An alternative approach to constructing the surface integral over a surface of an arbitrary codimension in Rn
Abstract
In this paper, the surface integral over a surface of an arbitrary codimension in Rn is constructed using an alternative approach. Surface measures are built with an infinitesimal procedure using a set of pairwise commuting vector fields that have global fluxes and are transversal to the given surface. Densities of the constructed measures with respect to the classical one are found and, based upon the comparative analysis, it is concluded that constructed measures present a generalization of the classical surface measures. It is deemed to be reasonable to generalize this approach to surfaces of finite codimensions in infinite-dimensional spaces.
References
Skorokhod А.V. Integrirovaniye v gil’bertovom prostranstve. — M.: Nauka, 1975. — 232 s.
Uglanov А.V. Poverkhnostnyye integraly v prostranstvakh Freshe // Mat. sbornik. — 1998. — 189, № 11. — S. 139–157.
Kudryavtsev L.D. Kurs matematicheskogo analiza (v dvukh tomakh) // Uchebnik dlya studentov universitetov i vuzov. — M.: Vysshaya shkola, 1981. — T. II. — 584 s.
Bogdanskiy YU.V. Banakhovy mnogoobraziya s ogranichennoy strukturoy i formula Gaussa-Ostrogradskogo // Ukrayins'kyy matematychnyy zhurnal, 2012. — 64, # 10. — S. 1299–1313.
Gromol D., Klingenberg V., MeyYEr V. Rimanova geometriya v tselom. — M.: Mir, 1971. — 343 s.
