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Viser: Modern Geometry — Methods and Applications

Modern Geometry — Methods and Applications, 2. udgave

Modern Geometry — Methods and Applications

(2022)
Sprog: Engelsk
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  • 2. Udgave
  • Udgiver: (April 2022)
  • ISBN: 9780387976631
1 Geometry in Regions of a Space. Basic Concepts.- §1. Co-ordinate systems.-
1.1. Cartesian co-ordinates in a space.-
1.2. Co-ordinate changes.- §2. Euclidean space.-
2.1. Curves in Euclidean space.-
2.2. Quadratic forms and vectors.- §3. Riemannian and pseudo-Riemannian spaces.-
3.1. Riemannian metrics.-
3.2. The Minkowski metric.- §4. The simplest groups of transformations of Euclidean space.-
4.1. Groups of transformations of a region.-
4.2. Transformations of the plane.-
4.3. The isometries of 3-dimensional Euclidean space.-
4.4. Further examples of transformation groups.-
4.5. Exercises.- §5. The Serret--Frenet formulae.-
5.1. Curvature of curves in the Euclidean plane.-
5.2. Curves in Euclidean 3-space. Curvature and torsion.-
5.3. Orthogonal transformations depending on a parameter.-
5.4. Exercises.- §6. Pseudo-Euclidean spaces.-
6.1. The simplest concepts of the special theory of relativity.-
6.2. Lorentz transformations.-
6.3. Exercises.- 2 The Theory of Surfaces.- §7. Geometry on a surface in space.-
7.1. Co-ordinates on a surface.-
7.2. Tangent planes.-
7.3. The metric on a surface in Euclidean space.-
7.4. Surface area.-
7.5. Exercises.- §8. The second fundamental form.-
8.1. Curvature of curves on a surface in Euclidean space.-
8.2. Invariants of a pair of quadratic forms.-
8.3. Properties of the second fundamental form.-
8.4. Exercises.- §9. The metric on the sphere.- §10. Space-like surfaces in pseudo-Euclidean space.-
10.1. The pseudo-sphere.-
10.2. Curvature of space-like curves in $$\mathbb{R}_1^3$$.- §11. The language of complex numbers in geometry.-
11.1. Complex and real co-ordinates.-
11.2. The Hermitian scalar product.-
11.3. Examples of complex transformation groups.- §12. Analytic functions.-
12.1. Complex notation for the element of length, and for the differential of a function.-
12.2. Complex co-ordinate changes.-
12.3. Surfaces in complex space.- §13. The conformal form of the metric on a surface.-
13.1. Isothermal co-ordinates. Gaussian curvature in terms of conformal co-ordinates.-
13.2. Conformal form of the metrics on the sphere and the Lobachevskian plane.-
13.3. Surfaces of constant curvature.-
13.4. Exercises.- §14. Transformation groups as surfaces in N-dimensional space.-
14.1. Co-ordinates in a neighbourhood of the identity.-
14.2. The exponential function with matrix argument.-
14.3. The quaternions.-
14.4. Exercises.- §15. Conformal transformations of Euclidean and pseudo-Euclidean spaces of several dimensions.- 3 Tensors: The Algebraic Theory.- §16. Examples of tensors.- §17. The general definition of a tensor.-
17.1. The transformation rule for the components of a tensor of arbitrary rank.-
17.2. Algebraic operations on tensors.-
17.3. Exercises.- §18. Tensors of type (0, k).-
18.1. Differential notation for tensors with lower indices only.-
18.2. Skew-symmetric tensors of type (0, k).-
18.3. The exterior product of differential forms. The exterior algebra.-
18.4. Skew-symmetric tensors of type (k, 0) (polyvectors). Integrals with respect to anti-commuting variables.-
18.5. Exercises.- §19. Tensors in Riemannian and pseudo-Riemannian spaces.-
19.1. Raising and lowering indices.-
19.2. The eigenvalues of a quadratic form.-
19.3. The operator '.-
19.4. Tensors in Euclidean space.-
19.5. Exercises.- §20. The crystallographic groups and the finite subgroups of the rotation group of Euclidean 3-space. Examples of invariant tensors.- §21. Rank 2 tensors in pseudo-Euclidean space, and their eigenvalues.-
21.1. Skew-symmetric tensors. The invariants of an electromagnetic field.-
21.2. Symmetric tensors and their eigenvalues. The energy-momentum tensor of an electromagnetic field.- §22. The behaviour of tensors under mappings.-
22.1. The general operation of restriction of tensors with lower indices.-
22.2. Mappings of tangent spaces.- §23. Vector fields.-
23.1. One-parameter groups of diffeomorphisms.-
23.2. The exponential function of a vector field.-
23.3. The Lie derivative.-
23.4. Exercises.- §24. Lie algebras.-
24.1. Lie algebras and vector fields.-
24.2. The fundamental matrix Lie algebras.-
24.3. Linear vector fields.-
24.4. Left-invariant fields defined on transformation groups.-
24.5. Invariant metrics on a transformation group.-
24.6. The classification of the 3-dimensional Lie algebras.-
24.7. The Lie algebras of the conformal groups.-
24.8. Exercises.- 4 The Differential Calculus of Tensors.- §25. The differential calculus of skew-symmetric tensors.-
25.1. The gradient of a skew-symmetric tensor.-
25.2. The exterior derivative of a form.-
25.3. Exercises.- §26. Skew-symmetric tensors and the theory of integration.-
26.1. Integration of differential forms.-
26.2. Examples of integrals of differential forms.-
26.3. The general Stokes formula. Examples.-
26.4. Proof of the general Stokes formula for the cube.-
26.5. Exercises.- §27. Differential forms on complex spaces.-
27.1. The operators d? and d'.-
27.2. Kählerian metrics. The curvature form.- §28. Covariant differentiation.-
28.1. Euclidean connexions.-
28.2. Covariant differentiation of tensors of arbitrary rank.- §29. Covariant differentiation and the metric.-
29.1. Parallel transport of vector fields.-
29.2. Geodesics.-
29.3. Connexions compatible with the metric.-
29.4. Connexions compatible with a complex structure (Hermitian metric).-
29.5. Exercises.- §30. The curvature tensor.-
30.1. The general curvature tensor.-
30.2. The symmetries of the curvature tensor. The curvature tensor defined by the metric.-
30.3. Examples: The curvature tensor in spaces of dimensions 2 and 3; the curvature tensor of transformation groups.-
30.4. The Peterson--Codazzi equations. Surfaces of constant negative curvature, and the "sine--Gordon" equation.-
30.5. Exercises.- 5 The Elements of the Calculus of Variations.- §31. One-dimensional variational problems.-
31.1. The Euler--Lagrange equations.-
31.2. Basic examples of functional.- §32. Conservation laws.-
32.1. Groups of transformations preserving a given variational problem.-
32.2. Examples. Applications of the conservation laws.- §33. Hamiltonian formalism.-
33.1. Legendre''s transformation.-
33.2. Moving co-ordinate frames.-
33.3. The principles of Maupertuis and Fermat.-
33.4. Exercises.- §34. The geometrical theory of phase space.-
34.1. Gradient systems.-
34.2. The Poisson bracket.-
34.3. Canonical transformations.-
34.4. Exercises.- §35. Lagrange surfaces.-
35.1. Bundles of trajectories and the Hamilton--Jacobi equation.-
35.2. Hamiltonians which are first-order homogeneous with respect to the momentum.- §36. The second variation for the equation of the geodesics.-
36.1. The formula for the second variation.-
36.2. Conjugate points and the minimality condition.- 6 The Calculus of Variations in Several Dimensions. Fields and Their Geometric Invariants.- §37. The simplest higher-dimensional variational problems.-
37.1. The Euler--Lagrange equations.-
37.2. The energy-momentum tensor.-
37.3. The equations of an electromagnetic field.-
37.4. The equations of a gravitational field.-
37.5. Soap films.-
37.6. Equilibrium equation for a thin plate.-
37.7. Exercises.- §38. Examples of Lagrangians.- §39. The simplest concepts of the general theory of relativity.- §40. The spinor representations of the groups SO(3) and O(3, 1). Dirac''s equation and its properties.-
40.1. Automorphisms of matrix algebras.-
40.2. The spinor representation of the group SO(3).-
40.3. The spinor representation of the Lorentz group.-
40.4. Dirac''s equation.-
40.5. Dirac''s equation in an electromagnetic field. The operation of charge conjugation.- §41. Covariant differentiation of fields with arbitrary symmetry.-
41.1. Gauge transformations. Gauge-invariant Lagrangians.-
41.2. The curvature form.-
41.3. Basic examples.- §42. Examples of gauge-invariant functionals. Maxwell''s equations and the Yang--Mills equation. Functionals with identically zero variational derivative (characteristic classes).
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