calculus 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...

exact and forms whose exterior derivative is 0 are called closed (see Closed and exact differential forms); the relationship d2 = 0 then says that exact forms...

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

heat and work, but is defined more generally within mathematics as a type of differential form. In contrast, an integral of an exact differential is always...

systolic geometry. Closed and exact differential forms Complex differential form Vector-valued differential form Equivariant differential form Calculus on Manifolds...

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...

list of real analysis topics, list of calculus topics. Closed and exact differential forms Contact (mathematics) Contour integral Contour line Critical...

and requires choosing a gauge. Fundamental theorem of vector calculus Magnetic vector potential Solenoidal vector field Closed and Exact Differential...

equation) (partial differential equations) Poincaré lemma of closed and exact differential forms Cotlar–Stein lemma Ehrling's lemma Riesz's lemma Abel's lemma...

learning Multivector Geometric algebra Clifford algebra Closed and exact differential forms Component-free treatment of tensors Cramer's rule Dual space...

properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions...

{\displaystyle \Omega _{c}^{1}(S^{1})} and d Ω 0 ( S 1 ) {\displaystyle d\Omega ^{0}(S^{1})} are the closed and exact differential forms on the circle group); in particular...

one-forms are described locally, particularly in local coordinates. In a local coordinate system, a one-form is a linear combination of the differentials...

d_{\mathfrak {g}}} -closed or d g {\displaystyle d_{\mathfrak {g}}} -exact forms are called equivariantly closed or equivariantly exact. The integral of...

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

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

manifolds. The theorem is foundational in differential topology and calculus on manifolds. Contact geometry studies 1-forms that maximally violates the assumptions...

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...

F(x,y_{1},\ldots ,y_{n})).} Some differential equations have solutions that can be written in an exact and closed form. Several important classes are given...

mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws...

In general relativity, an exact solution is a (typically closed form) solution of the Einstein field equations whose derivation does not invoke simplifying...

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

(or closed elements), and the elements in the image of d are called (co)boundaries (or exact elements). Right from the definition of the differential, all...

The linear variable differential transformer (LVDT) – also called linear variable displacement transformer, linear variable displacement transducer, or...

system of differential equations can be succinctly stated in terms of differential forms (i.e., for a form to be exact, it needs to be closed). See integrability...

manifold could be encoded by differential forms. As a particular form of this, he conjectured that a closed form is exact if it integrates to zero over...

\Omega ^{1}(M)} . Differential k {\displaystyle k} -forms, which we refer to simply as k {\displaystyle k} -forms here, are differential forms defined on T...

a differential form with the Euler class e. The Gysin sequence is a long exact sequence not only for the de Rham cohomology of differential forms, but...

In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form d d x [ p ( x ) d y...

a differential 0-form: f ∈ C 0 ( U ) = Ω 0 ( U ) {\displaystyle f\in C^{0}(U)=\Omega ^{0}(U)} . We first construct differential 1-forms from 0-forms and...