Ebook: Multivariable Analysis

This book contains an introduction to the theory of functions, with emphasis on functions of several variables. The central topics are the differentiation and integration of such functions. Although many of the topics are familiar, the treatment is new; the book developed from a new approach to the theory of differentiation. Iff is a function of two real variables x and y, its deriva­ tives at a point Po can be approximated and found as follows. Let PI' P2 be two points near Po such that Po, PI, P2 are not on a straight line. The linear function of x and y whose values at Po, PI' P2 are equal to those off at these points approximates f near Po; determinants can be used to find an explicit representation of this linear function (think of the equation of the plane through three points in three-dimensional space). The (partial) derivatives of this linear function are approximations to the derivatives of f at Po ; each of these (partial) derivatives of the linear function is the ratio of two determinants. The derivatives off at Po are defined to be the limits of these ratios as PI and P2 approach Po (subject to an important regularity condition). This simple example is only the beginning, but it hints at a m theory of differentiation for functions which map sets in IRn into IR which is both general and powerful, and which reduces to the standard theory of differentiation in the one-dimensional case.

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Content:
Front Matter....Pages i-xiv
Differentiable Functions and Their Derivatives....Pages 1-67
Uniform Differentiability and Approximations; Mappings....Pages 68-101
Simplexes, Orientations, Boundaries, and Simplicial Subdivisions....Pages 102-194
Sperner’s Lemma and the Intermediate-Value Theorem....Pages 195-236
The Inverse-Function Theorem....Pages 237-262
Integrals and the Fundamental Theorem of the Integral Calculus....Pages 263-367
Zero Integrals, Equal Integrals, and the Transformation of Integrals....Pages 368-406
The Evaluation of Integrals....Pages 407-442
The Kronecker Integral and the Sperner Degree....Pages 443-493
Differentiable Functions of Complex Variables....Pages 494-495
Back Matter....Pages 573-655

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Content:
Front Matter....Pages i-xiv
Differentiable Functions and Their Derivatives....Pages 1-67
Uniform Differentiability and Approximations; Mappings....Pages 68-101
Simplexes, Orientations, Boundaries, and Simplicial Subdivisions....Pages 102-194
Sperner’s Lemma and the Intermediate-Value Theorem....Pages 195-236
The Inverse-Function Theorem....Pages 237-262
Integrals and the Fundamental Theorem of the Integral Calculus....Pages 263-367
Zero Integrals, Equal Integrals, and the Transformation of Integrals....Pages 368-406
The Evaluation of Integrals....Pages 407-442
The Kronecker Integral and the Sperner Degree....Pages 443-493
Differentiable Functions of Complex Variables....Pages 494-495
Back Matter....Pages 573-655
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