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1.

図書

図書
by Hassler Whitney
出版情報: Princeton, N.J. : Princeton University Press, 1957  xv, 387 p ; 24 cm
シリーズ名: Princeton mathematical series ; 21
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Preface
Introduction
The general problem of integration / A:
The integral as a function of the domain / 1:
Polyhedral chains / 2:
Two continuity hypotheses / 3:
A further continuity hypothesis / 4:
Some examples / 5:
The case r = n / 6:
The r-vector of an oriented r-cell / 7:
On r-vectors and boundaries of (r + 1)-cells / 8:
Grassmann algebra / 9:
The dual algebra / 10:
Integration of differential forms / 11:
Some classical topics / B:
Grassmann algebra in metric oriented n-space / 12:
The same, n = 3 / 13:
The differential of a mapping / 14:
Jacobians / 15:
Transformation of the integral / 16:
Smooth manifolds / 17:
Particular forms of integrals in 3-space / 18:
The Theorem of Stokes / 19:
The exterior differential / 20:
Some special formulas in metric oriented E[superscript 3] / 21:
An existence theorem / 22:
De Rham's Theorem / 23:
Indications of general theory / C:
Normed spaces of chains and cochains / 24:
Continuous chains / 25:
On 0-dimensional integration / 26:
Classical theory / Part I:
Multivectors / Chapter I:
Multicovectors
Properties of V[subscript r] and V[superscript r]
Alternating r-linear functions
Use of coordinate systems
Exterior products
Interior products
n-vectors in n-space
Simple multivectors
Linear mappings of vector spaces
Duality
Euclidean vector spaces
Mass and comass
Mass and comass of products
On projections
Differential forms / Chapter II:
The differential of a smooth mapping
Some properties of differentials
Smooth mappings
The inverse and implicit function theorems
A representation of vectors and covectors
The tangent space of a smooth manifold
Differential forms in smooth manifolds
A characterization of the exterior differential
Riemann integration theory / Chapter III:
The r-vector of an oriented r-simplex
The r-vector of an r-chain
Integration over cellular chains
Some properties of integrals
Relation to the Riemann integral
Integration over open sets
The transformation formula
Proof of the transformation formula
Transformation of the Riemann integral
Integration in manifolds
Stokes' Theorem for a parallelepiped
A special case of Stokes' Theorem
Sets of zero s-extent
Stokes' Theorem for standard domains
Proof of the theorem
Regular forms in Euclidean space
Regular forms in smooth manifolds
Stokes' Theorem for standard manifolds
The iterated integral in Euclidean space
Manifolds in Euclidean space / Chapter IV:
The imbedding theorem
The compact case
Separation of subsets of E[superscript m]
Regular approximations
Proof of Theorem 1A, M compact
Admissible coordinate systems in M
Proof of Theorem 1A, M not compact
Local properties of M in E[superscript m]
On n-directions in E[superscript m]
The neighborhood of M in E[superscript m]
Projection along a plane
Triangulation of manifolds
The triangulation theorem
Outline of the proof
Fullness
Linear combinations of edge vectors of simplexes
The quantities used in the proof
The complex L
The complex L*
The intersections of M with L*
The complex K
Imbedding of simplexes in M
The complexes K[subscript p]
Cohomology in manifolds
[mu]-regular forms
Closed forms in star shaped sets
Extensions of forms
Elementary forms / 27:
Certain closed forms are derived / 28:
Isomorphism of cohomology rings / 29:
Periods of forms / 30:
The Hopf invariant / 31:
On smooth mappings of manifolds / 32:
Other expressions for the Hopf invariant / 33:
General theory / Part II:
Abstract integration theory / Chapter V:
Mass of polyhedral chains
The flat norm
Flat cochains
Examples
The sharp norm
Sharp cochains
Characterization of the norms
An algebraic criterion for a multicovector
Sharp r-forms
The semi-norms [vertical bar A vertical bar superscript flat], [vertical bar A vertical bar superscript sharp] are norms
Weak convergence
Some relations between the spaces of chains and cochains
The [rho]-norms
The mass of chains
Separability of spaces of chains
Non-separability of spaces of cochains
Some relations between chains and functions / Chapter VI:
Continuous chains on the real line
0-chains in E[superscript 1] defined by functions of bounded variation
Sharp functions times 0-chains
The part
Functions of bounded variation in E[superscript 1] defined by 0-chains
Some related analytical theorems
Continuous r-chains in E[superscript n]
On compact cochains
The boundary of a smooth chain
Continuous chains in smooth manifolds
General properties of chains and cochains / Chapter VII:
Sharp functions times chains
Sharp functions times cochains
Supports of chains and cochains
On non-compact chains
On polyhedral approximations
Sharp chains at a point
Molecular chains are dense
Flat r-chains in E[superscript r-k] are zero
Flat cochains in complexes
Elementary flat cochains in a complex
The isomorphism theorem
Chains and cochains in open sets / Chapter VIII:
Chains and cochains in open sets, elementary properties
Chains and cochains in open sets, further properties
Properties of mass
On the open sets to which a chain belongs
An expression for flat chains
An expression for sharp chains
Lebesgue theory / Part III:
Flat cochains and differential forms / Chapter IX:
n-cochains in E[superscript n]
Some properties of fullness
Properties of projections
Elementary properties of D[subscript X](p, [alpha])
The r-form defined by a flat r-cochain
Flat r-forms
Flat r-forms and flat r-cochains
Flat r-direction functions
Flat forms defined by components
Approximation to D[subscript X](p) by [characters not reproducible]
Total differentiability of Lipschitz functions
On the exterior differential of r-forms
On averages of r-forms
Products of cochains
Lebesgue chains
Products of cochains and chains
Products and weak limits
Characterization of the procducts
Lipschitz mappings / Chapter X:
Affine approximations to Lipschitz mappings
The approximation on the edges of a simplex
Approximation to the Jacobian
The volume of affine approximations
A continuity lemma
Lipschitz chains
Lipschitz mappings of open sets
Lipschitz mappings and flat cochains
Lipschitz mappings and flat forms
Lipschitz mappings and sharp functions
Lipschitz mappings and products
On the flat norm of Lipschitz chains
Deformations of chains
Chains and additive set functions / Chapter XI:
On finite dimensional Banach spaces
Vector valued additive set functions
Vector valued integration
Point functions times set functions
Relations between a set function and its variation
On positive linear functionals
On bounded linear functionals
Linear functions of sharp r-forms
The sharp norm of r-vector valued set functions
Molecular set functions
Sharp chains and set functions
Bounded Borel functions times chains
The part of a chain in a Borel set
Chains and point functions
Characterization of the sharp norm
Expression for the sharp norm
Other expressions for the norm
Vector and linear spaces / Appendix I:
Vector spaces
Linear transformations
Conjugate spaces
Direct sums, complements
Quotient spaces
Pairing of linear spaces
Abstract homology
Normed linear spaces
Euclidean linear spaces
Affine spaces
Barycentric coordinates
Affine mappings
Euclidean spaces
Banach spaces
Semi-conjugate spaces
Geometric and topological preliminaries / Appendix II:
Cells, simplexes
Polyhedra, complexes
Su bdivisions
Standard subdivisions
Orientation
Chains and cochains
Boundary and coboundary
Homology and cohomology
Products in a complex
Joins
Subdivisions of chains
Cartesian products of cells
Mappings of complexes
Some properties of planes
Mappings of n-pseudomanifolds into n-space
Distortion of triangulations of E[superscript m]
Analytical preliminaries / Appendix III:
Existence of certain functions
Partitions of unity
Smoothing functions by taking averages
The Weierstrass approximation theorem
The space L[superscript 1]
Index of symbols
Index of terms
Preface
Introduction
The general problem of integration / A:
2.

図書

図書
C.T.C. Wall
出版情報: London ; New York : Academic Press, 1970  x, 280 p. ; 23 cm
シリーズ名: L.M.S. monographs ; 1
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3.

図書

図書
V.I. Arnold ; translated and edited by Richard A. Silverman
出版情報: Cambridge : MIT Press, c1973  viii, 280 p. ; 23 cm
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目次情報: 続きを見る
Preface
Frequently Used Notation
Basic Concepts / 1:
Phase Spaces and Phase Flows
Vector Fields on the Line / 2:
Phase Flows on the Line / 3:
Vector Fields and Phase Flows in the Plane / 4:
Nonautonomous Equations / 5:
The Tangent Space / 6:
Basic Theorems
The Vector Field near a Nonsingular Point / 7:
Applications to the Nonautonomous Case / 8:
Applications to Equations of Higher Order / 9:
Phase Curves of Autonomous Systems / 10:
The Directional Derivative. First Integrals / 11:
Conservative Systems with One Degree of Freedom / 12:
Linear Systems
Linear Problems / 13:
The Exponential of an Operator / 14:
Properties of the Exponential / 15:
The Determinant of the Exponential / 16:
The Case of Distinct Real Eigenvalues / 17:
Complexification and Decomplexification / 18:
Linear Equations with a Complex Phase Space / 19:
Complexification of a Real Linear Equation / 20:
Classification of Singular Points of Linear Systems / 21:
Topological Classification of Singular Points / 22:
Stability of Equilibrium Positions / 23:
The Case of Purely Imaginary Eigenvalues / 24:
The Case of Multiple Eigenvalues / 25:
More on Quasi-Polynomials / 26:
Nonautonomous Linear Equations / 27:
Linear Equations with Periodic Coefficients / 28:
Variation of Constants / 29:
Proofs of the Basic Theorems
Contraction Mappings / 30:
The Existence, Uniqueness, and Continuity Theorems / 31:
The Differentiable Manifolds / 33:
The Tangent Bundle. Vector Fields on a Manifold / 34:
The Phase Flow Determined by a Vector Field / 35:
The Index of a Singular Point of a Vector Field / 36:
Sample Examination Problems
Bibliography
Index
Preface
Frequently Used Notation
Basic Concepts / 1:
4.

図書

図書
Robert C. Gunning, general editor
出版情報: Princeton, N.J. : Princeton University Press, 1970  x, 351 p. ; 24 cm
シリーズ名: Princeton mathematical series ; 31
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5.

図書

図書
editor, E.G.D. Cohen
出版情報: Amsterdam ; Tokyo : North-Holland , New York, N.Y., U.S.A. : Sole distributor for the U.S.A. and Canada, Elsevier Science Pub. Co., 1985  x, 398 p. ; 27 cm
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6.

図書

図書
by John von Neumann ; foreword by Israel Halperin
出版情報: Princeton, N.J. : Princeton University Press, 1960  xi, 299 p. ; 24 cm
シリーズ名: Princeton mathematical series ; 25
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7.

図書

図書
Sterling K. Berberian
出版情報: New York ; Berlin ; Tokyo : Springer, c1974  ix, 345 p. ; 25 cm
シリーズ名: Graduate texts in mathematics ; 15
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8.

図書

図書
Renfrey B. Potts, Robert M. Oliver
出版情報: New York : Academic Press, 1972  xi, 192 p. ; 24 cm
シリーズ名: Mathematics in science and engineering : a series of monographs and textbooks ; v. 90
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9.

図書

図書
Morris W. Hirsch and Stephen Smale
出版情報: New York : Academic Press, c1974  xi, 358 p. ; 23 cm
シリーズ名: Pure and applied mathematics ; v. 60
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目次情報: 続きを見る
Preface
First Examples
Newton's Equation and Kepler's Law
Linear Systems with Constant Coeffecients and Real Eigenvalues
Linear Systems with Constant Coefficients and Complex Eigenvalues
Linear Systems and Exponentials of Operators
Linear Systems and Canonical Forms of Operators
Contractions and Generic Properties of Operators
Fundamental Theory
Stability of Equilibria
Differential Equations for Electrical Circuits
The Poincare-Bendixson Theorem
Ecology
Periodic Attractors
Classical Mechanics
Nonautonomous Equations and Differentiability of Flows
Perturbation Theory and Structural Stability
Elementary Facts
Polynomials
On Canonical Forms
The Inverse Function Theorem
References
Answers to Selected Problems
Index
Preface
First Examples
Newton's Equation and Kepler's Law
10.

図書

図書
V.M. Popov ; [translated by Radu Georgescu]
出版情報: Berlin ; New York : Springer , Bucureşti : Editura Academiei, 1973  400 p. ; 24 cm
シリーズ名: Die Grundlehren der mathematischen Wissenschaften ; Bd. 204
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