Surface measures associated with a non-invariant measure in a finite-dimensional space
DOI:
https://doi.org/10.20535/SRIT.2308-8893.2019.4.12Keywords:
associated surface measure, smooth elementary surface, Jordan measure, Lebesgue measure, non-invariant measureAbstract
A generalization of the classical surface measure construction for smooth elementary surfaces of the arbitrary codimension embedded in a finite-dimensional Euclidean space is proposed. Namely, an approach to constructing a surface measure associated with a measure that is absolutely continuous with respect to the invariant Lebesgue measure is presented. This construction of the associated surface measure is correct in the sense that the value of the indicated surface measure does not depend on the choice of its parameterization in a class of equivalent parameterizations. An adequacy of the proposed approach is confirmed by the fact that the surface measure associated with the invariant Lebesgue measure coincides with the well-known classical surface measure construction, a particular case of which (area of a two-dimensional smooth parameterized surface in a three-dimensional space) is considered in the calculus course.References
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