Elementary differential geometry:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
London [u.a.]
Springer
2008
|
| Ausgabe: | 10. print. |
| Schriftenreihe: | Springer undergraduate mathematics series
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | IX, 332 S. Ill., graph. Darst. |
| ISBN: | 1852331526 9781852331528 |
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| 245 | 1 | 0 | |a Elementary differential geometry |c Andrew Pressley |
| 250 | |a 10. print. | ||
| 264 | 1 | |a London [u.a.] |b Springer |c 2008 | |
| 300 | |a IX, 332 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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Datensatz im Suchindex
| _version_ | 1804137650755469312 |
|---|---|
| adam_text | Contents
Preface
...................................................
v
1.
Curves
in the
Plane and in Space .................... 1
1.1
What is a Curve?
................................ 1
1.2
Arc-Length
...................................... 7
1.3
Reparametrization
................................ 10
1.4
Level Curves vs. Parametrized Curves
............... 16
2.
How Much Does a Curve Curve
? .................... 23
2.1
Curvature
....................................... 23
2.2
Plane Curves
..................................... 28
2.3
Space Curves
.................................... 36
3.
Global Properties of Curves
.......................... 47
3.1
Simple Closed Curves
............................. 47
3.2
The Isoperimetric Inequality
....................... 51
3.3
The Four Vertex Theorem
......................... 55
4.
Surfaces in Three Dimensions
........................ 59
4.1
What is a Surface?
............................... 59
4.2
Smooth Surfaces
.................................. 66
4.3
Tangents, Normals and Orientability
................ 74
4.4
Examples of Surfaces
.............................. 78
4.5
Quadric Surfaces
................................. 84
4.6
Triply Orthogonal Systems
........................ 90
4.7
Applications of the Inverse Function Theorem
........ 93
VII
Elementary Differential Geometry
5.
The First Fundamental Form
......................... 97
5.1
Lengths of Curves on Surfaces
...................... 97
5.2
Isometries of Surfaces
............................. 101
5.3
Conformai
Mappings of Surfaces
.................... 106
5.4
Surface Area
..................................... 112
5.5
Equiareal Maps and a Theorem of Archimedes
....... 116
6.
Curvature of Surfaces
................................ 123
6.1
The Second Fundamental Form
..................... 123
6.2
The Curvature of Curves on a Surface
............... 127
6.3
The Normal and Principal Curvatures
............... 130
6.4
Geometric Interpretation of Principal Curvatures
..... 141
7.
Gaussian Curvature and the Gauss Map
.............. 147
7.1
The Gaussian and Mean Curvatures
................ 147
7.2
The Pseudosphere
................................ 151
7.3
Flat Surfaces
..................................... 155
7.4
Surfaces of Constant Mean Curvature
............... 161
7.5
Gaussian Curvature of Compact Surfaces
............ 164
7.6
The Gauss map
.................................. 165
8.
Geodesies
............................................ 171
8.1
Definition and Basic Properties
..................... 171
8.2
Geodesic Equations
............................... 175
8.3
Geodesies on Surfaces of Revolution
................. 181
8.4
Geodesies as Shortest Paths
........................ 190
8.5
Geodesic Coordinates
............................. 197
9.
Minimal Surfaces
.....................................201
9.1
Plateau s Problem
................................ 201
9.2
Examples of Minimal Surfaces
...................... 207
9.3
Gauss map of a Minimal Surface
................... 217
9.4
Minimal Surfaces and Holomorphic Functions
........ 219
10.
Gauss s
Theorema
Egregium
.........................229
10.1
Gauss s Remarkable Theorem
...................... 229
10.2
Isometries of Surfaces
............................. 238
10.3
The Codazzi-Mainardi Equations
................... 240
10.4
Compact Surfaces of Constant Gaussian Curvature
... 244
11.
The Gauss-Bonnet Theorem
.........................247
11.1
Gauss-Bonnet for Simple Closed Curves
............. 247
11.2
Gauss-Bonnet for Curvilinear Polygons
.............. 252
11.3
Gauss-Bonnet for Compact Surfaces
................ 258
Contents
IX
11.4
Singularities of Vector Fields
....................... 269
11.5
Critical Points
.................................... 275
Solutions
.................................................. 281
Chapter
1 ............................................. 272
Chapter
2 ............................................. 275
Chapter
3 ............................................. 279
Chapter
4 ............................................. 280
Chapter
5 ............................................. 286
Chapter
6............................................. 289
Chapter
7 ............................................. 294
Chapter
8 ............................................. 299
Chapter
9 ............................................. 307
Chapter
10 ............................................ 311
Chapter
11 ............................................ 314
Index
...................................................... 329
|
| adam_txt |
Contents
Preface
.
v
1.
Curves
in the
Plane and in Space . 1
1.1
What is a Curve?
. 1
1.2
Arc-Length
. 7
1.3
Reparametrization
. 10
1.4
Level Curves vs. Parametrized Curves
. 16
2.
How Much Does a Curve Curve
? . 23
2.1
Curvature
. 23
2.2
Plane Curves
. 28
2.3
Space Curves
. 36
3.
Global Properties of Curves
. 47
3.1
Simple Closed Curves
. 47
3.2
The Isoperimetric Inequality
. 51
3.3
The Four Vertex Theorem
. 55
4.
Surfaces in Three Dimensions
. 59
4.1
What is a Surface?
. 59
4.2
Smooth Surfaces
. 66
4.3
Tangents, Normals and Orientability
. 74
4.4
Examples of Surfaces
. 78
4.5
Quadric Surfaces
. 84
4.6
Triply Orthogonal Systems
. 90
4.7
Applications of the Inverse Function Theorem
. 93
VII
Elementary Differential Geometry
5.
The First Fundamental Form
. 97
5.1
Lengths of Curves on Surfaces
. 97
5.2
Isometries of Surfaces
. 101
5.3
Conformai
Mappings of Surfaces
. 106
5.4
Surface Area
. 112
5.5
Equiareal Maps and a Theorem of Archimedes
. 116
6.
Curvature of Surfaces
. 123
6.1
The Second Fundamental Form
. 123
6.2
The Curvature of Curves on a Surface
. 127
6.3
The Normal and Principal Curvatures
. 130
6.4
Geometric Interpretation of Principal Curvatures
. 141
7.
Gaussian Curvature and the Gauss Map
. 147
7.1
The Gaussian and Mean Curvatures
. 147
7.2
The Pseudosphere
. 151
7.3
Flat Surfaces
. 155
7.4
Surfaces of Constant Mean Curvature
. 161
7.5
Gaussian Curvature of Compact Surfaces
. 164
7.6
The Gauss map
. 165
8.
Geodesies
. 171
8.1
Definition and Basic Properties
. 171
8.2
Geodesic Equations
. 175
8.3
Geodesies on Surfaces of Revolution
. 181
8.4
Geodesies as Shortest Paths
. 190
8.5
Geodesic Coordinates
. 197
9.
Minimal Surfaces
.201
9.1
Plateau's Problem
. 201
9.2
Examples of Minimal Surfaces
. 207
9.3
Gauss map of a Minimal Surface
. 217
9.4
Minimal Surfaces and Holomorphic Functions
. 219
10.
Gauss's
Theorema
Egregium
.229
10.1
Gauss's Remarkable Theorem
. 229
10.2
Isometries of Surfaces
. 238
10.3
The Codazzi-Mainardi Equations
. 240
10.4
Compact Surfaces of Constant Gaussian Curvature
. 244
11.
The Gauss-Bonnet Theorem
.247
11.1
Gauss-Bonnet for Simple Closed Curves
. 247
11.2
Gauss-Bonnet for Curvilinear Polygons
. 252
11.3
Gauss-Bonnet for Compact Surfaces
. 258
Contents
IX
11.4
Singularities of Vector Fields
. 269
11.5
Critical Points
. 275
Solutions
. 281
Chapter
1 . 272
Chapter
2 . 275
Chapter
3 . 279
Chapter
4 . 280
Chapter
5 . 286
Chapter
6. 289
Chapter
7 . 294
Chapter
8 . 299
Chapter
9 . 307
Chapter
10 . 311
Chapter
11 . 314
Index
. 329 |
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| illustrated | Illustrated |
| index_date | 2024-07-02T20:51:47Z |
| indexdate | 2024-07-09T21:15:40Z |
| institution | BVB |
| isbn | 1852331526 9781852331528 |
| language | English |
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| physical | IX, 332 S. Ill., graph. Darst. |
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| spelling | Pressley, Andrew Verfasser (DE-588)122153049 aut Elementary differential geometry Andrew Pressley 10. print. London [u.a.] Springer 2008 IX, 332 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer undergraduate mathematics series Geometria diferencial (textos elementares) larpcal Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Differentialgeometrie (DE-588)4012248-7 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016499272&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pressley, Andrew Elementary differential geometry Geometria diferencial (textos elementares) larpcal Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4123623-3 |
| 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 Andrew Pressley |
| title_fullStr | Elementary differential geometry Andrew Pressley |
| title_full_unstemmed | Elementary differential geometry Andrew Pressley |
| title_short | Elementary differential geometry |
| title_sort | elementary differential geometry |
| topic | Geometria diferencial (textos elementares) larpcal Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Geometria diferencial (textos elementares) Differentialgeometrie Lehrbuch |
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