CONTENTS
PREFACE
CHAPTER I. INTRODUCTORY CONSIDERATIONS
§1. The Main Theorems
§2. Some Basic Definitions for Curves
§3. Some Definitions for Non-Parametric Surfaces
§4. The Example of Schwarz and Peano
CHAPTER II. LEBESGUE AREA
§5. The Lebesgue Area L
§6. Some Alternate Definitions of Lebesgue Area
§7. Some Critical Considerations on Area
CHAPTER III. THE GEÖCZE AREAS V AND U AND THE PEANO AREA P
§8. The Topological Index
§9. The Geocze and Peano Areas V, U, P
§10. Continuous Mappings and Semicontinuous Collections
§11. Some Properties of the Euclidean Plane E2
CHAPTER IV. BV AND AC PLANE MAPPINGS
§12. BV Plane Mappings
§13. AC Plane Mappings
§14. Local Properties of Plane Mappings
§15. A Characterization of AC Plane Mappings
CHAPTER V. THE FIRST THEOREM
§16. An Analytical Property of Continuous Mappings
§17. Some Properties of Homotopy for Continuous Curves in E3
§18. The First Theorem
CHAPTER VI. THE CAVALIERI INEQUALITY
§19. On the Boundary of Open Sets (Carathéodory Theory)
§20. Contours of a Continuous Surface and the Cavalieri Inequality
CHAPTER VII. IDENTIFICATION OF LEBESGUE, GEÖCZE, PEANO AREAS
§21. The Equality V = U
§22. Some Limit Theorems for the Functions U and V
§23. Some Analytical Properties of Continuous Mappings
§24. The Equality V = L = P
§25. The Lebesgue Area as a Measure Function
CHAPTER VIII. GEOMETRICAL PROPERTIES AND THE SECOND THEOREM
§26. Regular Approximate Differentials
§27. Interval Functions
§28. Generalized Jacobians
§29. Formulas for the Transformation of Areas and Double Integrals
§30. The Second Theorem
CHAPTER IX. THE REPRESENTATION PROBLEM
§31. Fréchet Equivalence
§32. Mean Value Integrals of L-Integrable Functions
§33. Some Particular Types of Surfaces
§34. Representation Theorems for Non-Degenerate Surfaces
CHAPTER X. THE REPRESENTATION OF GENERAL SURFACES AND THE THIRD THEOREM
§35. Generalized Conformal Representations
§36. A Retraction Process for Surfaces
§37. Representation of General Surfaces, The Third Theorem
APPENDIX A. A DIRECT PROOF OF A PROPERTY OF CONTINUOUS SURFACES
APPENDIX B. WEIERSTRASS INTEGRAL OVER A SURFACE
BIBLIOGRAPHY
SPECIAL SIGNS AND ABBREVIATIONS