An Alternative Surface Measures Construction in Finite-Dimensional Spaces and its Consistency with the Classical Approach
DOI:
https://doi.org/10.20535/1810-0546.2017.4.96805Keywords:
Surface measure, Surface area, Vector fieldAbstract
Background. The area formulae are well known for surfaces embedded into a finite-dimensional Euclidean space. However, in the case of an infinite-dimensional Banach manifold, such formulae cannot be used. Thus, a problem of finding an alternative approach to the surface measures construction appears, that, on the one hand, leads to classical results in finite-dimensional case, and on the other hand, can be used for infinite-dimensional Banach manifolds.
Objective. The aim of the paper is to get a construction of surface measure induced by the Lebesgue measure and the associated form for a parametrically defined surface embedded into finite-dimensional Euclidean space. Show the consistency of surface area calculation by this construction with an area calculated by using well-known classical formulae.
Methods. Basic results of mathematical analyses, measure theory and differential geometry are used.
Results. An alternative construction of surface measures induced by the Lebesgue measure on surfaces in finite-dimensional space is obtained. It is shown that such approach is consistent with the classical definition of the surface area.
Conclusions. The construction of surface measures suggested for infinite-dimensional spaces is a generalization of the classical approach in finite-dimensional spaces. Therefore further investigation of the described approach seems to be reasonable.
References
A.V. Uglanov, “Surface integrals in the Fréchet spaces”, Matematicheskij Sbornik, vol. 189, no. 11, pp. 139–157, 1998 (in Russian).
A.V. Uglanov, Integration on Infinite-Dimensional Surfaces and its Application. Dordrecht, Netherlands: Kluwer Acad. Publ., 2000.
O.V. Pugachev, “Capacities and surface measures in locally convex spaces”, Teorija Verojatnostej i ee Primenenija, vol. 53, no. 1, pp. 178–188, 2008 (in Russian).
Yu.V. Bogdansky, “Banach manifolds with bounded structure and the Gauss–Ostrogradskii formula”, Ukr. Mat. Zhurnal, vol. 64, no. 10, pp. 1299–1313, 2012 (in Russian).
Yu.V. Bogdansky, “Laplacian with respect to a measure on a Hilbert space and of the Dirichlet problem for the Poisson equation”, Ukr. Mat. Zhurnal, vol. 63, no. 9, pp. 1169–1178, 2011 (in Russian).
Yu.V. Bogdansky and Ya.Yu. Sanzharevskii, “The Dirichlet problem with Laplacian with respect to a measure in the Hilbert space”, Ukr. Mat. Zhurnal, vol. 66, no. 6, pp. 733–739, 2014 (in Russian).
Yu.V. Bogdansky, “Boundary trace operator in a domain of Hilbert space and the characteristic property of its kernel”, Ukr. Mat. Zhurnal, vol. 67, no. 11, pp. 1450–1460, 20155 (in Russian).
Yu.V. Bogdansky and K.V. Moravetska, “Surface measures on Banach manifolds with uniform structure”, Ukr. Mat. Zhurnal, vol. 69, no. 8, pp. 1030–1048, 2017 (in Russian).
Kh.D. Ikramov, Nonsymmetric Eigenvalue Problem. Moscow, SU: Nauka, 1991 (in Russian).
V.A. Zorich, Mathematical Analysis, vol. 2. Moscow, SU: Nauka, 1984 (in Russian).
Downloads
Published
Issue
Section
License
Copyright (c) 2017 Igor Sikorsky Kyiv Polytechnic Institute
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under CC BY 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work
