Elementary differential geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam [u.a.]
Elsevier [u.a.]
2006
|
| Edition: | rev. 2. ed. |
| Subjects: | |
| Online Access: | Publisher description Table of contents only Inhaltsverzeichnis |
| Physical Description: | XI, 503 S. graph. Darst. 24 cm |
| ISBN: | 0120887355 9780120887354 |
Staff View
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Record in the Search Index
| _version_ | 1804136251767390208 |
|---|---|
| adam_text | Contents
Preface
to the Revised Second Edition
ix
Introduction
ι
1.
Calculus on Euclidean Space
1.1.
Euclidean Space
3
1.2.
Tangent Vectors
6
1.3.
Directional Derivatives
11
1.4.
Curves in R3
16
1.5. 1
-Forms
23
1.6.
Differential Forms
28
1.7.
Mappings
34
1.8.
Summary
41
2.
Frame Fields
2.1.
Dot Product
43
2.2.
Curves
52
2.3.
The Frenet Formulas
58
2.4.
Arbitrary-Speed Curves
69
2.5.
Covariant Derivatives
81
2.6.
Frame Fields
84
2.7.
Connection Forms
88
2.8.
The Structural Equations
94
2.9.
Summary
99
vi
Contents
3.
Euclidean Geometry
3.1.
Isometries of R3
100
3.2.
The Tangent Map of an Isometry
107
3.3.
Orientation
110
3.4.
Euclidean Geometry
116
3.5.
Congruence of Curves
121
3.6.
Summary
128
4.
Calculus on a Surface
4.1.
Surfaces in R3
130
4.2.
Patch Computations
139
4.3.
Differentiable Functions and Tangent Vectors
149
4.4.
Differential Forms on a Surface
158
4.5.
Mappings of Surfaces
166
4.6.
integration of Forms
174
4.7.
Topological Properties of Surfaces
184
4.8.
Manifolds
193
4.9.
Summary
201
5.
Shape Operators
5.1.
The Shape Operator ofMcR3
202
5.2.
Normal Curvature
209
5.3.
Gaussian Curvature
216
5.4.
Computational Techniques
224
5.5.
The Implicit Case
235
5.6.
Special Curves in a Surface
240
5.7.
Surfaces of Revolution
252
5.8.
Summary
262
6.
Geometry of Surfaces in R3
6.1.
The Fundamental Equations
263
6.2.
Form Computations
269
6.3.
Some Global Theorems
273
6.4.
Isometries and Local Isometries
281
6.5.
Intrinsic Geometry of Surfaces in R3
289
6.6.
Orthogonal Coordinates
294
6.7.
Integration and Orientation
297
6.8.
Total Curvature
304
6.9.
Congruence of Surfaces
314
6.10.
Summary
319
Contents
vii
7. Riemannian
Geometry
7.1. Geometrie
Surfaces
321
7.2.
Gaussian Curvature
329
7.3.
Covariant Derivative
337
7.4.
Geodesies
346
7.5.
Clairaut Parametrizations
353
7.6.
The Gauss-Bonnet Theorem
364
7.7.
Applications of Gauss-Bonnet
376
7.8.
Summary
386
8.
Global Structure of Surfaces
8.1.
Length-Minimizing Properties of Geodesies
388
8.2.
Complete Surfaces
400
8.3.
Curvature and Conjugate Points
405
8.4.
Covering Surfaces
416
8.5.
Mappings That Preserve Inner Products
425
8.6.
Surfaces of Constant Curvature
433
8.7.
Theorems of Bonnet and
Hadamard
442
8.8.
Summary
449
Appendix: Computer Formulas
451
Bibliography
467
Answers to Odd-Numbered Exercises
468
Index
495
|
| adam_txt |
Contents
Preface
to the Revised Second Edition
ix
Introduction
ι
1.
Calculus on Euclidean Space
1.1.
Euclidean Space
3
1.2.
Tangent Vectors
6
1.3.
Directional Derivatives
11
1.4.
Curves in R3
16
1.5. 1
-Forms
23
1.6.
Differential Forms
28
1.7.
Mappings
34
1.8.
Summary
41
2.
Frame Fields
2.1.
Dot Product
43
2.2.
Curves
52
2.3.
The Frenet Formulas
58
2.4.
Arbitrary-Speed Curves
69
2.5.
Covariant Derivatives
81
2.6.
Frame Fields
84
2.7.
Connection Forms
88
2.8.
The Structural Equations
94
2.9.
Summary
99
vi
Contents
3.
Euclidean Geometry
3.1.
Isometries of R3
100
3.2.
The Tangent Map of an Isometry
107
3.3.
Orientation
110
3.4.
Euclidean Geometry
116
3.5.
Congruence of Curves
121
3.6.
Summary
128
4.
Calculus on a Surface
4.1.
Surfaces in R3
130
4.2.
Patch Computations
139
4.3.
Differentiable Functions and Tangent Vectors
149
4.4.
Differential Forms on a Surface
158
4.5.
Mappings of Surfaces
166
4.6.
integration of Forms
174
4.7.
Topological Properties of Surfaces
184
4.8.
Manifolds
193
4.9.
Summary
201
5.
Shape Operators
5.1.
The Shape Operator ofMcR3
202
5.2.
Normal Curvature
209
5.3.
Gaussian Curvature
216
5.4.
Computational Techniques
224
5.5.
The Implicit Case
235
5.6.
Special Curves in a Surface
240
5.7.
Surfaces of Revolution
252
5.8.
Summary
262
6.
Geometry of Surfaces in R3
6.1.
The Fundamental Equations
263
6.2.
Form Computations
269
6.3.
Some Global Theorems
273
6.4.
Isometries and Local Isometries
281
6.5.
Intrinsic Geometry of Surfaces in R3
289
6.6.
Orthogonal Coordinates
294
6.7.
Integration and Orientation
297
6.8.
Total Curvature
304
6.9.
Congruence of Surfaces
314
6.10.
Summary
319
Contents
vii
7. Riemannian
Geometry
7.1. Geometrie
Surfaces
321
7.2.
Gaussian Curvature
329
7.3.
Covariant Derivative
337
7.4.
Geodesies
346
7.5.
Clairaut Parametrizations
353
7.6.
The Gauss-Bonnet Theorem
364
7.7.
Applications of Gauss-Bonnet
376
7.8.
Summary
386
8.
Global Structure of Surfaces
8.1.
Length-Minimizing Properties of Geodesies
388
8.2.
Complete Surfaces
400
8.3.
Curvature and Conjugate Points
405
8.4.
Covering Surfaces
416
8.5.
Mappings That Preserve Inner Products
425
8.6.
Surfaces of Constant Curvature
433
8.7.
Theorems of Bonnet and
Hadamard
442
8.8.
Summary
449
Appendix: Computer Formulas
451
Bibliography
467
Answers to Odd-Numbered Exercises
468
Index
495 |
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| illustrated | Illustrated |
| index_date | 2024-07-02T16:40:34Z |
| indexdate | 2024-07-09T20:53:26Z |
| institution | BVB |
| isbn | 0120887355 9780120887354 |
| language | English |
| lccn | 2005057176 |
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| physical | XI, 503 S. graph. Darst. 24 cm |
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| spelling | O'Neill, Barrett Verfasser aut Elementary differential geometry Barrett O'Neill rev. 2. ed. Amsterdam [u.a.] Elsevier [u.a.] 2006 XI, 503 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Courbe rasuqam Geometria diferencial (textos elementares) larpcal Geometría diferencial lemb Géométrie différentielle Géométrie différentielle rasuqam Surface (Mathématiques) rasuqam Geometry, Differential Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content Differentialgeometrie (DE-588)4012248-7 s DE-604 http://www.loc.gov/catdir/enhancements/fy0626/2005057176-d.html Publisher description http://www.loc.gov/catdir/enhancements/fy0626/2005057176-t.html Table of contents only Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015465088&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | O'Neill, Barrett Elementary differential geometry Courbe rasuqam Geometria diferencial (textos elementares) larpcal Geometría diferencial lemb Géométrie différentielle Géométrie différentielle rasuqam Surface (Mathématiques) rasuqam Geometry, Differential Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4151278-9 |
| title | Elementary differential geometry |
| title_auth | Elementary differential geometry |
| title_exact_search | Elementary differential geometry |
| title_exact_search_txtP | Elementary differential geometry |
| title_full | Elementary differential geometry Barrett O'Neill |
| title_fullStr | Elementary differential geometry Barrett O'Neill |
| title_full_unstemmed | Elementary differential geometry Barrett O'Neill |
| title_short | Elementary differential geometry |
| title_sort | elementary differential geometry |
| topic | Courbe rasuqam Geometria diferencial (textos elementares) larpcal Geometría diferencial lemb Géométrie différentielle Géométrie différentielle rasuqam Surface (Mathématiques) rasuqam Geometry, Differential Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Courbe Geometria diferencial (textos elementares) Geometría diferencial Géométrie différentielle Surface (Mathématiques) Geometry, Differential Differentialgeometrie Einführung |
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| work_keys_str_mv | AT oneillbarrett elementarydifferentialgeometry |