Open AccessAN EXAMPLE IN SURFACE AREA
Author(s) -
Casper Goffman
Publication year - 1969
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 5.011
H-Index - 771
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.63.1.38
Subject(s) - unit cube , lebesgue integration , surface (topology) , mathematics , cube (algebra) , value (mathematics) , geometry , unit (ring theory) , mathematical analysis , statistics , mathematics education
For length and area, a central fact is that the value of the length of a curve or the area of a surface, as given by the Lebesgue theory, is at least as great as that given by the classical formula, whenever the latter has meaning. This is now found not to be valid in higher dimensions. We give an example of a continuous mapping of the unit cube into itself for which the value given by the formula exceeds the three-dimensional Lebesgue area of the corresponding suface.