certain kind. The concept was introduced...

Gauss's law for magnetism can be written in two forms, a differential form and an integral form. These forms are equivalent due to the divergence theorem...

language of differential geometry and differential forms is used. The electric and magnetic fields are now jointly described by a 2-form F in a 4-dimensional...

putting the difference of two objects in normal form. Canonical form can also mean a differential form that is defined in a natural (canonical) way. Given...

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first...

geometrical significance if the differential is regarded as a particular differential form, or analytical significance if the differential is regarded as a linear...

multilinear forms defines a natural exterior product for differential forms. Differential forms play a major role in diverse areas of differential geometry...

equivalent forms: the integral form, in which we look at the amount of energy flowing into or out of a body as a whole, and the differential form, in which...

condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed p-form on an open ball...

of differential form. In contrast, an integral of an exact differential is always path independent since the integral acts to invert the differential operator...