Book:Lynn H. Loomis/Advanced Calculus/Second Edition

From ProofWiki
Jump to navigation Jump to search

Lynn H. Loomis and Shlomo Sternberg: Advanced Calculus (2nd Edition)

Published $\text {1990}$, Jones and Bartlett

ISBN 0-86720-122-3.


Subject Matter


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 1 Vector Spaces
1 Fundamental notions
2 Vector spaces and geometry
3 Product spaces and $\map {\operatorname {Hom} } {V, W}$
4 Affine subspaces and quotient spaces
5 Direct sums
6 Bilinearity
Chapter 2 Finite-Dimensional Vector Spaces
1 Bases
2 Dimension
3 The dual space
4 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 $\R^n$
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 transformations
5 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 an of alternating tensors
6 The determinant
7 The exterior algebra
8 Exterior powers of scalar product spaces
9 The star operator
Chapter 8 Integration
1 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
7 Linear differential forms
8 Computations with coordinates
9 Riemann metrics
Chapter 10 The Integral Calculus on Manifolds
1 Compactness
2 Partitions of unity
3 Densities
4 Volume density of a Riemann metric
5 Pullback and Lie derivatives of densities
6 The divergence theorem
7 More complicated domains
Chapter 11 Exterior Calculus
1 Exterior differential forms
2 Oriented manifolds and the integration of exterior differential forms
3 The operator $d$
4 Stokes' theorem
5 Some illustrations of Stokes' theorem
6 The Lie derivative of a differential form
Appendix $\text I$. "Vector analysis"
Appendix $\text {II}$. Elementary differential geometry of surfaces in $\mathbb E^3$
Chapter 12 Potential Theory in $\mathbb E^n$
1 Solid angle
2 Green's formulas
3 The maximum principle
4 Green's functions
5 The Poisson integral formula
6 Consequences of the Poisson integral formula
7 Harnack's theorem
8 Subharmonic functions
9 Dirichlet's problem
10 Behavior near the boundary
11 Dirichlet's principle
12 Physical applications
13 Problem set: The calculus of residues
Chapter 13 Classical Mechanics
1 The tangent and cotangent bundles
2 Equations of variation
3 The fundamental linear differential form on $\map {T*} M$
4 The fundamental exterior two-form on $\map {T*} M$
5 Hamiltonian mechanics
6 The central-force problem
7 The two-body problem
8 Lagrange's equations
9 Variational principles
10 Geodesic coordinates
11 Euler's equations
12 Rigid-body motion
13 Small oscillations
14 Small oscillations (continued)
15 Canonical transformations
Selected References
Notation Index
Index


Further Editions