Cargando…

Differential geometry and lie groups for physicists /

Covering subjects including manifolds, tensor fields, spinors, and differential forms, this textbook introduces geometrical topics useful in modern theoretical physics and mathematics. It develops understanding through over 1000 short exercises, and is suitable for advanced undergraduate or graduate...

Descripción completa

Detalles Bibliográficos
Clasificación:	Libro Electrónico
Autor principal:	Fecko, Marián
Formato:	Electrónico eBook
Idioma:	Inglés
Publicado:	Cambridge ; New York : Cambridge University Press, 2006.
Temas:	Geometry, Differential. Lie groups. Mathematical physics. Géométrie différentielle. Groupes de Lie. Physique mathématique. SCIENCE > Physics > Mathematical & Computational. Geometry, Differential Lie groups Mathematical physics
Acceso en línea:	Texto completo

Tabla de Contenidos:

Cover
Title
Copyright
Contents
Preface
Introduction
Chapter 1 The concept of a manifold
1.1 Topology and continuous maps
1.2 Classes of smoothness of maps of Cartesian spaces
1.3 Smooth structure, smooth manifold
1.4 Smooth maps of manifolds
1.5 A technical description of smooth surfaces in R
Summary of Chapter 1
Chapter 2 Vector and tensor fields
2.1 Curves and functions on M
2.2 Tangent space, vectors and vector fields
2.3 Integral curves of a vector field
2.4 Linear algebra of tensors (multilinear algebra)
2.5 Tensor fields on M
2.6 Metric tensor on a manifold
Summary of Chapter 2
Chapter 3 Mappings of tensors induced by mappings of manifolds
3.1 Mappings of tensors and tensor fields
3.2 Induced metric tensor
Summary of Chapter 3
Chapter 4 Lie derivative
4.1 Local flow of a vector field
4.2 Lie transport and Lie derivative
4.3 Properties of the Lie derivative
4.4 Exponent of the Lie derivative
4.5 Geometrical interpretation of the commutator [V, W], non-holonomic frames
4.6 Isometries and conformal transformations, Killing equations
Summary of Chapter 4
Chapter 5 Exterior algebra
5.1 Motivation: volumes of parallelepipeds
5.2 p-forms and exterior product
5.3 Exterior algebra Lambda L
5.4 Interior product iv
5.5 Orientation in L
5.6 Determinant and generalized Kronecker symbols
5.7 The metric volume form
5.8 Hodge (duality) operator
Summary of Chapter 5
Chapter 6 Differential calculus of forms
6.1 Forms on a manifold
6.2 Exterior derivative
6.3 Orientability, Hodge operator and volume form on M
6.4 V-valued forms
Summary of Chapter 6
Chapter 7 Integral calculus of forms
7.1 Quantities under the integral sign regarded as differential forms
7.2 Euclidean simplices and chains
7.3 Simplices and chains on a manifold
7.4 Integral of a form over a chain on a manifold
7.5 Stokes' theorem
7.6 Integral over a domain on an orientable manifold
7.7 Integral over a domain on an orientable Riemannian manifold
7.8 Integral and maps of manifolds
Summary of Chapter 7
Chapter 8 Particular cases and applications of Stokes' theorem
8.1 Elementary situations
8.2 Divergence of a vector field and Gauss' theorem
8.3 Codifferential and Laplace-deRham operator
8.4 Green identities
8.5 Vector analysis in E
8.6 Functions of complex variables
Summary of Chapter 8
Chapter 9 Poincaré lemma and cohomologies
9.1 Simple examples of closed non-exact forms
9.2 Construction of a potential on contractible manifolds
9.3 Cohomologies and deRham complex
Summary of Chapter 9
Chapter 10 Lie groups: basic facts
10.1 Automorphisms of various structures and groups
10.2 Lie groups: basic concepts
Summary of Chapter 10
Chapter 11 Differential geometry on Lie groups
11.1 Left-invariant tensor fields on a Lie group
11.2 Lie algebra G of a group G
11.3 One-parameter subgroups
11.4 Exponential map
11.5 Derived homomorphism of Lie algebras
11.6 Invariant integral on G
11.7 Matrix Lie groups: enjoy simplifications
Summary of Chapter 11
Chapter 12 Representations of Lie groups and Lie algebras
12.1 Basic concept.

Differential geometry and lie groups for physicists /

Ejemplares similares