Advanced calculus

Abstract

Intro; CONTENTS; PREFACE; Chapter 0 Introduction; 1 Logic: quantifiers; 2 The logical connectives; 3 Negations of quantifiers; 4 Sets; 5 Restricted variables; 6 Ordered pairs and relations; 7 Functions and mappings; 8 Product sets; index notation; 9 Composition; 10 Duality; 11 The Boolean operations; 12 Partitions and equivalence relations; Chapter I Vector Spaces; 1 Fundamental notions; 2 Vector spaces and geometry; 3 Product spaces and Hom(V, W); 4 Affine subspaces and quotient spaces; 5 Direct sums; 6 Bilinearity; Chapter 2 Finite-Dimensional Vector Spaces; 1 Basps; 2 Dimension. 3 The dual space4 Matrices; 5 Trace and determinant; 6 Matrix computations; *7 The diagonalization of a quadratic form; Chapter 3 The Differential Calculus; 1 Review in R; 2 Norms; 3 Continuity; 4 Equivalent norms; 5 Infinitesimals; 6 The differential; 7 Directional derivatives; the mean-value theorem; 8 The differential and product spaces; 9 The differential and Rn; 10 Elementary applications; 11 The implicit-function theorem; 12 Submanifolds and Lagrange multipliers; *13 Functional dependence; *14 Uniform continuity and function-valued mappings; *15 The calculus of variations. *16 The second differential and the classification of critical points*17 The Taylor formula; Chapter 4 Compactness and Completeness; 1 Metric spaces; open and closed sets; *2 Topology; 3 Sequential convergence; 4 Sequential compactness; 5 Compactness and uniformity; 6 Equicontinuity; 7 Completeness; 8 A first look at Banach algebras; 9 The contraction mapping fixed-point theorem; 10 The integral of a parametrized arc; 11 The complex number system; *12 Weak methods; Chapter 5 Scalar Product Spaces; 1 Scalar products; 2 Orthogonal projection; 3 Self-adjoint transformations. 4 Orthogonal transformations5 Compact transformations; Chapter 6 Differential Equations; 1 The fundamental theorem; 2 Differentiable dependence on parameters; 3 The linear equation; 4 The nth-order linear equation; 5 Solving the inhomogeneous equation; 6 The boundary-value problem; 7 Fourier series; Chapter 7 Multilinear Functionals; 1 Bilinear functionals; 2 Multilinear functionals; 3 Permutations; 4 The sign of a permutation; 5 The subspace n of alternating tensors; 6 The determinant; 7 The exterior algebra; 8 Exterior powers of scalar product spaces; 9 The star operator. Chapter 8 Integration1 Introduction; 2 Axioms; 3 Rectangles and paved sets; 4 The minimal theory; 5 The minimal theory (continued); 6 Contented sets; 7 When is a set contented?; 8 Behavior under linear distortions; 9 Axioms for integration; 10 Integration of contented functions; 11 The change of variables formula; 12 Successive integration; 13 Absolutely integrable functions; 14 Problem set: The Fourier transform; Chapter 9 Differentiable Manifolds; 1 Atlases; 2 Functions, convergence; 3 Differentiable manifolds; 4 The tangent space; 5 Flows and vector fields; 6 Lie derivatives