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

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## Lynn H. Loomis and Shlomo Sternberg:

## 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

- 1968: Lynn H. Loomis and Shlomo Sternberg:
*Advanced Calculus*