Ebook: Derivatives and Integrals of Multivariable Functions

This text is appropriate for a one-semester course in what is usually called ad­ vanced calculus of several variables. The approach taken here extends elementary results about derivatives and integrals of single-variable functions to functions in several-variable Euclidean space. The elementary material in the single- and several-variable case leads naturally to significant advanced theorems about func­ tions of multiple variables. In the first three chapters, differentiability and derivatives are defined; prop­ erties of derivatives reducible to the scalar, real-valued case are discussed; and two results from the vector case, important to the theoretical development of curves and surfaces, are presented. The next three chapters proceed analogously through the development of integration theory. Integrals and integrability are de­ fined; properties of integrals of scalar functions are discussed; and results about scalar integrals of vector functions are presented. The development of these lat­ ter theorems, the vector-field theorems, brings together a number of results from other chapters and emphasizes the physical applications of the theory.

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This work provides a systematic examination of derivatives and integrals of multivariable functions. The approach taken here is similar to that of the author’s previous text, "Continuous Functions of Vector Variables": specifically, elementary results from single-variable calculus are extended to functions in several-variable Euclidean space. Topics encompass differentiability, partial derivatives, directional derivatives and the gradient; curves, surfaces, and vector fields; the inverse and implicit function theorems; integrability and properties of integrals; and the theorems of Fubini, Stokes, and Gauss. Prerequisites include background in linear algebra, one-variable calculus, and some acquaintance with continuous functions and the topology of the real line.

Written in a definition-theorem-proof format, the book is replete with historical comments, questions, and discussions about strategy, difficulties, and alternate paths.  "Derivatives and Integrals of Multivariable Functions" is a rigorous introduction to multivariable calculus that will help students build a foundation for further explorations in analysis and differential geometry.

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This work provides a systematic examination of derivatives and integrals of multivariable functions. The approach taken here is similar to that of the author’s previous text, "Continuous Functions of Vector Variables": specifically, elementary results from single-variable calculus are extended to functions in several-variable Euclidean space. Topics encompass differentiability, partial derivatives, directional derivatives and the gradient; curves, surfaces, and vector fields; the inverse and implicit function theorems; integrability and properties of integrals; and the theorems of Fubini, Stokes, and Gauss. Prerequisites include background in linear algebra, one-variable calculus, and some acquaintance with continuous functions and the topology of the real line.

Written in a definition-theorem-proof format, the book is replete with historical comments, questions, and discussions about strategy, difficulties, and alternate paths.  "Derivatives and Integrals of Multivariable Functions" is a rigorous introduction to multivariable calculus that will help students build a foundation for further explorations in analysis and differential geometry.

Content:
Front Matter....Pages i-xi
Differentiability of Multivariable Functions....Pages 1-32
Derivatives of Scalar Functions....Pages 33-71
Derivatives of Vector Functions....Pages 73-103
Integrability of Multivariable Functions....Pages 105-134
Integrals of Scalar Functions....Pages 135-200
Vector Integrals and the Vector-Field Theorems....Pages 201-246
Back Matter....Pages 247-319

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This work provides a systematic examination of derivatives and integrals of multivariable functions. The approach taken here is similar to that of the author’s previous text, "Continuous Functions of Vector Variables": specifically, elementary results from single-variable calculus are extended to functions in several-variable Euclidean space. Topics encompass differentiability, partial derivatives, directional derivatives and the gradient; curves, surfaces, and vector fields; the inverse and implicit function theorems; integrability and properties of integrals; and the theorems of Fubini, Stokes, and Gauss. Prerequisites include background in linear algebra, one-variable calculus, and some acquaintance with continuous functions and the topology of the real line.

Written in a definition-theorem-proof format, the book is replete with historical comments, questions, and discussions about strategy, difficulties, and alternate paths.  "Derivatives and Integrals of Multivariable Functions" is a rigorous introduction to multivariable calculus that will help students build a foundation for further explorations in analysis and differential geometry.

Content:
Front Matter....Pages i-xi
Differentiability of Multivariable Functions....Pages 1-32
Derivatives of Scalar Functions....Pages 33-71
Derivatives of Vector Functions....Pages 73-103
Integrability of Multivariable Functions....Pages 105-134
Integrals of Scalar Functions....Pages 135-200
Vector Integrals and the Vector-Field Theorems....Pages 201-246
Back Matter....Pages 247-319
....