This textbook serves as an introduction to groups, rings, fields, vector and tensor spaces, algebras, topological spaces, differentiable manifolds and Lie groups --- mathematical structures which are foundational to modern theoretical physics. It is aimed primarily at undergraduate students in physics and mathematics with no previous background in these topics. Applications to physics --- such as the metric tensor of special relativity, the symplectic structures associated with Hamilton's equations and the Generalized Stokes's Theorem --- appear at appropriate places in the text. Worked examples, end-of-chapter problems (many with hints and some with answers) and guides to further reading make this an excellent book for self-study. Upon completing this book the reader will be well prepared to delve more deeply into advanced texts and specialized monographs in theoretical physics or mathematics.
Author(s): Steven P. Starkovich
Edition: 1
Publisher: Springer
Year: 2021
Language: English
Pages: 280
Tags: mathematical, structures, physics
Preface
Acknowledgements
References
Contents
Glossary of Symbols
Number Systems
Set Theory and General Topology
Relations and Maps
Basic Algebraic Structures and Groups
Rings
Vector Spaces and Algebras
Differentiable Manifolds
1 Sets, Relations and Maps
1.1 The Algebra of Sets
1.1.1 Set Inclusion, Subsets and Set Equality
1.1.2 The Algebra of Sets: Union, Intersection and Complement
1.2 Relations Within Sets
1.2.1 Order Relations
1.2.2 Equivalence Relations and Quotient Sets
1.3 Mappings Between Sets
1.3.1 Injective, Surjective and Bijective Maps
1.3.2 Continuous Maps
1.3.3 Countable and Uncountable Sets
1.4 Cartesian Products of Sets and Projection Maps
1.5 A Universal Construction for Quotient Sets
References
2 Groups
2.1 Groupoids, Semigroups and Monoids
2.2 Groups
2.3 Some Noteworthy Groups
2.3.1 Cyclic Groups
2.3.2 Symmetric Groups
2.3.3 Alternating Groups
2.3.4 Dihedral Groups
2.3.5 Continuous Groups
2.4 Morphisms, and a Glance at Algebraic Topology and Categories
2.5 Subgroups
2.6 Classes and Invariant Subgroups
2.7 Cosets and Quotient Groups
2.8 Group Products
References
3 Rings and Fields
3.1 Rings
3.1.1 Ring Axioms
3.1.2 Ring Morphisms
3.2 Subrings, Ideals and Quotient Rings
3.3 Product Rings
3.4 Fields
3.4.1 Completeness and Order
3.4.2 The Complex Field mathbbC and Hamilton's Search for Number Triplets
3.4.3 The Quaternion Skew Field mathbbH
References
4 Vector and Tensor Spaces
4.1 Modules and Vector Spaces
4.2 Linear Independence, Basis Vectors and Norms
4.2.1 Linear Independence
4.2.2 Basis and Dimension
4.2.3 Norms and Distance Functions on Vector Spaces
4.3 Inner Product Spaces
4.3.1 Inner Products in mathbbR2 Over mathbbR
4.3.2 Inner Products in Coordinate Spaces
4.3.3 Inner Products on Complex and Real Function Spaces—Sesquilinear and Bilinear Maps
4.4 Orthogonality, Normalization and Complete Sets of Vectors
4.4.1 Gram-Schmidt Orthogonalization—Coordinate Space
4.4.2 Orthonormalization in Function Spaces
4.4.3 Gram-Schmidt Orthogonalization—Function Space
4.4.4 Hilbert Spaces
4.5 Subspaces, Sums, and Products of Vector Spaces
4.5.1 Vector Subspaces
4.5.2 Unions, Sums and Direct Sums of Vector Spaces
4.5.3 Tensors and Tensor Spaces
4.5.4 Metric and Associated Tensors
4.6 Cosets and Quotient Spaces
References
5 Algebras and Operators
5.1 Algebras
5.2 Structure Constants
5.3 Lie and Poisson Algebras
5.3.1 Lie Algebras
5.3.2 Poisson Algebras
5.4 Subalgebras, Quotients and Sums of Algebras
5.4.1 Subalgebras, Algebra Ideals and Quotients
5.4.2 Direct Sums of Algebras
5.5 Associative Operator Algebras on Inner Product Spaces
5.5.1 Definitions, Notations and Basic Operations with Matrices
5.5.2 Linear Transformations, Images and Null Spaces
5.5.3 Eigenvectors, Similarity Transformations and Diagonalization of Matrices in Real Spaces
5.6 Hermitian Operators
5.7 Unitary, Orthogonal and Hermitian Transformations
5.8 Functions of Operators
References
6 Fundamental Concepts of General Topology
6.1 General Topology in a Geometric Context
6.2 Foundations of General Topology
6.3 Bases and Generators of a Topology
6.4 Separation and Connectedness
6.4.1 Separated and Connected Sets and Spaces
6.4.2 Separation Axioms and Metric Spaces
6.5 Compactness, Continuity, Convergence and Completeness
6.6 Product Spaces
6.7 Quotient Spaces
6.8 Topological Invariants
References
7 Differentiable Manifolds
7.1 Differentiation in mathbbRn
7.1.1 Review of Single-Variable Differentiation and Directional Derivatives
7.1.2 Multi-variable Differentiation and the Jacobian
7.2 Differentiable Manifolds in mathbbRn
7.3 Antisymmetric Tensors and p-Forms in mathbbRn
7.4 Differential Forms in mathbbRn
7.5 Correspondences Between Exterior and Vector Calculus in mathbbR3
7.6 Hamilton's Equations and Differential Forms
7.6.1 Lagrange's Equation, Legendre Transformations and Hamilton's Equations
7.6.2 Hamiltonian Phase Space as a Symplectic Manifold
7.7 Transformations of Vectors and Differential Forms
References
8 Aspects of Integration and Elements of Lie Groups
8.1 The Generalized Stokes's Theorem
8.2 Line Integrals and the Integration of One-Forms
8.3 Homotopy and the Cauchy Theorems of Complex Analysis
8.4 Integration of p-Forms and the Vector Integral Theorems
8.5 Lie Groups as Manifolds
8.6 Integral Transforms as Maps
References
Index
