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




Surface measure, Surface area, Vector field


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.


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).