Alternative construction of surface measure on a finite-dimensional space

Bohdan M. Snizhko

Abstract


To find surface measure in finite-dimensional spaces, the classical construction is used. However, the absence of an invariant Lebesgue measure in infinite-dimensional spaces prompts researchers to propose other methods of constructing surface measures. One of these methods, which will be referred to as the alternative, was proposed by Yuriy Bogdanskiy for spaces of an arbitrary (finite and infinite) dimension. An adaptation of this method for constructing surface measure, in the case of the space R^m, is presented in the article. The alternative construction is based on the concept of the phase flow of the autonomous Cauchy problem, in which the right side is a vector field that coincides with the field of the unit normal to the surface on the surface. Equivalence of the results of the classical and alternative methods for constructing surface measure for compact smooth elementary surfaces in R^m that have unit codimension is proved. In the research, modern instruments
of mathematical analysis, ordinary differential equations theory and differential geometry were used. Future investigations may apply to analysis of equivalence of classical and alternative methods for surfaces with arbitrary finite codimension.

Keywords


Surface measure; Finite-dimensional space; Classical construction of surface measure; Smooth elementary surface; Sectionally smooth surface; Alternative construction of surface measure; Phase flow of Cauchy problem.

References


Bogdanskii, Y. V. (2013). Banach manifolds with bounded structure and the Gauss–Ostrogradskii formula. Ukrainian Mathematical Journal, 64 (10), 1475–1494. https://doi.org/10.1007/s11253-013-0730-9

Bogdanskii, Y. V., & Moravetskaya, E. V. (2018). Surface measures on Banach manifolds with uniform structure. Ukrainian Mathematical Journal, 69 (8), 1196–1219. https://doi.org/10.1007/s11253-017-1425-4

Conlon, L. (2008). Differentiable manifolds. Basel: Birkhauser.

Fedoriuk, M. V. (1985). Ordinary differential equations [in Russian]. Moscow: Nauka.

Filippov, A. F. (2007). An introduction to differential equations [in Russian]. Moscow: KomKniga.

Zorich, V. A. (2004). Mathematical analysis II. Berlin: Springer.






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