Curves and surfaces:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English Spanish |
| Published: |
Providence, RI
American Mathem. Soc.
2009
|
| Edition: | 2. ed. |
| Series: | Graduate studies in mathematics
69 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XVI, 376 S. graph. Darst. |
| ISBN: | 9780821847633 |
Staff View
MARC
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| 240 | 1 | 0 | |a Curvas y superficies |
| 245 | 1 | 0 | |a Curves and surfaces |c Sebastián Montiel ; Antonio Ros |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Providence, RI |b American Mathem. Soc. |c 2009 | |
| 300 | |a XVI, 376 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 69 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Curves on surfaces | |
| 650 | 4 | |a Geometry, Differential | |
| 650 | 4 | |a Submanifolds | |
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Record in the Search Index
| _version_ | 1804140737406697472 |
|---|---|
| adam_text | Contents
Preface
to the Second Edition
xi
Preface to the English Edition
xiii
Preface
xv
Chapter
1.
Plane and Space Curves
1
§1.1.
Historical notes
1
§1.2.
Curves. Arc length
2
§1.3.
Regular curves and curves parametrized by arc length
8
§1.4.
Local theory of plane curves
9
§1.5.
Local theory of space curves
13
Exercises
19
Hints for solving the exercises
23
Chapter
2.
Surfaces in Euclidean Space
31
§2.1.
Historical notes
31
§2.2.
Definition of surface
32
§2.3.
Change of parameters
38
§2.4.
Differentiable functions
40
§2.5.
The tangent plane
44
§2.6.
Differential of a differentiable map
46
Exercises
55
Hints for solving the exercises
58
Chapter
3.
The Second Fundamental Form
67
vii
viii Contents
§3.1.
Introduction
and historical notes
67
§3.2.
Normal fields. Orientation
69
§3.3.
The Gauss map and the second fundamental form
76
§3.4.
Normal sections
81
§3.5.
The height function and the second fundamental form
85
§3.6.
Continuity of the curvatures
88
Exercises
94
Hints for solving the exercises
97
Chapter
4.
Separation and Orientability
107
§4.1.
Introduction
107
§4.2.
Local separation
108
§4.3.
Surfaces, straight lines, and planes 111
§4.4.
The Jordan-Brouwer separation theorem
116
§4.5.
Tubular neighbourhoods
120
Exercises
125
Hints for solving the exercises
126
§4.6.
Appendix: Proof of Sard s theorem
131
Chapter
5.
Integration on Surfaces
135
§5.1.
Introduction
135
§5.2.
Integrable
functions and integration on
S x M
136
§5.3.
Integrable
functions and integration on surfaces
139
§5.4.
Formula for the change of variables
144
§5.5.
Fubini s theorem and other properties
145
§5.6.
Area formula
152
§5.7.
The divergence theorem
157
§5.8.
The
Brouwer
fixed point theorem
161
Exercises
163
Hints for solving the exercises
166
Chapter
6.
Global Extrinsic Geometry
171
§6.1.
Introduction and historical notes
171
§6.2.
Positively curved surfaces
173
§6.3.
Minkowski formulas and ovaloids
182
§6.4.
The Alexandrov theorem
186
§6.5.
The isoperimetric inequality
188
Contents ix
Exercises
195
Hints for solving the exercises
198
Chapter
7.
Intrinsic Geometry of Surfaces
203
§7.1.
Introduction
203
§7.2.
Rigid motions and isometries
206
§7.3.
Gauss s
Theorema
Egregium
210
§7.4.
Rigidity of ovaloids
214
§7.5.
Geodesies
219
§7.6.
The exponential map
229
Exercises
239
Hints for solving the exercises
242
§7.7.
Appendix: Some additional results of an intrinsic type
252
Chapter
8.
The Gauss-Bonnet Theorem
275
§8.1.
Introduction
275
§8.2.
Degree of maps between compact surfaces
276
§8.3.
Degree and surfaces bounding the same domain
284
§8.4.
The index of a field at an isolated zero
288
§8.5.
The Gauss-Bonnet formula
295
§8.6.
Exercise: The
Euler
characteristic is even
304
Exercises: Steps of the proof
305
Chapter
9.
Global Geometry of Curves
309
§9.1.
Introduction and historical notes
309
§9.2.
Parametrized curves and simple curves
313
§9.3.
Results already shown on surfaces
319
§9.4.
Rotation index of plane curves
329
§9.5.
Periodic space curves
338
§9.6.
The four-vertices theorem
346
Exercises
350
Hints for solving the exercises
353
§9.7.
Appendix: One-dimensional degree theory
365
Bibliography
371
Index
373
|
| any_adam_object | 1 |
| author | Montiel, Sebastián Ros, Antonio 1957- |
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| dewey-ones | 516 - Geometry |
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| dewey-search | 516.3/62 |
| dewey-sort | 3516.3 262 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| isbn | 9780821847633 |
| language | English Spanish |
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| series | Graduate studies in mathematics |
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| spelling | Montiel, Sebastián Verfasser aut Curvas y superficies Curves and surfaces Sebastián Montiel ; Antonio Ros 2. ed. Providence, RI American Mathem. Soc. 2009 XVI, 376 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 69 Includes bibliographical references and index Curves on surfaces Geometry, Differential Submanifolds Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Kurve (DE-588)4033824-1 gnd rswk-swf Fläche (DE-588)4129864-0 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Fläche (DE-588)4129864-0 s Differentialgeometrie (DE-588)4012248-7 s DE-604 Kurve (DE-588)4033824-1 s Ros, Antonio 1957- Verfasser (DE-588)12383015X aut Graduate studies in mathematics 69 (DE-604)BV009739289 69 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018653858&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 | Montiel, Sebastián Ros, Antonio 1957- Curves and surfaces Graduate studies in mathematics Curves on surfaces Geometry, Differential Submanifolds Differentialgeometrie (DE-588)4012248-7 gnd Kurve (DE-588)4033824-1 gnd Fläche (DE-588)4129864-0 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4033824-1 (DE-588)4129864-0 (DE-588)4123623-3 |
| title | Curves and surfaces |
| title_alt | Curvas y superficies |
| title_auth | Curves and surfaces |
| title_exact_search | Curves and surfaces |
| title_full | Curves and surfaces Sebastián Montiel ; Antonio Ros |
| title_fullStr | Curves and surfaces Sebastián Montiel ; Antonio Ros |
| title_full_unstemmed | Curves and surfaces Sebastián Montiel ; Antonio Ros |
| title_short | Curves and surfaces |
| title_sort | curves and surfaces |
| topic | Curves on surfaces Geometry, Differential Submanifolds Differentialgeometrie (DE-588)4012248-7 gnd Kurve (DE-588)4033824-1 gnd Fläche (DE-588)4129864-0 gnd |
| topic_facet | Curves on surfaces Geometry, Differential Submanifolds Differentialgeometrie Kurve Fläche Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018653858&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
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