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

Subject Matter

 * Calculus

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