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

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...

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...

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...

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

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...

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...

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...

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

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

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

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...

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 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...

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...

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

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

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

algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted...

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

positive form refers to several classes of real differential forms of Hodge type (p, p). Real (p,p)-forms on a complex manifold M are forms which are...

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...

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...

In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold...

concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan...

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...