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Categories, Bundles and Spacetime Topology / Christopher Terence John Dodson / Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer (1988)
Titre : Categories, Bundles and Spacetime Topology Type de document : document électronique Auteurs : Christopher Terence John Dodson ; SpringerLink (Online service) Editeur : Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer Année de publication : 1988 Collection : Mathematics and Its Applications num. 45 Importance : XVII, 243 p Présentation : online resource ISBN/ISSN/EAN : 978-94-015-7776-2 Langues : Anglais (eng) Tags : Mathematics Category theory Homological algebra Geometry Physics Category Theory Homological Algebra Mathematical and Computational Physics Résumé : Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Gad in Crane Feathers' in R. Brown'The point of a Pin'. van Gulik's TheChinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging SUbdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics Note de contenu : I. Preliminaries -- II. Naive Category Theory -- III. Existence of Limiting Topologies -- IV. Manifolds and Bundles -- V. Spacetime Structure -- Supplementary Bibliography Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=122689 Categories, Bundles and Spacetime Topology [document électronique] / Christopher Terence John Dodson ; SpringerLink (Online service) . - Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer, 1988 . - XVII, 243 p : online resource. - (Mathematics and Its Applications; 45) .
ISBN : 978-94-015-7776-2
Langues : Anglais (eng)
Tags : Mathematics Category theory Homological algebra Geometry Physics Category Theory Homological Algebra Mathematical and Computational Physics Résumé : Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Gad in Crane Feathers' in R. Brown'The point of a Pin'. van Gulik's TheChinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging SUbdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics Note de contenu : I. Preliminaries -- II. Naive Category Theory -- III. Existence of Limiting Topologies -- IV. Manifolds and Bundles -- V. Spacetime Structure -- Supplementary Bibliography Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=122689 Differential Geometry of Frame Bundles / Luis A. Cordero / Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer (1989)
Titre : Differential Geometry of Frame Bundles Type de document : document électronique Auteurs : Luis A. Cordero ; Christopher Terence John Dodson ; SpringerLink (Online service) ; Manuel de León Editeur : Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer Année de publication : 1989 Collection : Mathematics and Its Applications num. 47 Importance : X, 234 p Présentation : online resource ISBN/ISSN/EAN : 978-94-009-1265-6 Langues : Anglais (eng) Tags : Mathematics Global analysis Manifolds Differential geometry Physics Differential Geometry Global Analysis and Analysis on Manifolds Mathematical and Computational Physics Résumé : It isn't that they can't see the solution. It is Approach your problems from the right end and begin with the answers. Then one day, that they can't see the problem perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gu!ik's The Chillese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics Note de contenu : 1 The Functor Jp1 -- 1.1 The Bundle Jp1M ? M -- 1.2 Jp1G for a Lie group G -- 1.3 Jp1V for a vector space V -- 1.4 The embedding jp -- 2 Prolongation of G-structures -- 2.1 Imbedding of Jn1FM into FFM -- 2.2 Prolongation of G-structures to FM -- 2.3 Integrability -- 2.4 Applications -- 3 Vector-valued differential forms -- 3.1 General Theory -- 3.2 Applications -- 4 Prolongation of linear connections -- 4.1 Forms with values in a Lie algebra -- 4.2 Prolongation of connections -- 4.3 Complete lift of linear connections -- 4.4 Connections adapted to G-structures -- 4.5 Geodesics of ?C -- 4.6 Complete lift of derivations -- 5 Diagonal lifts -- 5.1 Diagonal lifts -- 5.2 Applications -- 6 Horizontal lifts -- 6.1 General theory -- 6.2 Applications -- 7 Lift GD of a Riemannian G to FM -- 7.1 GD, G of type (0,2) -- 7.2 Levi-Civita connection of GD -- 7.3 Curvature of GD -- 7.4 Bundle of orthonormal frames -- 7.5 Geodesics of GD -- 7.6 Applications -- 8 Constructing G-structures on FM -- 8.1 ?-associated G-structures on FM -- 8.2 Defined by (1,1)-tensor fields -- 8.3 Application to polynomial structures on FM -- 8.4 G-structures defined by (0,2)-tensor fields -- 8.5 Applications to almost complex and Hermitian structures -- 8.6 Application to spacetime structure -- 9 Systems of connections -- 9.1 Connections on a fibred manifold -- 9.2 Principal bundle connections -- 9.3 Systems of connections -- 9.4 Universal Connections -- 9.5 Applications -- 10 The Functor Jp2 -- 10.1 The Bundle Jp2M ? M -- 10.2 The second order frame bundle -- 10.3 Second order connections -- 10.4 Geodesics of second order -- 10.5 G-structures on F2M -- 10.6 Vector fields on F2M -- 10.7 Diagonal lifts of tensor fields -- 10.8 Natural prolongations of G-structures -- 10.9 Diagonal prolongation of G-structures Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=122537 Differential Geometry of Frame Bundles [document électronique] / Luis A. Cordero ; Christopher Terence John Dodson ; SpringerLink (Online service) ; Manuel de León . - Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer, 1989 . - X, 234 p : online resource. - (Mathematics and Its Applications; 47) .
ISBN : 978-94-009-1265-6
Langues : Anglais (eng)
Tags : Mathematics Global analysis Manifolds Differential geometry Physics Differential Geometry Global Analysis and Analysis on Manifolds Mathematical and Computational Physics Résumé : It isn't that they can't see the solution. It is Approach your problems from the right end and begin with the answers. Then one day, that they can't see the problem perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gu!ik's The Chillese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics Note de contenu : 1 The Functor Jp1 -- 1.1 The Bundle Jp1M ? M -- 1.2 Jp1G for a Lie group G -- 1.3 Jp1V for a vector space V -- 1.4 The embedding jp -- 2 Prolongation of G-structures -- 2.1 Imbedding of Jn1FM into FFM -- 2.2 Prolongation of G-structures to FM -- 2.3 Integrability -- 2.4 Applications -- 3 Vector-valued differential forms -- 3.1 General Theory -- 3.2 Applications -- 4 Prolongation of linear connections -- 4.1 Forms with values in a Lie algebra -- 4.2 Prolongation of connections -- 4.3 Complete lift of linear connections -- 4.4 Connections adapted to G-structures -- 4.5 Geodesics of ?C -- 4.6 Complete lift of derivations -- 5 Diagonal lifts -- 5.1 Diagonal lifts -- 5.2 Applications -- 6 Horizontal lifts -- 6.1 General theory -- 6.2 Applications -- 7 Lift GD of a Riemannian G to FM -- 7.1 GD, G of type (0,2) -- 7.2 Levi-Civita connection of GD -- 7.3 Curvature of GD -- 7.4 Bundle of orthonormal frames -- 7.5 Geodesics of GD -- 7.6 Applications -- 8 Constructing G-structures on FM -- 8.1 ?-associated G-structures on FM -- 8.2 Defined by (1,1)-tensor fields -- 8.3 Application to polynomial structures on FM -- 8.4 G-structures defined by (0,2)-tensor fields -- 8.5 Applications to almost complex and Hermitian structures -- 8.6 Application to spacetime structure -- 9 Systems of connections -- 9.1 Connections on a fibred manifold -- 9.2 Principal bundle connections -- 9.3 Systems of connections -- 9.4 Universal Connections -- 9.5 Applications -- 10 The Functor Jp2 -- 10.1 The Bundle Jp2M ? M -- 10.2 The second order frame bundle -- 10.3 Second order connections -- 10.4 Geodesics of second order -- 10.5 G-structures on F2M -- 10.6 Vector fields on F2M -- 10.7 Diagonal lifts of tensor fields -- 10.8 Natural prolongations of G-structures -- 10.9 Diagonal prolongation of G-structures Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=122537 Experiments In Mathematics Using Maple / Christopher Terence John Dodson / Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer (1995)
Titre : Experiments In Mathematics Using Maple Type de document : document électronique Auteurs : Christopher Terence John Dodson ; Elizabeth A. Gonzalez ; SpringerLink (Online service) Editeur : Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer Année de publication : 1995 Importance : XIX, 465 p. 5 illus Présentation : online resource ISBN/ISSN/EAN : 978-3-642-79758-3 Langues : Anglais (eng) Tags : Mathematics Mathematical analysis Analysis Computer mathematics Computer software Computational Mathematics and Numerical Analysis Mathematical Software Résumé : This book is designed for use in school computer labs or with home computers, running the computer algebra system Maple, or its student version. It supports the interactive Maple worksheets that we have developed and which are available free of charge from various sites. For example consult the anonymous ftp site ftp.utirc.utoronto.ca (/pub/ednet/maths/maple), or the University of Toronto Instructional and Research Computing World Wide Web home page (hhtp://www.utirc.utoronto.ca/home.html), over the Internet. The topics proceed through the full mathematics syllabus for the two senior years, from basic algebra, functions and sequences, to calculus and its additional explanatory text, answers to exercises, cross-referencing, bibliography and index Note de contenu : I Pre-Calculus Mathematics -- 1 Introduction to Maple -- 2 Functions -- 3 Quadratic Functions -- 4 Solving Quadratic Equations -- 5 Polynomial Functions -- 6 Exponential Functions -- 7 Logarithmic Functions -- 8 Circular Functions -- 9 Trigonometry -- 10 Similar Figures -- 11 Circles and Spheres -- 12 Loci -- 13 Sequences and Series -- 14 Statistics and Probability -- II Beginning Calculus -- 15 Secants and Tangents -- 16 Sequences and Limits -- 17 Derivatives of Functions -- 18 Functions and Graphs -- 19 Rates -- 20 Integration -- 21 Trigonometry -- 22 Exponents and Logarithms -- 23 Polar Coordinates -- Appendices -- A Solutions to Part I Exercises -- A.1 Functions -- A.2 Quadratics -- A.3 Solving Quadratics -- A.4 Polynomials -- A.5 Exponential Functions -- A.6 Logarithmic Functions -- A.8 Trigonometry -- A.9 Similar Figures -- A.10 Circles and Spheres -- A.11 Loci -- A.12 Sequences and Series -- A.13 Statistics and Probability -- B Solutions to Part II Exercises -- B.1 Secants and Tangents -- B.2 Sequences and Limits -- B.3 Derivatives of Functions -- B.4 Functions and Graphs -- B.5 Rates -- B.6 Integration -- B.7 Trigonometry -- B.8 Exponents and Logarithms -- B.9 Polar Coordinates -- C.1 angles -- C.2 showAntiderivative Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=128982 Experiments In Mathematics Using Maple [document électronique] / Christopher Terence John Dodson ; Elizabeth A. Gonzalez ; SpringerLink (Online service) . - Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer, 1995 . - XIX, 465 p. 5 illus : online resource.
ISBN : 978-3-642-79758-3
Langues : Anglais (eng)
Tags : Mathematics Mathematical analysis Analysis Computer mathematics Computer software Computational Mathematics and Numerical Analysis Mathematical Software Résumé : This book is designed for use in school computer labs or with home computers, running the computer algebra system Maple, or its student version. It supports the interactive Maple worksheets that we have developed and which are available free of charge from various sites. For example consult the anonymous ftp site ftp.utirc.utoronto.ca (/pub/ednet/maths/maple), or the University of Toronto Instructional and Research Computing World Wide Web home page (hhtp://www.utirc.utoronto.ca/home.html), over the Internet. The topics proceed through the full mathematics syllabus for the two senior years, from basic algebra, functions and sequences, to calculus and its additional explanatory text, answers to exercises, cross-referencing, bibliography and index Note de contenu : I Pre-Calculus Mathematics -- 1 Introduction to Maple -- 2 Functions -- 3 Quadratic Functions -- 4 Solving Quadratic Equations -- 5 Polynomial Functions -- 6 Exponential Functions -- 7 Logarithmic Functions -- 8 Circular Functions -- 9 Trigonometry -- 10 Similar Figures -- 11 Circles and Spheres -- 12 Loci -- 13 Sequences and Series -- 14 Statistics and Probability -- II Beginning Calculus -- 15 Secants and Tangents -- 16 Sequences and Limits -- 17 Derivatives of Functions -- 18 Functions and Graphs -- 19 Rates -- 20 Integration -- 21 Trigonometry -- 22 Exponents and Logarithms -- 23 Polar Coordinates -- Appendices -- A Solutions to Part I Exercises -- A.1 Functions -- A.2 Quadratics -- A.3 Solving Quadratics -- A.4 Polynomials -- A.5 Exponential Functions -- A.6 Logarithmic Functions -- A.8 Trigonometry -- A.9 Similar Figures -- A.10 Circles and Spheres -- A.11 Loci -- A.12 Sequences and Series -- A.13 Statistics and Probability -- B Solutions to Part II Exercises -- B.1 Secants and Tangents -- B.2 Sequences and Limits -- B.3 Derivatives of Functions -- B.4 Functions and Graphs -- B.5 Rates -- B.6 Integration -- B.7 Trigonometry -- B.8 Exponents and Logarithms -- B.9 Polar Coordinates -- C.1 angles -- C.2 showAntiderivative Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=128982 Information Geometry / Khadiga A. Arwini / Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer (2008)
Tensor Geometry / Christopher Terence John Dodson / Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer (1991)
Titre : Tensor Geometry : The Geometric Viewpoint and its Uses Type de document : document électronique Auteurs : Christopher Terence John Dodson ; Timothy Poston ; SpringerLink (Online service) Editeur : Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer Année de publication : 1991 Collection : Graduate Texts in Mathematics, ISSN 0072-5285 num. 130 Importance : XIV, 434 p Présentation : online resource ISBN/ISSN/EAN : 978-3-642-10514-2 Langues : Anglais (eng) Tags : Mathematics Matrix theory Algebra Differential geometry Physics Differential Geometry Linear and Multilinear Algebras Matrix Theory Mathematical and Computational Physics Résumé : We have been very encouraged by the reactions of students and teachers using our book over the past ten years and so this is a complete retype in TEX, with corrections of known errors and the addition of a supplementary bibliography. Thanks are due to the Springer staff in Heidelberg for their enthusiastic sup port and to the typist, Armin Kollner for the excellence of the final result. Once again, it has been achieved with the authors in yet two other countries. November 1990 Kit Dodson Toronto, Canada Tim Poston Pohang, Korea Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI O. Fundamental Not(at)ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3. Physical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 I. Real Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1. Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Subspace geometry, components 2. Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Linearity, singularity, matrices 3. Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Projections, eigenvalues, determinant, trace II. Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1. Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Tangent vectors, parallelism, coordinates 2. Combinations of Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Midpoints, convexity 3. Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Linear parts, translations, components III. Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 1. Contours, Co- and Contravariance, Dual Basis . . . . . . . . . . . . . . 57 IV. Metric Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1. Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Basic geometry and examples, Lorentz geometry 2. Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Isometries, orthogonal projections and complements, adjoints 3. Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Orthonormal bases Contents VIII 4. Diagonalising Symmetric Operators 92 Principal directions, isotropy V. Tensors and Multilinear Forms 98 1. Multilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Tensor Products, Degree, Contraction, Raising Indices VE Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 1. Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Metrics, topologies, homeomorphisms 2. Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Convergence and continuity 3. The Usual Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Note de contenu : 0. Fundamental Not(at)ions -- I. Real Vector Spaces -- II. Affine Spaces -- III. Dual Spaces -- IV. Metric Vector Spaces -- V. Tensors and Multilinear Forms -- VI Topological Vector Spaces -- VII. Differentiation and Manifolds -- VIII. Connections and Covariant Differentiation -- IX. Geodesics -- X. Curvature -- XI. Special Relativity -- XII. General Relativity -- Index of Notations Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=122423 Tensor Geometry : The Geometric Viewpoint and its Uses [document électronique] / Christopher Terence John Dodson ; Timothy Poston ; SpringerLink (Online service) . - Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer, 1991 . - XIV, 434 p : online resource. - (Graduate Texts in Mathematics, ISSN 0072-5285; 130) .
ISBN : 978-3-642-10514-2
Langues : Anglais (eng)
Tags : Mathematics Matrix theory Algebra Differential geometry Physics Differential Geometry Linear and Multilinear Algebras Matrix Theory Mathematical and Computational Physics Résumé : We have been very encouraged by the reactions of students and teachers using our book over the past ten years and so this is a complete retype in TEX, with corrections of known errors and the addition of a supplementary bibliography. Thanks are due to the Springer staff in Heidelberg for their enthusiastic sup port and to the typist, Armin Kollner for the excellence of the final result. Once again, it has been achieved with the authors in yet two other countries. November 1990 Kit Dodson Toronto, Canada Tim Poston Pohang, Korea Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI O. Fundamental Not(at)ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3. Physical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 I. Real Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1. Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Subspace geometry, components 2. Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Linearity, singularity, matrices 3. Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Projections, eigenvalues, determinant, trace II. Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1. Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Tangent vectors, parallelism, coordinates 2. Combinations of Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Midpoints, convexity 3. Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Linear parts, translations, components III. Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 1. Contours, Co- and Contravariance, Dual Basis . . . . . . . . . . . . . . 57 IV. Metric Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1. Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Basic geometry and examples, Lorentz geometry 2. Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Isometries, orthogonal projections and complements, adjoints 3. Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Orthonormal bases Contents VIII 4. Diagonalising Symmetric Operators 92 Principal directions, isotropy V. Tensors and Multilinear Forms 98 1. Multilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Tensor Products, Degree, Contraction, Raising Indices VE Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 1. Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Metrics, topologies, homeomorphisms 2. Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Convergence and continuity 3. The Usual Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Note de contenu : 0. Fundamental Not(at)ions -- I. Real Vector Spaces -- II. Affine Spaces -- III. Dual Spaces -- IV. Metric Vector Spaces -- V. Tensors and Multilinear Forms -- VI Topological Vector Spaces -- VII. Differentiation and Manifolds -- VIII. Connections and Covariant Differentiation -- IX. Geodesics -- X. Curvature -- XI. Special Relativity -- XII. General Relativity -- Index of Notations Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=122423