Handbook of geometrical methods for scientists and engineers:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Nova Science Publ.
2010
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XIII, 630 S. Ill., graph. Darst. |
| ISBN: | 9781607417699 |
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MARC
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| 010 | |a 2009018844 | ||
| 020 | |a 9781607417699 |c hardcover |9 978-1-60741-769-9 | ||
| 035 | |a (OCoLC)705500810 | ||
| 035 | |a (DE-599)BVBBV036446417 | ||
| 040 | |a DE-604 |b ger |e aacr | ||
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| 044 | |a xxu |c US | ||
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| 084 | |a SK 950 |0 (DE-625)143273: |2 rvk | ||
| 100 | 1 | |a Ivancevic, Vladimir G. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Handbook of geometrical methods for scientists and engineers |c Vladimir G. Ivancevic and Tijana Ivancevic |
| 264 | 1 | |a New York |b Nova Science Publ. |c 2010 | |
| 300 | |a XIII, 630 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Geometric analysis | |
| 650 | 4 | |a Geometrical models | |
| 650 | 0 | 7 | |a Ingenieurwissenschaften |0 (DE-588)4137304-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Geometrische Methode |0 (DE-588)4156715-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Ingenieurwissenschaften |0 (DE-588)4137304-2 |D s |
| 689 | 0 | 1 | |a Geometrische Methode |0 (DE-588)4156715-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Ivancevic, Tijana T. |e Verfasser |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020318667&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-020318667 | ||
Record in the Search Index
| _version_ | 1804142931269910528 |
|---|---|
| adam_text | Contents
Preface
xiii
Acknowledgments
xv
I Geometrical Preliminaries
1
1
Classical Local Calculation Tools
3
1.1
Elements of Vector Analysis
....................... 3
1.1.1
The Hamilton V—operator
................... 4
1.1.2
The Laplace
Δ—
operator and the Fundamental PDEs
.... 6
1.1.3
Integrals of Vector-Fields over Paths and Surfaces
...... 7
1.1.4
Integral Theorems of Vector Analysis
.............. 9
1.1.5
Some More Dynamical PDEs
.................. 11
1.1.6
Euler
Rigid Body Equations
................... 12
1.2
Transformation of Coordinates and Elementary Tensors
....... 14
1.2.1
Transformation of Coordinates
................. 14
1.2.2
Scalar Invariants
......................... 16
1.2.3
Vectors and Covectors
...................... 16
1.2.4
Second-Order Tensors
...................... 17
1.2.5
Higher-Order Tensors
...................... 19
1.2.6
Tensor Symmetry
......................... 20
1.3
Euclidean Tensors
............................ 21
1.3.1
Basis Vectors and the Metric Tensor in K
.......... 21
1.3.2
Tensor Products in W1
...................... 22
1.4
Covariant Differentiation
......................... 23
1.4.1
Christoffeľs
Symbols
....................... 23
1.4.2
Geodesies
............................. 23
1.4.3
Covariant Derivative
....................... 24
1.4.4
Covariant Form of Vector Differential Operators
....... 25
1.4.5
Absolute Derivative
....................... 26
1.4.6 3D
Curve Geometry: Frenet-Serret Formulae
......... 27
VI
Contents
1.4.7
Mechanical Acceleration and Force
............... 27
1.5
Euler-Lagrangian vs. Hamiltonian Dynamics
............. 29
1.6
Kirchhoff s and Newton-Euler s Rigid Body Equations
........ 31
1.7
Covariant Mechanics
........................... 33
1.8
Covariant Fluid Dynamics
........................ 37
1.8.1
Continuity Equation
....................... 37
1.8.2
Forces Acting on a Fluid
..................... 39
1.8.3
Constitutive and Dynamical Equations
............. 40
1.8.4
Navier-Stokes Equations
..................... 41
1.9
Riemannian Curvature Tensor
...................... 42
1.10
Various PDEs with Their Lagrangians
................. 43
1.11
Exterior Differential Forms
....................... 43
1.11.1
From Green to Stokes
...................... 44
1.11.2
Exterior Derivative
........................ 46
1.11.3
Exterior p-Forms in R4
..................... 46
1.11.4
Stokes Theorem in M4
...................... 48
1.11.5
Hodge Star Operator
....................... 48
1.11.6
Topological Duality of p—Forms and p—Chains
........ 49
1.11.7 De Rham
Cochain Complex
................... 49
1.11.8 De
Rham Cohomology of a Smooth Manifold
......... 50
1.12
The Covariant Force Law
........................ 50
1.13
Basic
Neurodynamics
........................... 52
Modern Global Algebraic Framework
55
2.1
Sets
.................................... 55
2.2
Maps
.................................... 56
2.2.1
Algebra of Maps
......................... 57
2.2.2
Compositions of Maps
...................... 57
2.2.3
η
-tuple Chain Rule
....................... 58
2.2.4
n—tuple Integration and Change of Variables
......... 58
2.3
General Topology
............................. 59
2.4
Homotopy
................................. 50
2.4.1
Commutative Diagrams
..................... 62
2.5
Groups
................................... 53
2.6
Group-Related Algebraic Structures
.................. 65
2.7
Categories
................................. 57
2.8
Functors
. ........................... 59
2.9
Natural Transformations
........................ 72
2.9.1
Compositions of Natural Transformations
........... 72
2.9.2
Dinatural
Transformations
................... 73
2.10
Limits and Colimits
.......................... 74
2.11
Adjunction
......................... 74
Contents
vii
2.12 Application:
Physiological Sensory-Motor Adjunction
........ 76
2.13
те—
Categories
............................... 76
2.13.1
From Small Categories to Big
те—
Categories
........ 76
2.13.2
Topological Structure of n—Categories
............. 80
Brief Review of Tools from Modern Analysis and Geometry
87
3.1
Banah and Hubert Spaces
........................ 87
3.1.1
Topological Space
........................ 87
3.1.2
Metric Space
........................... 88
3.1.3
Banah Space
........................... 89
3.1.4
Hubert Space
........................... 89
3.2
Manifolds, Bundles and Jets
....................... 92
3.2.1
Manifolds
............................. 92
3.2.2
Bundles
.............................. 93
3.2.3
Jets
................................ 94
II Basic Geometrical Methods
99
4
Smooth Manifolds and (Co)Tangent Bundles
101
4.1
Smooth Manifolds
............................ 101
4.1.1
Introduction to Smooth Manifolds
............... 101
4.1.2
Formal Definition of a Smooth Manifold
............ 105
4.1.3
Smooth Maps Between Smooth Manifolds
........... 106
4.2
(Co)Tangent Bundles of Smooth Manifolds
.............. 107
4.2.1
Tangent Bundle and Lagrangian Dynamics
.......... 107
4.2.2
Cotangent Bundle and Hamiltonian Dynamics
........ 109
4.2.3
Introducing Fibre-, Tensor-, and Jet-Bundles
......... 110
4.2.4
Example: Command/Control in Human-Robot Interactions
.
Ill
4.3
Tensor Fields on Smooth Manifolds
................... 114
4.3.1
Tensor Bundle
.......................... 114
4.3.2
Differential Forms on Smooth Manifolds
......... . . . 122
4.3.3
Exterior Derivative and (Co)Homology
............. 126
4.3.4
The Paradigm of Dynamical Modelling
............. 138
5
Lie Derivatives and Lie Groups
141
5.1
Lie Derivatives on Smooth Manifolds
.................. 141
5.1.1
Lie Derivative Operating on Functions
............. 141
5.1.2
Lie Derivative of Vector Fields
................. 143
5.1.3
Time Derivative of the Evolution Operator
.......... 145
5.1.4
Lie Derivative of Differential Forms
............... 146
5.1.5
Lie Derivative of Various Tensor Fields
............. 147
5.1.6
Example: Lie-Derivative
Neurodynamics
........... 149
viii Contents
5.1.7
Lie Algebras
........................... 149
5.2
Nonlinear Control
............................ 150
5.2.1
Feedback Linearization
...................... 151
5.2.2
Controllability
.......................... 15
5.3
Lie Groups
................................ 161
5.3.1
Definition of a Lie Group
.................... 162
5.3.2
Actions of Lie Groups on Smooth Manifolds
.......... 165
5.3.3
Basic Dynamical Lie Groups
.................. 167
5.3.4
Lie Groups in Human/Humanoid
Biodynamics
........ 169
5.3.5
Euclidean Groups of Rigid Body Motion
............ 173
5.3.6
Basic Mechanical Examples
................... 176
5.3.7
Summary on Newton-Euler SE(S)—Dynamics
........ 178
5.3.8
Symplectic Group in Hamiltonian Mechanics
......... 182
5.3.9
Group Structure of a Biodynamical Manifold
......... 183
5.4
Biomedical
Application: Brain Injury Mechanics
........... 187
5.4.1
Introduction to Traumatic Brain Injury
............ 187
5.4.2
SE(3)-jolt: the Cause of TBI
................. 189
Riemann—Finsler and Symplectic Geometry
197
6.1
Riemannian Manifolds
.......................... 197
6.1.1
Local Riemannian Geometry
.................. 197
6.1.2
Global Riemannian Geometry
.................. 206
6.1.3
Ricci
Flow
............................. 209
6.1.4
Structure Equations
....................... 210
6.1.5
Example: Autonomous Lagrangian Dynamics
......... 212
6.1.6
Basics of Morse and (Co)Bordism Theories
.......... 214
6.2
Soft Introduction to
De Rham-Hodge
Theory
............. 218
6.2.1
Exact and Closed Forms and Chains
.............. 218
6.2.2 De Rham
Duality of Forms and Chains
............ 219
6.2.3
De Rham
Cochain and Chain Complex
............ 219
6.2.4
De Rham
Cohomology versus Chain Homology
........ 220
6.2.5
Hodge Star Operator
....................... 222
6.2.6
Hodge Inner Product
....................... 223
6.2.7
Hodge Codifferential Operator
................. 223
6.2.8
Hodge Laplacian Operator
.................... 224
6.2.9
Hodge
Adjoints
and Self-Adjoints
............... 226
6.2.10
The Hodge Decomposition Theorem
.............. 226
6.3
Ricci
Flow and Reaction-Diffusion Processes
............. 227
6.3.1
Bio-Reaction-Diffusion Systems
................ 230
6.3.2
Reactive
Neurodynamics
..................... 241
6.3.3
Dissipative Evolution under the
Ricci
Flow
.......... 249
6.4
Finsler Manifolds
............................. 260
Contents ix
6.4.1 Definition
of a Finsler Manifold
................. 260
6.4.2
Energy
Functional, Variations and
Extrema..........
261
6.4.3
Example: Finsler-Lagrangian Field Theory
.......... 264
6.5
Symplectic Manifolds
........................... 265
6.5.1
Symplectic Algebra
........................ 265
6.5.2
Symplectic Geometry
...................... 266
6.6
Autonomous Hamiltonian Dynamics
.................. 268
6.6.1
Basics of Hamiltonian Dynamics
................ 268
6.6.2
Real
1
DOF
Hamiltonian Dynamics
............... 270
6.6.3
Complex
1
DOF
Hamiltonian Dynamics
............ 276
6.6.4
Library of Basic Hamiltonian Systems
............. 279
6.6.5
Hamilton-Poisson Mechanics
.................. 286
6.6.6
Completely
Integrable
Hamiltonian Systems
.......... 288
6.6.7
Momentum Map and Symplectic Reduction
.......... 294
6.7
Example: Human versus Humanoid-Robot
Biodynamics
....... 296
6.7.1
Idealistic Configuration Manifold of Humanoid-Robot Motion
296
6.7.2
Realistic Configuration Manifold of Human Motion
...... 296
6.7.3
Generalized Lagrangian and Hamiltonian
Biodynamics
.... 297
7
Path-Integral and Quantum Fields
301
7.1
Soft Introduction to Quantum Field Theory
.............. 301
7.1.1
Basics of Non-Relativistic Quantum Mechanics
........ 301
7.1.2
Transition to Quantum Fields
.................. 313
7.2
Feynman s Sum-Over Histories
..................... 322
7.2.1
Intuition Behind a Path Integral
................ 322
7.2.2
Basic Path-Integral Calculations
................ 332
7.2.3
Path-Integral Quantization
................... 338
7.2.4
Statistical Mechanics via Path Integrals
............ 345
7.2.5
Path Integrals and Green s Functions
............. 347
7.2.6
Monte Carlo Simulation of the Path Integral
......... 352
7.2.7
Feynman s Action-Amplitude Formalism in QFT
....... 357
7.2.8
Path-Integral TQFT
....................... 363
7.2.9
Non-Abelian Gauge Theories
.................. 368
III Advanced Geometrical Methods
381
8 Kahler
and
Conformai
Geometry
383
8.1
Complex and
Kahler
Manifolds
..................... 383
8.1.1
Hermitian and
Kahler
Metrics
................. 385
8.1.2 Kähler-Ricci
Flow
........................ 389
8.1.3 Kahler Orbifolds......................... 392
8.1.4 Kähler-Ricci
Flow on
Kähler-Einstein
Orbifolds
....... 394
Contents
8.1.5
Induced
Evolution
Equations
.................. 395
8.1.6 Calabi-Yau
Manifolds
...................... 395
8.2
Conformai Killing-Riemannian
Geometry
............... 396
8.2.1
Conformai
Killing Vector-Fields and Forms
.......... 396
8.2.2
Conformai
Killing Tensors and Laplacian Symmetry
..... 397
8.3
Gauges and
Monopoles
.......................... 399
8.3.1
Gauge Theories
.......................... 399
8.3.2
Monopoles
............................ 406
8.4
Monge-Ampère
Equations
........................ 413
8.4.1
Monge-Ampère
Equations and Hitchin Pairs
......... 415
8.4.2
The
Ђ-
Operator
......................... 419
9
Geometry and Topology of Fibre Bundles
425
9.1
Fibre Bundles
............................... 425
9.1.1
Definition of a Fibre Bundle
................... 425
9.2
Vector and
Affine
Bundles
........................ 429
9.2.1
Vertical Tangent and Cotangent Bundles
........... 434
9.2.2
Distributions and Foliations on Manifolds
........... 436
9.2.3
Principal Bundles
........................ 437
9.3
Multivector-Fields and Tangent-Valued Forms
............ 439
9.4
Semi-Riemannian Dynamics
....................... 446
9.4.1
Vector-Fields and Connections
................. 446
9.4.2
Hamiltonian Structures on a Tangent Bundle
......... 448
9.5
A -Theory
................................ 451
9.5.1
Topological K-Theory
..................... 451
9.5.2
Algebraic A -Theory
....................... 452
9.5.3
Chern Classes and Chern Character
.............. 453
9.5.4
Atiyah s View on AT-Theory
.................. 456
9.5.5
Atiyah-Singer Index Theorem
.................. 458
9.5.6
The Infinite-Order Case
..................... 460
10
Jet Geometry and
Non-
Autonomous Dynamics
463
10.1
Definition of a 1-Jet Manifold
...................... 463
10.2
Connections as 1-Jet Fields
....................... 467
10.3
Definition of a 2-Jet Manifold
...................... 472
10.4
Jets and Non-Autonomous Dynamics
................. 475
10.4.1
Geodesies
............................. 479
10.4.2
Quadratic Dynamical Equations
................ 480
10.4.3
Equation of Free-Motion
.................... 481
10.4.4
Quadratic Lagrangian and Newtonian Systems
........ 481
10.4.5
Jacobi Fields
........................... 482
10.4.6
Constraints
............................ 484
10.4.7
Time-Dependent Lagrangian Dynamics
............ 487
Contents xi
10.4.8
Time-Dependent Hamiltonian
Dynamics............ 488
10.4.9
Time-Dependent Constraints
.................. 491
10.4.10 Lagrangian
Constraints
..................... 493
10.4.11
Quadratic Degenerate
Lagrangian Systems .......... 494
10.4.12
Time-Dependent
Integrable
Hamiltonian Systems
...... 497
10.4.13
Time-Dependent Action-Angle Coordinates
.......... 499
10.4.14
Lyapunov Stability
........................ 500
10.4.15
First-Order Dynamical Equations
............... 501
10.4.16
Lyapunov Tensor and Stability
................. 502
10.4.17
Multi-Time Rheonomic Dynamics
............... 506
10.5
Jets and Action Principles
........................ 508
10.6
Jets and Lagrangian Field Theory
................... 512
10.6.1
Lagrangian Conservation Laws
................. 516
10.6.2
General Covariance Condition
.................. 520
10.7
Jets and Hamiltonian Field Theory
................... 523
10.7.1
Covariant Hamiltonian Field Systems
............. 524
10.7.2
Associated Lagrangian and Hamiltonian Systems
....... 526
10.7.3
Evolution Operator
........................ 528
10.7.4
Quadratic Degenerate Systems
................. 532
11
Advanced Path-Integral Methods
535
11.1
Sum over Geometries and Topologies
.................. 535
11.1.1
Simplicial Quantum Geometry
................. 536
11.1.2
Discrete Gravitational Path Integrals
.............. 537
11.1.3
Regge
Calculus
.......................... 539
11.1.4
Lorentzian Path Integral
..................... 541
11.1.5
Non-Perturbative Quantum Gravity
.............. 545
11.2
Cerebellum as an oo—Dimensional Recurrent Neural Network
.... 565
11.2.1
Spinal Reflex Control of
Biodynamics
............. 567
11.2.2
Local Muscle-Joint Mechanics
................. 567
11.2.3
Cerebellum: Adaptive Path-Integral Comparator
....... 570
11.3
Quantum Geometrical Psychodynamics
................ 574
11.3.1
Classical versus Quantum Probability
............. 577
11.3.2
The Life Space Foam
....................... 581
11.3.3
Geometric Chaos and Topological Phase Transitions
..... 587
11.3.4
Joint Action of Several Agents
................. 592
11.3.5
Chaos and Bernstein-Brooks Adaptation
........... 594
11.4
Geometrical Model for Crowd Behavioral Dynamics
......... 596
11.4.1
Cognition and Crowd Behavior
................. 597
11.4.2
Generic Three-Step Crowd Behavioral Dynamics
....... 598
11.4.3
Formal Individual, Aggregate and Crowd dynamics
...... 600
11.4.4
Crowd Entropy, Chaos and Phase Transitions
......... 610
xii Contents
11.4.5
Crowd
Ricci
Flow
and Perelinan
Entropy
........... 610
11.4.6
Chaotic Inter-Phase in Crowd Dynamics
............ 613
11.4.7
Crowd Phase Transitions
.................... 614
Index
617
|
| any_adam_object | 1 |
| author | Ivancevic, Vladimir G. Ivancevic, Tijana T. |
| author_facet | Ivancevic, Vladimir G. Ivancevic, Tijana T. |
| author_role | aut aut |
| author_sort | Ivancevic, Vladimir G. |
| author_variant | v g i vg vgi t t i tt tti |
| building | Verbundindex |
| bvnumber | BV036446417 |
| callnumber-first | Q - Science |
| callnumber-label | QA300 |
| callnumber-raw | QA300 |
| callnumber-search | QA300 |
| callnumber-sort | QA 3300 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 950 |
| ctrlnum | (OCoLC)705500810 (DE-599)BVBBV036446417 |
| dewey-full | 516 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516 |
| dewey-search | 516 |
| dewey-sort | 3516 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV036446417 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:39:36Z |
| institution | BVB |
| isbn | 9781607417699 |
| language | English |
| lccn | 2009018844 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020318667 |
| oclc_num | 705500810 |
| open_access_boolean | |
| owner | DE-703 DE-19 DE-BY-UBM |
| owner_facet | DE-703 DE-19 DE-BY-UBM |
| physical | XIII, 630 S. Ill., graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Nova Science Publ. |
| record_format | marc |
| spelling | Ivancevic, Vladimir G. Verfasser aut Handbook of geometrical methods for scientists and engineers Vladimir G. Ivancevic and Tijana Ivancevic New York Nova Science Publ. 2010 XIII, 630 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Geometric analysis Geometrical models Ingenieurwissenschaften (DE-588)4137304-2 gnd rswk-swf Geometrische Methode (DE-588)4156715-8 gnd rswk-swf Ingenieurwissenschaften (DE-588)4137304-2 s Geometrische Methode (DE-588)4156715-8 s DE-604 Ivancevic, Tijana T. Verfasser aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020318667&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ivancevic, Vladimir G. Ivancevic, Tijana T. Handbook of geometrical methods for scientists and engineers Geometric analysis Geometrical models Ingenieurwissenschaften (DE-588)4137304-2 gnd Geometrische Methode (DE-588)4156715-8 gnd |
| subject_GND | (DE-588)4137304-2 (DE-588)4156715-8 |
| title | Handbook of geometrical methods for scientists and engineers |
| title_auth | Handbook of geometrical methods for scientists and engineers |
| title_exact_search | Handbook of geometrical methods for scientists and engineers |
| title_full | Handbook of geometrical methods for scientists and engineers Vladimir G. Ivancevic and Tijana Ivancevic |
| title_fullStr | Handbook of geometrical methods for scientists and engineers Vladimir G. Ivancevic and Tijana Ivancevic |
| title_full_unstemmed | Handbook of geometrical methods for scientists and engineers Vladimir G. Ivancevic and Tijana Ivancevic |
| title_short | Handbook of geometrical methods for scientists and engineers |
| title_sort | handbook of geometrical methods for scientists and engineers |
| topic | Geometric analysis Geometrical models Ingenieurwissenschaften (DE-588)4137304-2 gnd Geometrische Methode (DE-588)4156715-8 gnd |
| topic_facet | Geometric analysis Geometrical models Ingenieurwissenschaften Geometrische Methode |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020318667&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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