Advanced Calculus of Several Variables

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