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Parametric surfaces

I. Area

Published online by Cambridge University Press:  24 October 2008

E. R. Reifenberg
E. R. Reifenberg
Affiliation:
Trinity CollegeCambridge
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Extract

1. In this paper I consider various properties of the area of a surface including the relationship between Hausdorff measure and the Lebesgue area. The results here obtained will enable me, in a subsequent paper, to investigate the tangential properties of parametric surfaces. For simplicity of exposition I confine my attention to surfaces homoeomorphic to a disk, but the general case follows easily from cyclic element theory.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 47 , Issue 4 , October 1951 , pp. 687 - 698
Copyright
Copyright © Cambridge Philosophical Society 1951

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References

REFERENCES

(1)Besicovitch, A. S.A general form of the covering principle and the relative differentiation of additive functions. I. Proc. Cambridge Phil. Soc. 41 (1945), 103–10.Google Scholar
(2)Besicovitch, A. S.On the fundamental geometrical properties of linearly measurable plane sets of points. II. Math. Ann. 115 (1938), 296–329.Google Scholar
(3)Besicovitch, A. S. On the definition and value of the area of a surface. Quart. J. Math. (Oxford), 16 (1945), 86–102.Google Scholar
(4)Besicovitch, A. S.On surfaces of minimum area. Proc. Cambridge Phil. Soc. 44 (1948), 313–34.Google Scholar
(5)Besicovitch, A. S.Parametric surfaces. I. Compactness. Proc. Cambridge Phil. Soc. 45 (1949), 5–13.CrossRefGoogle Scholar
(6)Besicovitch, A. S.Parametric surfaces. II. Lower-semi-continuity. Proc. Cambridge Phil. Soc. 45 (1949), 14–23.Google Scholar
(7)Besicovitch, A. S.Parametric surfaces. III. Surfaces of minimum area. J. London Math. Soc. 23 (1948), 241–6.Google Scholar
(8)Federer, H.Hausdorff measure and Lebesgue area. Bull. American Math. Soc. (to be published).Google Scholar
(9)Youngs, J. W. T.The topological theory of Fréchet surfaces. Ann. Math. 45 (1944), 753–85.Google Scholar
(10)Courant, R. and Hilbert, D.Methoden der Mathematischen Physik 1 (Berlin, 1924), Chapter ii, § 2.Google Scholar