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Differential geometry and lie groups for physicists / Marián Fecko.

By: Fecko, MariánMaterial type: TextTextPublication details: Cambridge ; New York : Cambridge University Press, 2006. Description: 1 online resource (xv, 697 pages) : illustrationsContent type: text Media type: computer Carrier type: online resourceISBN: 9780511648656; 0511648650; 0511245211; 9780511245213; 0511244460; 9780511244469; 0511242964; 9780511242960; 1107164052; 9781107164055; 0511567677; 9780511567674; 0511755597; 9780511755590; 9780521187961; 0521187966Subject(s): Geometry, Differential | Lie groups | Mathematical physics | Géométrie différentielle | Lie, Groupes de | Physique mathématique | SCIENCE -- Physics -- Mathematical & Computational | Geometry, Differential | Lie groups | Mathematical physicsGenre/Form: Electronic books. Additional physical formats: Print version:: Differential geometry and lie groups for physicists.DDC classification: 530.15636 LOC classification: QC20.7.D52 | F43 2006ebOnline resources: Click here to access online
Contents:
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.
Summary: 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 courses in physics, mathematics and engineering.
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Includes bibliographical references (pages 685-686)-and indexes.

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.

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 courses in physics, mathematics and engineering.

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