Geometry and topology:
Gespeichert in:
| Beteiligte Personen: | , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2005
|
| Ausgabe: | 1. publ. |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014278832&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XVIII, 196 S. graph. Darst. |
| ISBN: | 9780521848893 9780521613255 0521613256 052184889x |
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0202 MAT 500f 2009 A 909 |
|---|---|
| DE-BY-TUM_katkey | 1657121 |
| DE-BY-TUM_location | 02 |
| DE-BY-TUM_media_number | 040020779102 |
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| adam_text | GEOMETRY AND TOPOLOGY MILES REID MATHEMATICS INSTITUTE, UNIVERSITY OF
WARWICK, COVENTRY CV4 7AL, UK BALAZS SZENDROI MATHEMATICAL INSTITUTE,
UNIVERSITY OF OXFORD, 24-29 ST GILES, OXFORD 0X1 3LB, UK CAMBRIDGE
UNIVERSITY PRESS CONTENTS LIST OF FIGURES PAGE X PREFACE XIII EUCLIDEAN
GEOMETRY 1 1.1 THE METRIC ON W 1 1.2 LINES AND COLLINEARITY IN W 3 1.3
EUCLIDEAN SPACE E 4 1.4 DIGRESSION: SHORTEST DISTANCE 4 1.5 ANGLES Y 5
1.6 MOTIONS 6 1.7 MOTIONS AND COLLINEARITY 7 1.8 A MOTION IS AFFINE
LINEAR ON LINES 7 1.9 MOTIONS ARE AFFINE TRANSFORMATIONS 8 1.10
EUCLIDEAN MOTIONS AND ORTHOGONAL TRANSFORMATIONS 9 1.11 NORMAL FORM OF
AN ORTHOGONAL MATRIX 10 1.11.1 THE 2X2 ROTATION AND REFLECTION MATRIXES
10 1.11.2 THE GENERAL CASE 12 1.12 EUCLIDEAN FRAMES AND MOTIONS 14 1.13
FRAMES AND MOTIONS OF E 2 14 1.14 EVERY MOTION OF E 2 IS A TRANSLATION,
ROTATION, REFLECTION OR GLIDE 15 1.15 CLASSIFICATION OF MOTIONS OF E 3
17 1.16 SAMPLE THEOREMS OF EUCLIDEAN GEOMETRY 19 1.16.1 PONS ASINORUM 19
1.16.2 THE ANGLE SUM OF TRIANGLES 19 1.16.3 PARALLEL LINES AND SIMILAR
TRIANGLES 20 1.16.4 FOUR CENTRES OF A TRIANGLE 21 1.16.5 THE FEUERBACH
9-POINT CIRCLE 23 EXERCISES 24 COMPOSING MAPS 26 2.1 COMPOSITION IS
THE BASIC OPERATION 26 2.2 COMPOSITION OF AFFINE LINEAR MAPS X I- AX +
B 27 CONTENTS 2.3 COMPOSITION OF TWO REFLECTIONS OF E 2 27 2.4
COMPOSITION OF MAPS IS ASSOCIATIVE 28 2.5 DECOMPOSING MOTIONS 28 2.6
REFLECTIONS GENERATE ALL MOTIONS 29 2.7 AN ALTERNATIVE PROOF OF THEOREM
1.14 31 2.8 PREVIEW OF TRANSFORMATION GROUPS 31 EXERCISES 32 3 SPHERICAL
AND HYPERBOLIC NON-EUCLIDEAN GEOMETRY 34 3.1 BASIC DEFINITIONS OF
SPHERICAL GEOMETRY 35 3.2 SPHERICAL TRIANGLES AND TRIG 37 3.3 THE
SPHERICAL TRIANGLE INEQUALITY 38 3.4 SPHERICAL MOTIONS 38 3.5 PROPERTIES
OF S 2 LIKE E 2 39 3.6 PROPERTIES OF S 2 UNLIKE E 2 40 3.7 PREVIEW OF
HYPERBOLIC GEOMETRY 41 3.8 HYPERBOLIC SPACE 42 3.9 HYPERBOLIC DISTANCE
43 3.10 HYPERBOLIC TRIANGLES AND TRIG 44 3.11 HYPERBOLIC MOTIONS 46 3.12
INCIDENCE OF TWO LINES IN 7I 2 47 3.13 THE HYPERBOLIC PLANE IS
NON-EUCLIDEAN 49 3.14 ANGULAR DEFECT 51 3.14.1 THE FIRST PROOF 51 3.14.2
AN EXPLICIT INTEGRAL 51 3.14.3 PROOF BY SUBDIVISION 53 3.14.4 AN
ALTERNATIVE SKETCH PROOF 54 EXERCISES 56 4 AFFINE GEOMETRY 62 4.1
MOTIVATION FOR AFFINE SPACE 62 4.2 BASIC PROPERTIES OF AFFINE SPACE 63
4.3 THE GEOMETRY OF AFFINE LINEAR SUBSPACES 65 4.4 DIMENSION OF
INTERSECTION 67 4.5 AFFINE TRANSFORMATIONS 68 4.6 AFFINE FRAMES AND
AFFINE TRANSFORMATIONS 68 4.7 THECENTROID 69 EXERCISES 69 5 PROJECTIVE
GEOMETRY 72 5.1 MOTIVATION FOR PROJECTIVE GEOMETRY 72 5.1.1
INHOMOGENEOUS TO HOMOGENEOUS 72 5.1.2 PERSPECTIVE 73 5.1.3 ASYMPTOTES 73
5.1.4 COMPACTIFICATION 75 CONTENTS VII 5.2 DEFINITION OF PROJECTIVE
SPACE 75 5.3 PROJECTIVE LINEAR SUBSPACES 76 5.4 DIMENSION OF
INTERSECTION 77 5.5 PROJECTIVE LINEAR TRANSFORMATIONS AND PROJECTIVE
FRAMES OF REFERENCE 77 5.6 PROJECTIVE LINEAR MAPS OF P 1 AND THE
CROSS-RATIO 79 5.7 PERSPECTIVITIES 81 5.8 AFFINE SPACE A AS A SUBSET OF
PROJECTIVE SPACE P 81 5.9 DESARGUES THEOREM 82 5.10 PAPPUS THEOREM 84
5.11 PRINCIPLE OF DUALITY 85 5.12 AXIOMATIC PROJECTIVE GEOMETRY 86
EXERCISES 88 6 GEOMETRY AND GROUP THEORY 92 6.1 TRANSFORMATIONS FORM A
GROUP 93 6.2 TRANSFORMATION GROUPS 94 6.3 KLEIN S ERLANGEN PROGRAM 95
6.4 CONJUGACY IN TRANSFORMATION GROUPS 96 6.5 APPLICATIONS OF CONJUGACY
98 6.5.1 NORMAL FORMS 98 6.5.2 FINDING GENERATORS 100 6.5.3 THE
ALGEBRAIC STRUCTURE OF TRANSFORMATION GROUPS 101 6.6 DISCRETE REFLECTION
GROUPS 103 EXERCISES 104 7 TOPOLOGY 107 7.1 DEFINITION OF A TOPOLOGICAL
SPACE 108 7.2 MOTIVATION FROM METRIC SPACES 108 7.3 CONTINUOUS MAPS AND
HOMEOMORPHISMS 111 7.3.1 DEFINITION OF A CONTINUOUS MAP 111 7.3.2
DEFINITION OF A HOMEOMORPHISM 111 7.3.3 HOMEOMORPHISMS AND THE ERLANGEN
PROGRAM 112 7.3.4 THE HOMEOMORPHISM PROBLEM 113 7.4 TOPOLOGICAL
PROPERTIES 113 7.4.1 CONNECTED SPACE 113 7.4.2 COMPACT SPACE 115 7.4.3
CONTINUOUS IMAGE OF A COMPACT SPACE IS COMPACT 116 7.4.4 AN APPLICATION
OF TOPOLOGICAL PROPERTIES 117 7.5 SUBSPACE AND QUOTIENT TOPOLOGY 117 7.6
STANDARD EXAMPLES OF GLUEING 118 7.7 TOPOLOGY OF PJ 121 7.8 NONMETRIC
QUOTIENT TOPOLOGIES 122 7.9 BASIS FOR A TOPOLOGY 124 CONTENTS 7.10
PRODUCT TOPOLOGY 126 7.11 THE HAUSDORFF PROPERTY 127 7.12 COMPACT VERSUS
CLOSED 128 7.13 CLOSED MAPS 129 7.14 A CRITERION FOR HOMEOMORPHISM 130
7.15 LOOPS AND THE WINDING NUMBER 130 7.15.1 PATHS, LOOPS AND FAMILIES
131 7.15.2 THE WINDING NUMBER 133 7.15.3 WINDING NUMBER IS CONSTANT IN A
FAMILY 135 7.15.4 APPLICATIONS OF THE WINDING NUMBER 13 6 EXERCISES 137
QUATERNIONS, ROTATIONS AND THE GEOMETRY OF TRANSFORMATION GROUPS 142 8.1
TOPOLOGY ON GROUPS , 143 8.2 DIMENSION COUNTING 144 8.3 COMPACT AND
NONCOMPACT GROUPS 146 8.4 COMPONENTS 148 8.5 QUATERNIONS, ROTATIONS AND
THE GEOMETRY OF SO() 149 8.5.1 QUATERNIONS 149 8.5.2 QUATERNIONS AND
ROTATIONS 151 8.5.3 SPHERES AND SPECIAL ORTHOGONAL GROUPS 152 8.6 THE
GROUP SU(2) 153 8.7 THE ELECTRON SPIN IN QUANTUM MECHANICS 154 8.7.1 THE
STORY OF THE ELECTRON SPIN 154 8.7.2 MEASURING SPIN: THE STERN-GERLACH
DEVICE 155 8.7.3 THE SPIN OPERATOR 156 8.7.4 ROTATE THE DEVICE 157 8.7.5
THE SOLUTION 158 8.8 PREVIEW OF LIE GROUPS 159 EXERCISES 161 CONCLUDING
REMARKS 164 9.1 ON THE HISTORY OF GEOMETRY 165 9.1.1 GREEK GEOMETRY AND
RIGOUR 165 9.1.2 THE PARALLEL POSTULATE 165 9.1.3 COORDINATES VERSUS
AXIOMS 168 9.2 GROUP THEORY 169 9.2.1 ABSTRACT GROUPS VERSUS
TRANSFORMATION GROUPS 169 9.2.2 HOMOGENEOUS AND PRINCIPAL HOMOGENEOUS
SPACES 169 9.2.3 THE ERLANGEN PROGRAM REVISITED 170 9.2.4 AFFINE SPACE
AS A TORSOR 171 CONTENTS IX 9.3 GEOMETRY IN PHYSICS 172 9.3.1 THE
GALILEAN GROUP AND NEWTONIAN DYNAMICS 172 9.3.2 THE POINCARE GROUP AND
SPECIAL RELATIVITY 173 9.3.3 WIGNER S CLASSIFICATION: ELEMENTARY
PARTICLES 175 9.3.4 THE STANDARD MODEL AND BEYOND 176 9.3.5 OTHER
CONNECTIONS 176 9.4 THE FAMOUS TRICHOTOMY 177 9.4.1 THE CURVATURE
TRICHOTOMY IN GEOMETRY 177 9.4.2 ON THE SHAPE AND FATE OF THE UNIVERSE
178 9.4.3 THE SNACK BAR AT THE END OF THE UNIVERSE 179 APPENDIX A
METRICS 180 EXERCISES 181 APPENDIX B LINEAR ALGEBRA 183 B.I BILINEAR
FORM AND QUADRATIC FORM 183 B. 2 EUCLID AND LORENTZ 184 B.3 COMPLEMENTS
AND BASES 185 B.4 SYMMETRIES 186 B.5 ORTHOGONAL AND LORENTZ MATRIXES 187
B.6 HERMITIAN FORMS AND UNITARY MATRIXES 188 EXERCISES 189 REFERENCES
190 INDEX 193
|
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| id | DE-604.BV021236067 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T12:26:12Z |
| institution | BVB |
| isbn | 9780521848893 9780521613255 0521613256 052184889x |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014278832 |
| oclc_num | 60794239 |
| open_access_boolean | |
| owner | DE-824 DE-384 DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-11 DE-188 DE-29T |
| owner_facet | DE-824 DE-384 DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-11 DE-188 DE-29T |
| physical | XVIII, 196 S. graph. Darst. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| spellingShingle | Reid, Miles 1948- Szendrői, Balázs Geometry and topology Géométrie Topologie Geometry Topology Geometrie (DE-588)4020236-7 gnd Topologie (DE-588)4060425-1 gnd |
| subject_GND | (DE-588)4020236-7 (DE-588)4060425-1 (DE-588)4123623-3 |
| title | Geometry and topology |
| title_auth | Geometry and topology |
| title_exact_search | Geometry and topology |
| title_full | Geometry and topology Miles Reid ; Balázs Szendrői |
| title_fullStr | Geometry and topology Miles Reid ; Balázs Szendrői |
| title_full_unstemmed | Geometry and topology Miles Reid ; Balázs Szendrői |
| title_short | Geometry and topology |
| title_sort | geometry and topology |
| topic | Géométrie Topologie Geometry Topology Geometrie (DE-588)4020236-7 gnd Topologie (DE-588)4060425-1 gnd |
| topic_facet | Géométrie Topologie Geometry Topology Geometrie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014278832&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT reidmiles geometryandtopology AT szendroibalazs geometryandtopology |
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Teilbibliothek Physik
| Signatur: |
0202 MAT 500f 2009 A 909
Lageplan |
|---|---|
| Exemplar 1 | Ausleihbar Am Standort |