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Higher dimensional mappings for which the area formula holds.

C Goffman1, W P Ziemer

  • 1PURDUE UNIVERSITY, LAFAYETTE, INDIANA.

Proceedings of the National Academy of Sciences of the United States of America
|March 1, 1970
PubMed
Summary

This study confirms the classical formula for Lebesgue area in higher dimensions. It proves that for continuous mappings from m-dimensional space to n-dimensional space, the formula holds if partial derivatives are in L(m).

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Area of Science:

  • Differential geometry
  • Geometric measure theory
  • Real analysis

Background:

  • The classical formula for Lebesgue area applies to mappings from 2D to nD space under specific derivative conditions.
  • The generalization of this formula to higher dimensions (mD to nD) has remained an open problem.

Purpose of the Study:

  • To address the long-standing question regarding the Lebesgue area formula for mappings between m-dimensional and n-dimensional spaces.
  • To establish the validity of the classical formula in a more general setting.

Main Methods:

  • The study investigates continuous mappings from m-dimensional space to n-dimensional space.
  • It analyzes the properties of partial derivatives of these mappings, specifically their membership in the L(m) class.

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Main Results:

  • The research confirms that the classical Lebesgue area formula is applicable to continuous mappings from m-dimensional space to n-dimensional space (m <= n).
  • It is shown that this formula holds when the partial derivatives of the mapping exist almost everywhere and belong to the L(m) class.

Conclusions:

  • The findings generalize the classical Lebesgue area formula to higher-dimensional spaces.
  • This work resolves a significant open problem in geometric measure theory and real analysis.