Continuous Transformations in Analysis

1. Front Matter

   Pages II-VII

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2. Background in topology

   • T. Rado, P. V. Reichelderfer

   Pages 1-110

3. Topological study of continuous transformations in Rⁿ

   • T. Rado, P. V. Reichelderfer

   Pages 110-190

4. Background in Analysis

   • T. Rado, P. V. Reichelderfer

   Pages 190-212

5. Bounded variation and absolute continuity in Rⁿ

   • T. Rado, P. V. Reichelderfer

   Pages 212-292

6. Differentiable transformations in Rⁿ

   • T. Rado, P. V. Reichelderfer

   Pages 292-377

7. Continuous transformations in R²

   • T. Rado, P. V. Reichelderfer

   Pages 377-438

8. Back Matter

   Pages 439-442

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The general objective of this treatise is to give a systematic presenta­ tion of some of the topological and measure-theoretical foundations of the theory of real-valued functions of several real variables, with particular emphasis upon a line of thought initiated by BANACH, GEOCZE, LEBESGUE, TONELLI, and VITALI. To indicate a basic feature in this line of thought, let us consider a real-valued continuous function I(u) of the single real variable tt. Such a function may be thought of as defining a continuous translormation T under which x = 1 (u) is the image of u. About thirty years ago, BANACH and VITALI observed that the fundamental concepts of bounded variation, absolute continuity, and derivative admit of fruitful geometrical descriptions in terms of the transformation T: x = 1 (u) associated with the function 1 (u). They further noticed that these geometrical descriptions remain meaningful for a continuous transformation T in Euclidean n-space Rff, where T is given by a system of equations of the form 1-/(1 ff) X-I U, . . . ,tt ,. ", and n is an arbitrary positive integer. Accordingly, these geometrical descriptions can be used to define, for continuous transformations in Euclidean n-space Rff, n-dimensional concepts 01 bounded variation and absolute continuity, and to introduce a generalized Jacobian without reference to partial derivatives. These ideas were further developed, generalized, and modified by many mathematicians, and significant applications were made in Calculus of Variations and related fields along the lines initiated by GEOCZE, LEBESGUE, and TONELLI.

• Algebraic
• Analysis
• Continuous function
• Topology
• calculus
• equation
• function
• variable

• The Ohio State University, USA

  T. Rado, P. V. Reichelderfer

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