Ebook: Continuous Functions of Vector Variables

This text is appropriate for a one-semester course in what is usually called ad­ vanced calculus of several variables. The focus is on expanding the concept of continuity; specifically, we establish theorems related to extreme and intermediate values, generalizing the important results regarding continuous functions of one real variable. We begin by considering the function f(x, y, ... ) of multiple variables as a function of the single vector variable (x, y, ... ). It turns out that most of the n treatment does not need to be limited to the finite-dimensional spaces R , so we will often place ourselves in an arbitrary vector space equipped with the right tools of measurement. We then proceed much as one does with functions on R. First we give an algebraic and metric structure to the set of vectors. We then define limits, leading to the concept of continuity and to properties of continuous functions. Finally, we enlarge upon some topological concepts that surface along the way. A thorough understanding of single-variable calculus is a fundamental require­ ment. The student should be familiar with the axioms of the real number system and be able to use them to develop elementary calculus, that is, to define continuous junction, derivative, and integral, and to prove their most important elementary properties. Familiarity with these properties is a must. To help the reader, we provide references for the needed theorems.

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This text is an axiomatic treatment of the properties of continuous multivariable functions and related results from topology. In the context of normed vector spaces, the author covers boundedness, extreme values, and uniform continuity of functions, along with the connections between continuity and topological concepts such as connectedness and compactness. The order of topics deliberately mimics the order of development in elementary calculus. This sequencing allows for an elementary approach, with analogies to and generalizations from such familiar ideas as the Pythagorean theorem. The reader is frequently reminded that the pictures suggested by geometry are powerful guides and tools. The definition-theorem-proof format resides within an informal exposition, containing numerous historical comments and questions within and between the proofs. The objective is to present precise proofs, but in a structure and tone that teach the student to plan and write proofs, both in general and specifically for the real analysis course that will follow this one. Applications are included where they provide interesting illustrations of the principles and theorems presented. Problems, solutions, bibliography and index complete this book. `Continuous Functions of Vector Variables' is suitable for a course in multivariable calculus aimed at advanced undergraduates preparing for graduate programs in pure mathematics. Required background includes a course in the theory of single-variable calculus and the elements of linear algebra. Also by the author: 'Derivatives and Integrals of Multivariable Functions,' ISBN 0-8176-4274-9.

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This text is an axiomatic treatment of the properties of continuous multivariable functions and related results from topology. In the context of normed vector spaces, the author covers boundedness, extreme values, and uniform continuity of functions, along with the connections between continuity and topological concepts such as connectedness and compactness. The order of topics deliberately mimics the order of development in elementary calculus. This sequencing allows for an elementary approach, with analogies to and generalizations from such familiar ideas as the Pythagorean theorem. The reader is frequently reminded that the pictures suggested by geometry are powerful guides and tools. The definition-theorem-proof format resides within an informal exposition, containing numerous historical comments and questions within and between the proofs. The objective is to present precise proofs, but in a structure and tone that teach the student to plan and write proofs, both in general and specifically for the real analysis course that will follow this one. Applications are included where they provide interesting illustrations of the principles and theorems presented. Problems, solutions, bibliography and index complete this book. `Continuous Functions of Vector Variables' is suitable for a course in multivariable calculus aimed at advanced undergraduates preparing for graduate programs in pure mathematics. Required background includes a course in the theory of single-variable calculus and the elements of linear algebra. Also by the author: 'Derivatives and Integrals of Multivariable Functions,' ISBN 0-8176-4274-9.
Content:
Front Matter....Pages i-x
Euclidean Space....Pages 1-31
Sequences in Normed Spaces....Pages 33-53
Limits and Continuity in Normed Spaces....Pages 55-84
Characteristics of Continuous Functions....Pages 85-118
Topology in Normed Spaces....Pages 119-153
Back Matter....Pages 155-209

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This text is an axiomatic treatment of the properties of continuous multivariable functions and related results from topology. In the context of normed vector spaces, the author covers boundedness, extreme values, and uniform continuity of functions, along with the connections between continuity and topological concepts such as connectedness and compactness. The order of topics deliberately mimics the order of development in elementary calculus. This sequencing allows for an elementary approach, with analogies to and generalizations from such familiar ideas as the Pythagorean theorem. The reader is frequently reminded that the pictures suggested by geometry are powerful guides and tools. The definition-theorem-proof format resides within an informal exposition, containing numerous historical comments and questions within and between the proofs. The objective is to present precise proofs, but in a structure and tone that teach the student to plan and write proofs, both in general and specifically for the real analysis course that will follow this one. Applications are included where they provide interesting illustrations of the principles and theorems presented. Problems, solutions, bibliography and index complete this book. `Continuous Functions of Vector Variables' is suitable for a course in multivariable calculus aimed at advanced undergraduates preparing for graduate programs in pure mathematics. Required background includes a course in the theory of single-variable calculus and the elements of linear algebra. Also by the author: 'Derivatives and Integrals of Multivariable Functions,' ISBN 0-8176-4274-9.
Content:
Front Matter....Pages i-x
Euclidean Space....Pages 1-31
Sequences in Normed Spaces....Pages 33-53
Limits and Continuity in Normed Spaces....Pages 55-84
Characteristics of Continuous Functions....Pages 85-118
Topology in Normed Spaces....Pages 119-153
Back Matter....Pages 155-209
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