Modern geometry, methods and applications: 1 The geometry of surfaces, transformation groups, and fields
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY [u.a.]
Springer
1984
|
| Schriftenreihe: | Graduate texts in mathematics
93 |
| Online-Zugang: | Inhaltsverzeichnis |
Internformat
MARC
| LEADER | 00000nam a2200000 cc4500 | ||
|---|---|---|---|
| 001 | BV024182777 | ||
| 003 | DE-604 | ||
| 005 | 20090910 | ||
| 007 | t | ||
| 008 | 940826s1984 |||| 00||| eng d | ||
| 035 | |a (OCoLC)220720690 | ||
| 035 | |a (DE-599)BVBBV024182777 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-83 | ||
| 100 | 1 | |a Dubrovin, Boris Anatol'evič |d 1950-2019 |e Verfasser |0 (DE-588)115874417X |4 aut | |
| 240 | 1 | 0 | |a Sovremennaja geometria |
| 245 | 1 | 0 | |a Modern geometry, methods and applications |n 1 |p The geometry of surfaces, transformation groups, and fields |c B. A. Dubrovin ; A. T. Fomenko ; S. P. Novikov |
| 264 | 1 | |a New York, NY [u.a.] |b Springer |c 1984 | |
| 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 | |
| 773 | 0 | 8 | |w (DE-604)BV023815797 |g 1 |
| 830 | 0 | |a Graduate texts in mathematics |v 93 |w (DE-604)BV000000067 |9 93 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018306290&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
Datensatz im Suchindex
| _version_ | 1805076308530561024 |
|---|---|
| adam_text |
Contents
CHAPTER 1
Geometry in Regions of a Space. Basic Concepts 1
§1. Co-ordinate systems ]
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-dimensional 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 Rf 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
11.3. 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 conformal form of the metric on a surface 109
13.1. Isothermal co-ordinates. Gaussian curvature in terms of conformal
co-ordinates 109
13.2. Conformal 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 jV-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. Conformal 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 xjjj
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 (0, k) 164
18.3. The exterior product of differential forms. The exterior algebra 166
18.4. Exercises 167
§19. Tensors in Riemannian and pseudo-Riemannian spaces 168
19.1. Raising and lowering indices 168
19.2. The eigenvalues of a quadratic form 170
19.3. The operator * 171
19.4. Tensors in Euclidean space 172
19.5. Exercises 173
§20. The crystallographic groups and the finite subgroups of the rotation group
of Euclidean 3-space. Examples of invariant tensors 173
§21. Rank 2 tensors in pseudo-Euclidean space, and their eigenvalues 194
21.1. Skew-symmetric tensors. The invariants of an electromagnetic field 194
21.2. Symmetric tensors and their eigenvalues. The energy-momentum
tensor of an electromagnetic field 199
§22. The behaviour of tensors under mappings 203
22.1. The general operation of restriction of tensors with lower indices 203
22.2. Mappings of tangent spaces 204
§23. Vector fields 205
23.1. One-parameter groups of diffeomorphisms 205
23.2. The Lie derivative 207
23.3. Exercises 211
§24. Lie algebras 212
24.1. Lie algebras and vector fields 212
24.2. The fundamental matrix Lie algebras 214
24.3. Linear vector fields 219
24.4. The Killing metric 224
24.5. The classification of the 3-dimensional Lie algebras 226
24.6. The Lie algebras of the conformal groups 227
24.7. Exercises 232
CHAPTER 4
The Differential Calculus of Tensors 234
§25. The differential calculus of skew-symmetric tensors 234
25.1. The gradient of a skew-symmetric tensor 234
25.2. The exterior derivative of a form 237
25.3. Exercises 243
§26. Skew-symmetric tensors and the theory of integration 244
26.1. Integration of differential forms 244
26.2. Examples of integrals of differential forms 250
26.3. The general Stokes formula. Examples 255
26.4. Proof of the general Stokes formula for the cube 263
26.5. Exercises 265
xiv Contents
§27. Differential forms on complex spaces 266
27.1. The operators d' and d" 266
27.2. Kahlerian metrics. The curvature form 269
§28. Covariant differentiation 271
28.1. Euclidean connexions 271
28.2. Covariant differentiation of tensors of arbitrary rank 280
§29. Covariant differentiation and the metric 284
29.1. Parallel transport of vector fields 284
29.2. Geodesies 286
29.3. Connexions compatible with the metric 287
29.4. Connexions compatible with a complex structure (Hermitian metric) 291
29.5. Exercises 293
§30. The curvature tensor 295
30.1. The general curvature tensor 295
30.2. The symmetries of the curvature tensor. The curvature tensor defined
by the metric 300
30.3. Examples: the curvature tensor in spaces of dimensions 2 and 3; the
curvature tensor defined by a Killing metric 302
30.4. The Peterson-Codazzi equations. Surfaces of constant negative
curvature, and the "sine-Gordon" equation 307
30.5. Exercises 310
CHAPTER 5
The Elements of the Calculus of Variations 313
§31. One-dimensional variational problems 313
31.1. The Euler-Lagrange equations 313
31.2. Basic examples of functionals 317
§32. Conservation laws 320
32.1. Groups of transformations preserving a given variational problem 320
32.2. Examples. Applications of the conservation laws 322
§33. Hamiltonian formalism 333
33.1. Legendre's transformation 333
33.2. Moving co-ordinate frames 336
33.3. The principles of Maupertuis and Fermat 341
33.4. Exercises 344
§34. The geometrical theory of phase space 344
34.1. Gradient systems 344
34.2. The Poisson bracket 348
34.3. Canonical transformations 354
34.4. Exercises 358
§35. Lagrange surfaces 358
35.1. Bundles of trajectories and the Hamilton-Jacobi equation 358
35.2. Hamiltonians which are first-order homogeneous with respect to the
momentum 363
§36. The second variation for the equation of the geodesies 367
36.1. The formula for the second variation 367
36.2. Conjugate points and the minimality condition 371
Contents XV
CHAPTER 6
The Calculus of Variations in Several Dimensions. Fields and
Their Geometric Invariants 375
§37. The simplest higher-dimensional variational problems 375
37.1. The Euler-Lagrange equations 375
37.2. The energy-momentum tensor 379
37.3. The equations of an electromagnetic field 384
37.4. The equations of a gravitational field 390
37.5. Soap films 397
37.6. Equilibrium equation for a thin plate 403
37.7. Exercises 408
§38. Examples of Lagrangians 409
§39. The simplest concepts of the general theory of relativity 412
§40. The spinor representations of the groups SO(3) and 0(3, 1). Dirac's
equation and its properties 427
40.1. Automorphisms of matrix algebras 427
40.2. The spinor representation of the group SO(3) 429
40.3. The spinor representation of the Lorentz group 431
40.4. Dirac's equation 435
40.5. Dirac's equation in an electromagnetic field. The operation of charge
conjugation 437
§41. Covariant differentiation of fields with arbitrary symmetry 439
41.1. Gauge transformations. Gauge-invariant Lagrangians 439
41.2. The curvature form 443
41.3. Basic examples 444
§42. Examples of gauge-invariant functionals. Maxwell's equations and the
Yang-Mills equation. Functionals with identically zero variational
derivative (characteristic classes) 449
Bibliography 455
Index 459 |
| any_adam_object | 1 |
| author | Dubrovin, Boris Anatol'evič 1950-2019 Fomenko, Anatolij Timofeevič 1945- Novikov, Sergej P. 1938-2024 |
| author_GND | (DE-588)115874417X (DE-588)119092689 (DE-588)118786490 |
| author_facet | Dubrovin, Boris Anatol'evič 1950-2019 Fomenko, Anatolij Timofeevič 1945- Novikov, Sergej P. 1938-2024 |
| author_role | aut aut aut |
| author_sort | Dubrovin, Boris Anatol'evič 1950-2019 |
| author_variant | b a d ba bad a t f at atf s p n sp spn |
| building | Verbundindex |
| bvnumber | BV024182777 |
| ctrlnum | (OCoLC)220720690 (DE-599)BVBBV024182777 |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>00000nam a2200000 cc4500</leader><controlfield tag="001">BV024182777</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20090910</controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">940826s1984 |||| 00||| eng d</controlfield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)220720690</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV024182777</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-83</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Dubrovin, Boris Anatol'evič</subfield><subfield code="d">1950-2019</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)115874417X</subfield><subfield code="4">aut</subfield></datafield><datafield tag="240" ind1="1" ind2="0"><subfield code="a">Sovremennaja geometria</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Modern geometry, methods and applications</subfield><subfield code="n">1</subfield><subfield code="p">The geometry of surfaces, transformation groups, and fields</subfield><subfield code="c">B. A. Dubrovin ; A. T. Fomenko ; S. P. Novikov</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">New York, NY [u.a.]</subfield><subfield code="b">Springer</subfield><subfield code="c">1984</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">Graduate texts in mathematics</subfield><subfield code="v">93</subfield></datafield><datafield tag="490" ind1="0" ind2=" "><subfield code="a">Graduate texts in mathematics</subfield><subfield code="v">...</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Fomenko, Anatolij Timofeevič</subfield><subfield code="d">1945-</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)119092689</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Novikov, Sergej P.</subfield><subfield code="d">1938-2024</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)118786490</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="w">(DE-604)BV023815797</subfield><subfield code="g">1</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">Graduate texts in mathematics</subfield><subfield code="v">93</subfield><subfield code="w">(DE-604)BV000000067</subfield><subfield code="9">93</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">HBZ Datenaustausch</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018306290&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield></record></collection> |
| id | DE-604.BV024182777 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-20T05:55:14Z |
| institution | BVB |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018306290 |
| oclc_num | 220720690 |
| open_access_boolean | |
| owner | DE-83 |
| owner_facet | DE-83 |
| publishDate | 1984 |
| publishDateSearch | 1984 |
| publishDateSort | 1984 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Dubrovin, Boris Anatol'evič 1950-2019 Verfasser (DE-588)115874417X aut Sovremennaja geometria Modern geometry, methods and applications 1 The geometry of surfaces, transformation groups, and fields B. A. Dubrovin ; A. T. Fomenko ; S. P. Novikov New York, NY [u.a.] Springer 1984 txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 93 Graduate texts in mathematics ... Fomenko, Anatolij Timofeevič 1945- Verfasser (DE-588)119092689 aut Novikov, Sergej P. 1938-2024 Verfasser (DE-588)118786490 aut (DE-604)BV023815797 1 Graduate texts in mathematics 93 (DE-604)BV000000067 93 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018306290&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| 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 geometria |
| 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, methods and applications |
| title_sort | modern geometry methods and applications the geometry of surfaces transformation groups and fields |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018306290&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV023815797 (DE-604)BV000000067 |
| work_keys_str_mv | AT dubrovinborisanatolevic sovremennajageometria AT fomenkoanatolijtimofeevic sovremennajageometria AT novikovsergejp sovremennajageometria AT dubrovinborisanatolevic moderngeometrymethodsandapplications1 AT fomenkoanatolijtimofeevic moderngeometrymethodsandapplications1 AT novikovsergejp moderngeometrymethodsandapplications1 |