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El. knyga: Introduction to Lorentz Geometry: Curves and Surfaces [Taylor & Francis e-book]

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  • Formatas: 340 pages, 94 Illustrations, black and white
  • Išleidimo metai: 30-Dec-2020
  • Leidėjas: Chapman & Hall/CRC
  • ISBN-13: 9781003031574
Kitos knygos pagal šią temą:
  • Taylor & Francis e-book
  • Kaina: 170,80 €*
  • * this price gives unlimited concurrent access for unlimited time
  • Standartinė kaina: 244,00 €
  • Sutaupote 30%
Introduction to Lorentz Geometry: Curves and SurfacesDidesnis paveikslėlis
  • Formatas: 340 pages, 94 Illustrations, black and white
  • Išleidimo metai: 30-Dec-2020
  • Leidėjas: Chapman & Hall/CRC
  • ISBN-13: 9781003031574
Kitos knygos pagal šią temą:
Taylor & Francis e-booksMore info about Taylor & Francis e-books
Partnerių interneto svetainė: https://www.taylorfrancis.com/books/9781003031574
"Lorentz Geometry is a very important intersection between Mathematics and Physics, being the mathematical language of General Relativity. Learning this type of geometry is the first step in properly understanding questions regarding the structure of theuniverse, such as: What is the shape of the universe? What is a spacetime? What is the relation between gravity and curvature? Why exactly is time treated in a different manner than other spatial dimensions? Introduction to Lorentz Geometry: Curves and Surfaces intends to provide the reader with the minimum mathematical background needed to pursue these very interesting questions, by presenting the classical theory of curves and surfaces in both Euclidean and Lorentzian ambient spaces simultaneously. Features Over 300 exercises Suitable for senior undergraduates and graduates studying Mathematics and Physics Written in an accessible style without loss of precision or mathematical rigour Solution manual available on www.routledge.com/9780367468644"--

Lorentz Geometry is a very important intersection between Mathematics and Physics, being the mathematical language of General Relativity.

Learning this type of geometry is the first step in properly understanding questions regarding the structure of the universe, such as: What is the shape of the universe? What is a spacetime? What is the relation between gravity and curvature? Why exactly is time treated in a different manner than other spatial dimensions?

Introduction to Lorentz Geometry: Curves and Surfaces

intends to provide the reader with the minimum mathematical background needed to pursue these very interesting questions, by presenting the classical theory of curves and surfaces in both Euclidean and Lorentzian ambient spaces simultaneously.

Features

  • Over 300 exercises

  • Suitable for senior undergraduates and graduates studying Mathematics and Physics

  • Written in an accessible style without loss of precision or mathematical rigour

  • Solution manual available on www.routledge.com/9780367468644

Preface of the Portuguese Version vii
Preface ix
Chapter 1 Welcome to Lorentz-Minkowski Space
1(62)
1.1 PSEUDO-EUCLIDEAN SPACES
2(2)
1.1.1 Defining Rnv
2(1)
1.1.2 The causal character of a vector in Rnv
3(1)
1.2 SUBSPACES OF Rnv
4(15)
1.3 CONTEXTUALIZATION IN SPECIAL RELATIVITY
19(10)
1.4 ISOMETRIES IN Rnv
29(14)
1.5 INVESTIGATING O1a (2, R) AND O1 (3, R)
43(10)
1.5.1 The group O1 (2, R) in detail
43(1)
1.5.2 The group O1(3, R) in (a little less) detail
44(3)
1.5.3 Rotations and boosts
47(6)
1.6 CROSS PRODUCT IN Rnv
53(10)
1.6.1 Completing the toolbox
57(6)
Chapter 2 Local Theory of Curves
63(66)
2.1 PARAMETRIZED CURVES IN Rnv
64(13)
2.2 CURVES IN THE PLANE
77(20)
2.3 CURVES IN SPACE
97(32)
2.3.1 The Frenet-Serret trihedron
98(8)
2.3.2 Geometric effects of curvature and torsion
106(12)
2.3.3 Curves with degenerate osculating plane
118(11)
Chapter 3 Surfaces in Space
129(130)
3.1 BASIC TOPOLOGY OF SURFACES
130(25)
3.2 CAUSAL TYPE OF SURFACES, FIRST FUNDAMENTAL FORM
155(23)
3.2.1 Isometries between surfaces
169(9)
3.3 SECOND FUNDAMENTAL FORM AND CURVATURES
178(12)
3.4 THE DIAGONALIZATION PROBLEM
190(17)
3.4.1 Interpretations for curvatures
194(13)
3.5 CURVES IN A SURFACE
207(12)
3.6 GEODESICS, VARIATIONAL METHODS AND ENERGY
219(31)
3.6.1 Darboux-Ribaucour frame
222(6)
3.6.2 Christoffel symbols
228(4)
3.6.3 Critical points of the energy functional
232(18)
3.7 THE FUNDAMENTAL THEOREM OF SURFACES
250(9)
3.7.1 The compatibility equations
250(9)
Chapter 4 Abstract Surfaces and Further Topics
259(66)
4.1 PSEUDO-RIEMANNIAN METRICS
260(20)
4.2 RIEMANN'S CLASSIFICATION THEOREM
280(6)
4.3 SPLIT-COMPLEX NUMBERS AND CRITICAL SURFACES
286(28)
4.3.1 A brief introduction to split-complex numbers
286(12)
4.3.2 Bonnet rotations
298(6)
4.3.3 Enneper-Weierstrass representation formulas
304(10)
4.4 DIGRESSION: COMPLETENESS AND CAUSALITY
314(11)
Appendix Some Results from Differential Calculus 325(6)
Bibliography 331(4)
Index 335
Ivo Terek Couto, born in Sćo Paulo, graduated with a B.Sc. and a M.Sc. in Mathematics from the Institute of Mathematics and Statistics of the University of Sćo Paulo (IMEUSP). Hes currently pursuing PhD at The Ohio State University in Columbus, Ohio. His study and research interests lie mainly in Differential Geometry and its applications in other areas of Mathematics and Physics, particularly in General Relativity and Classical Mechanics.

Alexandre Lymberopoulos, born in Sćo Paulo, has a PhD in Mathematics from the Institute of Mathematics and Statistics of the University of Sćo Paulo (IMEUSP). He has taught in several higher education institutes in Sćo Paulo and returned to IMEUSP as an Assistant Professor in 2011. His main research interest is in Differential Geometry, particularly in immersions and its interactions with other branches of Science.
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