Open AccessOn the area of surfaces
Author(s) -
William Young
Publication year - 1919
Publication title -
proceedings of the royal society of london. series a, containing papers of a mathematical and physical character
Language(s) - English
Resource type - Journals
eISSN - 2053-9150
pISSN - 0950-1207
DOI - 10.1098/rspa.1919.0039
Subject(s) - surface (topology) , mathematics , surface integral , mathematical analysis , geometry , pure mathematics , integral equation , calculus (dental) , medicine , dentistry
The necessary and sufficient condition that a curve should possess a length, this length being given by the usual integral formula, is well known. The curve being defined by the equationsx =x (u ),y =y (u ), the condition is thatx (u ) andy (u ) should be expressible as integrals with respect tou . It may seem scarcely credible that no corresponding theorem is known with regard to the area of a surface. Such is, however, the case. And what is more surprising, no one has hitherto succeeded in giving such a definition of the area of a curved surface as permits of a determination of a sufficient condition of a general nature that the surface should possess an area, this area being given by the integral formula known to hold in the simplest cases.