DOI: https://doi.org/10.20535/1810-0546.2017.4.96805

### An Alternative Surface Measures Construction in Finite-Dimensional Spaces and its Consistency with the Classical Approach

#### Abstract

**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.

#### Keywords

#### Full Text:

PDF (Українська)#### 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).

#### GOST Style Citations

Copyright (c) 2017 Igor Sikorsky Kyiv Polytechnic Institute

This work is licensed under a Creative Commons Attribution 4.0 International License.