Modern geometry: methods and applications 1 The geometry of surfaces, transformation groups, and fields
Gespeichert in:
| Beteiligte Personen: | , , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
New York u.a.
Springer
1992
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Graduate texts in mathematics
93 |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003274768&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XV, 468 S. graph. Darst. |
| ISBN: | 0387976639 3540976639 |
Internformat
MARC
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| 100 | 1 | |a Dubrovin, Boris Anatol'evič |d 1950-2019 |e Verfasser |0 (DE-588)115874417X |4 aut | |
| 240 | 1 | 0 | |a Sovremennaja geometrija |
| 245 | 1 | 0 | |a Modern geometry |b methods and applications |n 1 |p The geometry of surfaces, transformation groups, and fields |c B. A. Dubrovin ; A. T. Fomenko ; S. P. Novikov |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York u.a. |b Springer |c 1992 | |
| 300 | |a XV, 468 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate texts in mathematics |v 93 | |
| 490 | 0 | |a Graduate texts in mathematics |v ... | |
| 700 | 1 | |a Fomenko, Anatolij Timofeevič |d 1945- |e Verfasser |0 (DE-588)119092689 |4 aut | |
| 700 | 1 | |a Novikov, Sergej P. |d 1938-2024 |e Verfasser |0 (DE-588)118786490 |4 aut | |
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| 830 | 0 | |a Graduate texts in mathematics |v 93 |w (DE-604)BV000000067 |9 93 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003274768&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-003274768 | |
Datensatz im Suchindex
| DE-BY-TUM_call_number | 0202 MAT 500f 2007 A 5832-1(2) |
|---|---|
| DE-BY-TUM_katkey | 552598 |
| DE-BY-TUM_location | 02 |
| DE-BY-TUM_media_number | 040020773413 |
| _version_ | 1821936893603872768 |
| adam_text | Contents
Preface
to the First Edition
v
CHAPTER
1
Geometry in Regions of a Space. Basic Concepts
1
§1.
Co-ordinate systems
1
1.1.
Cartesian co-ordinates in a space
2
1.2.
Co-ordinate changes
3
§2.
Euclidean space
9
2.1.
Curves in Euclidean space
9
2.2.
Quadratic forms and vectors
14
§3.
Riemannian and pseudo-Riemannian spaces
17
3.1.
Riemannian metrics
17
3.2.
The Minkowski metric
20
§4.
The simplest groups of transformations of Euclidean space
23
4.1.
Groups of transformations of a region
23
4.2.
Transformations of the plane
25
4.3.
The isometries of
3-dímensíonaI
Euclidean space
31
4.4.
Further examples of transformation groups
34
4.5.
Exercises
37
§5.
The Serret-Frenet formulae
38
5.1.
Curvature of curves in the Euclidean plane
38
5.2.
Curves in Euclidean 3-space. Curvature and torsion
42
5.3.
Orthogonal transformations depending on a parameter
47
5.4.
Exercises
48
§6.
Pseudo-Euclidean spaces
50
6.1.
The simplest concepts of the special theory of relativity
50
6.2.
Lorentz
transformations
52
6.3.
Exercises
60
xii Contents
CHAPTER
2
The Theory of Surfaces
61
§7.
Geometry on a surface in space
61
7.1.
Co-ordinates on a surface
61
7.2.
Tangent planes
66
7.3.
The metric on a surface in Euclidean space
68
7.4.
Surface area
72
7.5.
Exercises
76
§8.
The second fundamental form
76
8.1.
Curvature of curves on a surface in Euclidean space
76
8.2.
Invariants of a pair of quadratic forms
79
8.3.
Properties of the second fundamental form
80
8.4.
Exercises
86
§9.
The metric on the sphere
86
§10.
Space-like surfaces in pseudo-Euclidean space
90
10.1.
The pseudo-sphere
90
10.2.
Curvature of space-like curves in
R J
94
§11.
The language of complex numbers in geometry
95
11.1.
Complex and real co-ordinates
95
11.2.
The Hermitian scalar product
97
Л.З.
Examples of complex transformation groups
99
§12.
Analytic functions
100
12.1.
Complex notation for the element of length, and for
the differential of a function
100
12.2.
Complex co-ordinate changes
104
12.3.
Surfaces in complex space
106
§13.
The
conformai
form of the metric on a surface
109
13.1.
Isothermal co-ordinates. Gaussian curvature in terms of
conformai
co-ordinates
109
13.2.
Conformai
form of the metrics on the sphere and
the Lobachevskian plane
114
13.3.
Surfaces of constant curvature
117
13.4.
Exercises
120
§14.
Transformation groups as surfaces in iV-dimensional space
120
14.1.
Co-ordinates in a neighbourhood of the identity
120
14.2.
The exponential function with matrix argument
127
14.3.
The quaternions
131
14.4.
Exercises
136
§15.
Conformai
transformations of Euclidean and pseudo-Euclidean
spaces of several dimensions
136
CHAPTER
3
Tensors: The Algebraic Theory
145
§16.
Examples of tensors
145
§17.
The general definition of a tensor
151
17.1.
The transformation rule for the components of a tensor
of arbitrary rank
151
Contents xiii
17.2.
Algebraic operations on tensors
157
17.3.
Exercises
161
§18.
Tensors of type
(0,
k)
161
18.1.
Differential notation for tensors with lower indices only
161
18.2.
Skew-symmetric tensors of type
(ОД)
164
18.3.
The exterior product of differential forms. The exterior algebra
166
18.4.
Skew-symmetric tensors of type (k,
0)
(poly vectors). Integrals
with respect to anti-commuting variables
167
18.5.
Exercises
170
§19.
Tensors in Riemannian and pseudo-Riemannian spaces
170
19.1.
Raising and lowering indices
170
19.2.
The eigenvalues of a quadratic form
172
19.3.
The operator*
174
19.4.
Tensors in Euclidean space
174
19.5.
Exercises
175
§20.
The crystallographic groups and the finite subgroups of the rotation
group of Euclidean 3-space. Examples of invariant tensors
176
§21.
Rank
2
tensors in pseudo-Euclidean space, and their eigenvalues
197
21.1.
Skew-symmetric tensors. The invariants of an electromagnetic field
197
21.2.
Symmetric tensors and their eigenvalues. The energy-momentum
tensor of an electromagnetic field
202
§22.
The behaviour of tensors under mappings
205
22.1.
The general operation of restriction of tensors with lower indices
205
22.2.
Mappings of tangent spaces
207
§23.
Vector fields
208
23.1.
One-parameter groups of diffeomorphisms
208
23.2.
The exponential function of a vector field
210
23.3.
The Lie derivative
211
23.4.
Exercises
215
§24.
Lie algebras
216
24.1.
Lie algebras and vector fields
216
24.2.
The fundamental matrix Lie algebras
218
24.3.
Linear vector fields
223
24.4.
Left-invariant fields defined on transformation groups
225
24.5.
Invariant metrics on a transformation group
227
24.6.
The classification of the 3-dimensional Lie algebras
229
24.7.
The Lie algebras of the
conformai
groups
231
24.8.
Exercises
236
CHAPTER
4
The Differential Calculus of Tensors
238
§25.
The differential calculus of skew-symmetric tensors
238
25.1.
The gradient of a skew-symmetric tensor
238
25.2.
The exterior derivative of a form
241
25.3.
Exercises
247
§26.
Skew-symmetric tensors and the theory of integration
248
26.1.
Integration of differential forms
248
xiv
Contents
26.2.
Examples of integrals of differential forms
254
26.3.
The general Stokes formula. Examples
259
26.4.
Proof of the general Stokes formula for the cube
267
26.5.
Exercises
269
§27.
Differential forms on complex spaces
270
27.1.
The operators
ď
and d
270
27.2. Kählerian
metrics. The curvature form
273
§28.
Covariant differentiation
275
28.1.
Euclidean connexions
275
28.2.
Covariant differentiation of tensors of arbitrary rank
284
§29.
Covariant differentiation and the metric
288
29.1.
Parallel transport of vector fields
288
29.2.
Geodesies
290
29.3.
Connexions compatible with the metric
291
29.4.
Connexions compatible with a complex structure (Hermitian metric)
295
29.5.
Exercises
297
§30.
The curvature tensor
299
30.1.
The general curvature tensor
299
30.2.
The symmetries of the curvature tensor. The curvature tensor
defined by the metric
304
30.3.
Examples: The curvature tensor in spaces of dimensions
2
and
3;
the curvature tensor of transformation groups
306
30.4.
The Peterson-Codazzi equations. Surfaces of constant negative
curvature, and the
sine-Gordon
equation
311
30.5.
Exercises
314
CHAPTER
5
The Elements of the Calculus of Variations
317
§31.
One-dimensional variational problems
317
31.1.
The Euler-Lagrange equations
317
31.2.
Basic examples of functionals
321
§32.
Conservation laws
324
32.
1
.
Groups of transformations preserving a given variational problem
324
32.2.
Examples. Applications of the conservation laws
326
§33.
Hamiltonian formalism
337
33.1.
Legendre s transformation
337
33.2.
Moving co-ordinate frames
340
33.3.
The principles of Maupertuis and
Fermat
345
33.4.
Exercises
348
§34.
The geometrical theory of phase space
348
34.1.
Gradient systems
348
34.2.
The
Poisson
bracket
352
34.3.
Canonical transformations
358
34.4.
Exercises
362
§35. Lagrange
surfaces
362
35.1.
Bundles of trajectories and the Hamilton-Jacobi equation
362
35.2.
Hamiltonians which are first-order homogeneous with
respect to the momentum
367
Contents
XV
§36.
The second variation for the equation of the geodesies
371
36.1.
The formula for the second variation
371
36.2.
Conjugate points and the minimality condition
375
CHAPTER
6
The Calculus of Variations in Several Dimensions.
Fields and Their Geometric Invariants
379
§37.
The simplest higher-dimensional variational problems
379
37.1.
The Euler-Lagrange equations
379
37.2.
The energy-momentum tensor
383
37.3.
The equations of an electromagnetic field
388
37.4.
The equations of a gravitational field
394
37.5.
Soap films
401
37.6.
Equilibrium equation for a thin plate
407
37.7.
Exercises
412
§38.
Examples of Lagrangians
413
§39.
The simplest concepts of the general theory of relativity
416
§40.
The spinor representations of the groups SO(3) and
0(3,1).
Dirac s equation and its properties
431
40.1.
Automorphisms of matrix algebras
431
40.2.
The spinor representation of the group SO(3)
433
40.3.
The spinor representation of the
Lorentz
group
435
40.4.
Dirac s equation
439
40.5.
Dirac s equation in an electromagnetic field. The operation
of charge conjugation
441
§41.
Covariant differentiation of fields with arbitrary symmetry
443
41.1.
Gauge transformations. Gauge-invariant Lagrangians
443
41.2.
The curvature form
447
41.3.
Basic examples
448
§42.
Examples of gauge-invariant functionals. Maxwell s equations and
the Yang—Mills equation. Functionals with identically zero
variational derivative (characteristic classes)
453
Bibliography
459
Index
463
|
| any_adam_object | 1 |
| author | Dubrovin, Boris Anatol'evič 1950-2019 Fomenko, Anatolij Timofeevič 1945- Novikov, Sergej P. 1938-2024 |
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| id | DE-604.BV005260744 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T08:30:33Z |
| institution | BVB |
| isbn | 0387976639 3540976639 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-003274768 |
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| physical | XV, 468 S. graph. Darst. |
| publishDate | 1992 |
| publishDateSearch | 1992 |
| publishDateSort | 1992 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spellingShingle | Dubrovin, Boris Anatol'evič 1950-2019 Fomenko, Anatolij Timofeevič 1945- Novikov, Sergej P. 1938-2024 Modern geometry methods and applications Graduate texts in mathematics |
| title | Modern geometry methods and applications |
| title_alt | Sovremennaja geometrija |
| title_auth | Modern geometry methods and applications |
| title_exact_search | Modern geometry methods and applications |
| title_full | Modern geometry methods and applications 1 The geometry of surfaces, transformation groups, and fields B. A. Dubrovin ; A. T. Fomenko ; S. P. Novikov |
| title_fullStr | Modern geometry methods and applications 1 The geometry of surfaces, transformation groups, and fields B. A. Dubrovin ; A. T. Fomenko ; S. P. Novikov |
| title_full_unstemmed | Modern geometry methods and applications 1 The geometry of surfaces, transformation groups, and fields B. A. Dubrovin ; A. T. Fomenko ; S. P. Novikov |
| title_short | Modern geometry |
| title_sort | modern geometry methods and applications the geometry of surfaces transformation groups and fields |
| title_sub | methods and applications |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003274768&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV005835289 (DE-604)BV000000067 |
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