Mathematics for Physics: An Illustrated Handbook

https://kirkmanandjourdain.com/92o6s48v Cover Halftitle Title Copyright Preface Notation Contents 1. Mathematical structures 1.1 Classifying mathematical concepts 1.2 Defining mathematical structures and mappings 2. Abstract algebra 2.1 Generalizing numbers 2.1.1 Groups 2.1.2 Rings 2.2 Generalizing vectors 2.2.1 Inner products of vectors 2.2.2 Norms of vectors 2.2.3 Multilinear forms on vectors 2.2.4 Orthogonality of vectors 2.2.5 Algebras: multiplication of vectors 2.2.6 Division algebras 2.3 Combining algebraic objects 2.3.1 The direct product and direct sum 2.3.2 The free product 2.3.3 The tensor product 2.4 Dividing algebraic objects 2.4.1 Quotient groups 2.4.2 Semidirect products 2.4.3 Quotient rings 2.4.4 Related constructions and facts 2.5 Summary 3. Vector algebras 3.1 Constructing algebras from a vector space 3.1.1 The tensor algebra 3.1.2 The exterior algebra 3.1.3 Combinatorial notations 3.1.4 The Hodge star 3.1.5 Graded algebras 3.1.6 Clifford algebras 3.1.7 Geometric algebra 3.2 Tensor algebras on the dual space 3.2.1 The structure of the dual space 3.2.2 Tensors 3.2.3 Tensors as multilinear mappings 3.2.4 Abstract index notation 3.2.5 Tensors as multi-dimensional arrays 3.3 Exterior forms 3.3.1 Exterior forms as multilinear mappings 3.3.2 Exterior forms as completely anti-symmetric tensors 3.3.3 Exterior forms as anti-symmetric arrays 3.3.4 The Clifford algebra of the dual space 3.3.5 Algebra-valued exterior forms 3.3.6 Related constructions and facts 4. Topological spaces 4.1 Generalizing surfaces 4.1.1 Spaces 4.1.2 Generalizing dimension 4.1.3 Generalizing tangent vectors 4.1.4 Existence and uniqueness of additional structure 4.1.5 Summary 4.2 Generalizing shapes 4.2.1 Defining spaces 4.2.2 Mapping spaces 4.3 Constructing spaces 4.3.1 Cell complexes 4.3.2 Projective spaces 4.3.3 Combining spaces 4.3.4 Classifying spaces 5. Algebraic topology 5.1 Constructing surfaces within a space 5.1.1 Simplices 5.1.2 Triangulations 5.1.3 Orientability 5.1.4 Chain complexes 5.2 Counting holes that aren’t boundaries 5.2.1 The homology groups 5.2.2 Examples 5.2.3 Calculating homology groups 5.2.4 Related constructions and facts 5.3 Counting the ways a sphere maps to a space 5.3.1 The fundamental group 5.3.2 The higher homotopy groups 5.3.3 Calculating the fundamental group 5.3.4 Calculating the higher homotopy groups 5.3.5 Related constructions and facts 6. Manifolds 6.1 Defining coordinates and tangents 6.1.1 Coordinates 6.1.2 Tangent vectors and differential forms 6.1.3 Frames 6.1.4 Tangent vectors in terms of frames 6.2 Mapping manifolds 6.2.1 Diffeomorphisms 6.2.2 The differential and pullback 6.2.3 Immersions and embeddings 6.2.4 Critical points 6.3 Derivatives on manifolds 6.3.1 Derivations 6.3.2 The Lie derivative of a vector field 6.3.3 The Lie derivative of an exterior form 6.3.4 The exterior derivative of a 1-form 6.3.5 The exterior derivative of a k-form 6.3.6 Relationships between derivations 6.4 Homology on manifolds 6.4.1 The Poincaré lemma 6.4.2 de Rham cohomology 6.4.3 Poincaré duality 7. Lie groups 7.1 Combining algebra and geometry 7.1.1 Spaces with multiplication of points 7.1.2 Vector spaces with topology 7.2 Lie groups and Lie algebras 7.2.1 The Lie algebra of a Lie group 7.2.2 The Lie groups of a Lie algebra 7.2.3 Relationships between Lie groups and Lie algebras 7.2.4 The universal cover of a Lie group 7.3 Matrix groups 7.3.1 Lie algebras of matrix groups 7.3.2 Linear algebra 7.3.3 Matrix groups with real entries 7.3.4 Other matrix groups 7.3.5 Manifold properties of matrix groups 7.3.6 Matrix group terminology in physics 7.4 Representations 7.4.1 Group actions 7.4.2 Group and algebra representations 7.4.3 Lie group and Lie algebra representations 7.4.4 Combining and decomposing representations 7.4.5 Other representations 7.5 Classification of Lie groups 7.5.1 Compact Lie groups 7.5.2 Simple Lie algebras 7.5.3 Classifying representations 8. Clifford groups 8.1 Classification of Clifford algebras 8.1.1 Isomorphisms 8.1.2 Representations and spinors 8.1.3 Pauli and Dirac matrices 8.1.4 Chiral decomposition 8.2 Clifford groups and representations 8.2.1 Reflections 8.2.2 Rotations 8.2.3 Lie group properties 8.2.5 Representations in spacetime 8.2.6 Spacetime and spinors in geometric algebra 9. Riemannian manifolds 9.1 Introducing parallel transport of vectors 9.1.1 Change of frame 9.1.2 The parallel transporter 9.1.3 The covariant derivative 9.1.4 The connection 9.1.5 The covariant derivative in terms of the connection 9.1.6 The parallel transporter in terms of the connection 9.1.7 Geodesics and normal coordinates 9.1.8 Summary 9.2 Manifolds with connection 9.2.1 The covariant derivative on the tensor algebra 9.2.2 The exterior covariant derivative of vector-valued forms 9.2.3 The exterior covariant derivative of algebra-valued forms 9.2.4 Torsion 9.2.5 Curvature 9.2.6 First Bianchi identity 9.2.7 Second Bianchi identity 9.2.8 The holonomy group 9.3 Introducing lengths and angles 9.3.1 The Riemannian metric 9.3.2 The Levi-Civita connection 9.3.3 Independent quantities and dependencies 9.3.4 The divergence and conserved quantities 9.3.5 Ricci and sectional curvature 9.3.6 Curvature and geodesics 9.3.7 Jacobi fields and volumes 9.3.8 Summary 9.3.9 Related constructions and facts 10. Fiber bundles 10.1 Gauge theory 10.1.1 Matter fields and gauges 10.1.2 The gauge potential and field strength 10.1.3 Spinor fields 10.2 Defining bundles 10.2.1 Fiber bundles 10.2.2 G-bundles 10.2.3 Principal bundles 10.3 Generalizing tangent spaces 10.3.1 Associated bundles 10.3.2 Vector bundles 10.3.3 Frame bundles 10.3.4 Gauge transformations on frame bundles 10.3.5 Smooth bundles and jets 10.3.6 Vertical tangents and horizontal equivariant forms 10.4 Generalizing connections 10.4.1 Connections on bundles 10.4.2 Parallel transport on the frame bundle 10.4.3 The exterior covariant derivative on bundles 10.4.4 Curvature on principal bundles 10.4.5 The tangent bundle and solder form 10.4.6 Torsion on the tangent frame bundle 10.4.7 Spinor bundles 10.5 Characterizing bundles 10.5.1 Universal bundles 10.5.2 Characteristic classes 10.5.3 Related constructions and facts Appendix A Categories and functors A.1 Generalizing sets and mappings A.2 Mapping mappings Bibliography Index