Advanced Calculus of Several Variables (1973)

PREFACE

Part I. Euclidean Space and Linear Mappings

Chapter 1. THE VECTOR SPACE Rn

Chapter 2. SUBSPACES OF Rn

Chapter 3. INNER PRODUCTS AND ORTHOGONALITY

Chapter 4. LINEAR MAPPINGS AND MATRICES

Chapter 5. THE KERNEL AND IMAGE OF A LINEAR MAPPING

Chapter 6. DETERMINANTS

Chapter 7. LIMITS AND CONTINUITY

Chapter 8. ELEMENTARY TOPOLOGY OF Rn

Part II. Multivariable Differential Calculus

Chapter 1. CURVES IN R

Chapter 2. DIRECTIONAL DERIVATIVES AND THE DIFFERENTIAL

Chapter 3. THE CHAIN RULE

Chapter 4. LAGRANGE MULTIPLIERS AND THE CLASSIFICATION OF CRITICAL POINTS FOR FUNCTIONS OF TWO VARIABLES

Chapter 5. MAXIMA AND MINIMA, MANIFOLDS, AND LAGRANGE MULTIPLIERS

Chapter 6. TAYLOR'S FORMULA FOR SINGLE-VARIABLE FUNCTIONS

Chapter 7. TAYLOR'S FORMULA IN SEVERAL VARIABLES

Chapter 8. THE CLASSIFICATION OF CRITICAL POINTS

Part III. Successive Approximations and Implicit Functions

Chapter 1. NEWTON'S METHOD AND CONTRACTION MAPPINGS

Chapter 2. THE MULTIVARIABLE MEAN VALUE THEOREM

Chapter 3. THE INVERSE AND IMPLICIT MAPPING THEOREMS/a>

Chapter 4. MANIFOLDS IN Rn

Chapter 5. HIGHER DERIVATIVES

Part IV. Multiple Integrals

Chapter 1. AREA AND THE 1-DIMENSIONAL INTEGRAL

Chapter 2. VOLUME AND THE n-DIMENSIONAL INTEGRAL

Chapter 3. STEP FUNCTIONS AND RIEMANN SUMS

Chapter 4. ITERATED INTEGRALS AND FUBINI'S THEOREM

Chapter 5. CHANGE OF VARIABLES

Chapter 6. IMPROPER INTEGRALS AND ABSOLUTELY INTEGRABLE FUNCTIONS

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

Chapter 1. PATHLENGTH AND LINE INTEGRALS

Chapter 2. GREEN'S THEOREM

Chapter 3. MULTILINEAR FUNCTIONS AND THE AREA OF A PARALLELEPIPED

Chapter 4. SURFACE AREA

Chapter 5. DIFFERENTIAL FORMS

Chapter 6. STOKES' THEOREM

Chapter 7. THE CLASSICAL THEOREMS OF VECTOR ANALYSIS

Chapter 8. CLOSED AND EXACT FORMS

Part VI. The Calculus of Variations

Chapter 1. NORMED VECTOR SPACES AND UNIFORM CONVERGENCE

Chapter 2. CONTINUOUS LINEAR MAPPINGS AND DIFFERENTIALS

Chapter 3. THE SIMPLEST VARIATIONAL PROBLEM

Chapter 4. THE ISOPERIMETRIC PROBLEM

Chapter 5. MULTIPLE INTEGRAL PROBLEMS

APPENDIX

REFERENCES