Advanced Calculus of Several Variables

Advanced Calculus of Several Variables

1st Edition - January 1, 1973

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  • Author: C. H. Edwards
  • eBook ISBN: 9781483268057

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Advanced Calculus of Several Variables provides a conceptual treatment of multivariable calculus. This book emphasizes the interplay of geometry, analysis through linear algebra, and approximation of nonlinear mappings by linear ones. The classical applications and computational methods that are responsible for much of the interest and importance of calculus are also considered. This text is organized into six chapters. Chapter I deals with linear algebra and geometry of Euclidean n-space Rn. The multivariable differential calculus is treated in Chapters II and III, while multivariable integral calculus is covered in Chapters IV and V. The last chapter is devoted to venerable problems of the calculus of variations. This publication is intended for students who have completed a standard introductory calculus sequence.

Table of Contents

  • Preface

    I Euclidean Space and Linear Mappings

    1 The Vector Space ℛn

    2 Subspaces of ℛn

    3 Inner Products and Orthogonality

    4 Linear Mappings and Matrices

    5 The Kernel and Image of a Linear Mapping

    6 Determinants

    7 Limits and Continuity

    8 Elementary Topology of ℛn

    II Multivariable Differential Calculus

    1 Curves in ℛn

    2 Directional Derivatives and the Differential

    3 The Chain Rule

    4 Lagrange Multipliers and the Classification of Critical Points for Functions of Two Variables

    5 Maxima and Minima, Manifolds, and Lagrange Multipliers

    6 Taylor's Formula for Single-Variable Functions

    7 Taylor's Formula in Several Variables

    8 The Classification of Critical Points

    III Successive Approximations and Implicit Functions

    1 Newton's Method and Contraction Mappings

    2 The Multivariable Mean Value Theorem

    3 The Inverse and Implicit Mapping Theorems

    4 Manifolds in ℛn

    5 Higher Derivatives

    IV Multiple Integrals

    1 Area and the 1-Dimensional Integral

    2 Volume and the n-Dimensional Integral

    3 Step Functions and Riemann Sums

    4 Iterated Integrals and Fubini's Theorem

    5 Change of Variables

    6 Improper Integrals and Absolutely Integrable Functions

    V Line and Surface Integrals; Differential Forms and Stokes' Theorem

    1 Pathlength and Line Integrals

    2 Green's Theorem

    3 Multilinear Functions and the Area of a Parallelepiped

    4 Surface Area

    5 Differential Forms

    6 Stokes' Theorem

    7 The Classical Theorems of Vector Analysis

    8 Closed and Exact Forms

    VI The Calculus of Variations

    1 Normed Vector Spaces and Uniform Convergence

    2 Continuous Linear Mappings and Differentials

    3 The Simplest Variational Problem

    4 The Isoperimetric Problem

    5 Multiple Integral Problems

    Appendix: The Completeness of ℛ

    Suggested Reading

    Subject Index

Product details

  • No. of pages: 470
  • Language: English
  • Copyright: © Academic Press 1973
  • Published: January 1, 1973
  • Imprint: Academic Press
  • eBook ISBN: 9781483268057

About the Author

C. H. Edwards

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