### 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.

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

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