Advanced Calculus of Several Variables

PrefaceI 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 RnII 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 PointsIII 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 DerivativesIV 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 FunctionsV 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 FormsVI 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 ProblemsAppendix: The Completeness of RSuggested ReadingSubject Index