This book is composed of two parts: Part I （Chaps. I through 3） is an introduction to tensors and their physical applications， and Part II （Chaps. 4 through 6） introduces group theory and intertwines it with the earlier material. Both parts are written at the advanced-undergraduate/beginning graduate level， although in the course of Part II the sophistication level rises somewhat. Though the two parts differ somewhat in flavor，l have aimed in both to fill a （perceived） gap in the literaiure by connecting
the component formalisms prevalent in physics calculations to the abstract but more conceptual formulations found in the math literature. My firm beliefis that we need to see tensors and groups in coordinates to get a sense of how they work， but also need an abstract formulation to understand their essential nature and organize our thinking about them.

Part I Linear Algebra and Tensors
I A Quicklntroduction to Tensors
2 VectorSpaces
2.1 Definition and Examples
2.2 Span，Linearlndependence，and Bases
2.3 Components
2.4 LinearOperators
2.5 DuaISpaces
2.6 Non-degenerate Hermitian Forms
2.7 Non-degenerate Hermitian Forms and Dual Spaces
2.8 Problems
3 Tensors
3.1 Definition and Examples
3.2 ChangeofBasis
3.3 Active and Passive TransformationsPart I Linear Algebra and Tensors
I A Quicklntroduction to Tensors
2 VectorSpaces
2.1 Definition and Examples
2.2 Span，Linearlndependence，and Bases
2.3 Components
2.4 LinearOperators
2.5 DuaISpaces
2.6 Non-degenerate Hermitian Forms
2.7 Non-degenerate Hermitian Forms and Dual Spaces
2.8 Problems
3 Tensors
3.1 Definition and Examples
3.2 ChangeofBasis
3.3 Active and Passive Transformations
3.4 The Tensor Product-Definition and Properties
3.5 Tensor Products of V and V*
3.6 Applications ofthe Tensor Product in Classical Physics
3.7 Applications of the Tensor Product in Quantum Physics
3.8 Symmetric Tensors
3.9 Antisymmetric Tensors
3.10 ProblemsPartll GroupTheory
4 Groups， Lie Groups，and Lie Algebras
4.1 Groups-Definition and Examples
4.2 The Groups ofClassical and Quantum Physics
4.3 Homomorphismandlsomorphism
4.4 From Lie Groups to Lie Algebras
4.5 Lie Algebras-Definition，Properties，and Examples
4.6 The Lie Algebras ofClassical and Quantum Physics
4.7 AbstractLieAlgebras
4.8 Homomorphism andlsomorphism Revisited
4.9 Problems
5 Basic Representation Theory
5.1 Representations: Definitions and Basic Examples
5.2 FurtherExamples
5.3 TensorProduet Representations
5.4 Symmetric and Antisymmetric Tensor Product Representations
5.5 Equivalence ofRepresentations
5.6 Direct Sums andlrreducibility
5.7 Moreonlrreducibility
5.8 Thelrreducible Representations ofsu（2），SU（2） and S0（3）
5.9 ReaIRepresentations andComplexifications
5.10 The Irreducible Representations of st（2， C）nk， SL（2， C） andS0（3，1）o
5.11 Irreducibility and the Representations of 0（3， 1） and Its Double Covers
5.12 Problems
6 The Wigner-Eckart Theorem and Other Applications
6.1 Tensor Operators， Spherical Tensors and Representation Operators
6.2 Selection Rules and the Wigner-Eckart Theorem
6.3 Gamma Matrices and Dirac Bilinears
6.4 Problems
AppendixComplexifications of Real Lie Algebras and the Tensor
Product Decomposition ofsl（2，C）rt Representations
A.1 Direct Sums and Complexifications ofLie Algebras
A.2 Representations of Complexified Lie Algebras and the Tensor
Product Decomposition ofst（2，C）R Representations
References
Index

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