Differential completions and compactifications of a differential space

Dziewa-Dawidczyk, Diana; Pasternak-Winiarski, Zbigniew

doi:10.1515/dema-2013-0417

Cited by 1 publication

(10 citation statements)

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“…It is easy to see that if G is an arbitrary family of generators of the differential structure C then in the formulation of Theorem 1 and Proposition 1 we can put G instead of C preserving only the condition: ω i 1 ...i k ∈ C. For the definition and basic properties of a set of generators of a differential structure see [1], [2], [3] or [4].…”

Section: Remarkmentioning

confidence: 99%

“…We have to give the generalization of n-dimensional cubes and chains. In order to do this we will use the theory of completions and compactifications of a differential space described in [2]. At first we will extend smooth functions.…”

Section: Remarkmentioning

confidence: 99%

“…Let (M, C) be a differential space such that M is Hausdorff space and C is generated by the family of functions G, which defines uniform structure U G on M (see [2]). Any function f ∈ C which is uniform with respect to U G and standard uniform structure on R can be extended in the univocal way to the continuous function on the completion compl G M of the uniform space (M, U G ).…”

Section: Remarkmentioning

confidence: 99%

“…Let π : T M → M be the natural projection, i.e. π(v) = m for any v ∈ T m M. We endow T M with the differential structure T C generated by the family of function {α • π, α ∈ C} ∪ {dα, α ∈ C}, where dα : T M → R, dα(v) = v(α) (see [2], Definition 2.4 and remarks after this definition). Then the map π is smooth with respect to differential structures C and T C. Definition 1.…”

Section: Point Formsmentioning

confidence: 99%

“…, [2], [3] or [4]) and " ∧ " means that the variable disappears in that expression. For any smooth mapping F : [0; 1] n → [−1, 1] G and any smooth point kform ω smoothly extendable with respect to G operator d fulfils the equality:…”

mentioning

confidence: 99%

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Integration on Differential Spaces

Dziewa-Dawidczyk

Pasternak-Winiarski²

2014

Demonstratio Mathematica

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Abstract. The generalization of the n-dimensional cube, an n-dimensional chain, the exterior derivative and the integral of a differential n-form on it are introduced and investigated. The analogue of Stokes theorem for the differential space is given. IntroductionThis article is the fifth of the series of papers concerning integration of differential forms and densities on differential spaces. We describe our generalization of the theory of integration of smooth skew-symmetric forms on cubes and chains.In Section 2, we present definition and properties of so called point differential forms on differential spaces. We prove the theorem about the local representation of such forms (Theorem 1). Section 3 of the paper contains basic definitions and the description of preliminary facts concerning generalized cubes and chains. We define the notion of a generalized ndimensional smooth cube on the differential space pM, Cq, the notion of a smooth n-dimensional chain of generalized smooth cubes on pM, Cq and the integral of a smooth point n-form on a smooth n-dimensional chain. In Section 4, we prove the existence of integrals for a wide class of smooth chains and skew-symmetric forms. To formulate these results we introduce the classes of cubes, chains and point forms smoothly extendable with respect to the family G of generators of a differential structure C on a set M . At the end of the paper we give an analogue of Stokes theorem in the case of smoothly extendable forms and chains (Theorem 3). To do that, we introduce the boundary of generalized n-dimensional chain (Definition 13).Without any other explanation we use the following symbols: N -the set of natural numbers; Q -the set of rational numbers; R -the set of reals.2010 Mathematics Subject Classification: 58A40, 26A18.

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Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Point Formsmentioning

confidence: 99%

mentioning

confidence: 99%

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