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Mathematics for Physics
| An Illustrated Handbook
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Intro
Preface
What this book is, and what it is not
Who this book is written for
Organization of the book
Notation
Standard notations
Defined notations
Notation conventions
Formatting
Contents
Mathematical structures
Classifying mathematical concepts
Defining mathematical structures and mappings
Abstract algebra
Generalizing numbers
Groups
Rings
Generalizing vectors
Inner products of vectors
Norms and angles of vectors
Multilinear forms on vectors
Orthogonality of vectors
Algebras: multiplication of vectors
Division algebras
Combining algebraic objects
The direct product and direct sum
The free product
The tensor product
Dividing algebraic objects
Quotient groups
Semidirect products
Quotient rings
Related constructions and facts
Summary
Vector algebras
Constructing algebras from a vector space
The tensor algebra
The exterior algebra
Combinatorial notations
The Hodge star
Graded algebras
Clifford algebras
Geometric algebra
Tensor algebras on the dual space
The structure of the dual space
Tensors
Tensors as multilinear mappings
Abstract index notation
Tensors as multi-dimensional arrays
Exterior forms
Exterior forms as multilinear mappings
Exterior forms as completely anti-symmetric tensors
Exterior forms as anti-symmetric arrays
The Clifford algebra of the dual space
Algebra-valued exterior forms
Related constructions and facts
Topological spaces
Generalizing surfaces
Spaces
Generalizing dimension
Generalizing tangent vectors
Existence and uniqueness of additional structure
Summary
Generalizing shapes
Defining spaces
Mapping spaces
Constructing spaces
Cell complexes
Projective spaces
Combining spaces
Classifying spaces
Algebraic topology
Constructing surfaces within a space
Simplices
Triangulations
Orientability
Chain complexes
Counting holes that aren’t boundaries
The homology groups
Examples
Calculating homology groups
Related constructions and facts
Counting the ways a sphere maps to a space
The fundamental group
The higher homotopy groups
Calculating the fundamental group
Calculating the higher homotopy groups
Related constructions and facts
Manifolds
Defining coordinates and tangents
Coordinates
Tangent vectors and differential forms
Frames
Tangent vectors in terms of frames
Mapping manifolds
Diffeomorphisms
The differential and pullback
Immersions, embeddings, and foliations
Critical points
Derivatives on manifolds
Derivations
The Lie derivative of a vector field
The Lie derivative of forms and tensors
The exterior derivative of a 1-form
The exterior derivative of a k-form
Relationships between derivations
Homology on manifolds
The PoincarĂ© lemma
de Rham cohomology
PoincarĂ© duality
Lie groups
Combining algebra and geometry
Spaces with multiplication of points
Vector spaces with topology
Lie groups and Lie algebras
The Lie algebra of a Lie group
The Lie groups of a Lie algebra
Relationships between Lie groups and Lie algebras
The universal cover of a Lie group
Matrix groups
Lie algebras of matrix groups
Linear algebra
Matrix groups with real entries
Other matrix groups
Manifold properties of matrix groups
Matrix group terminology in physics
Representations
Group actions
Group and algebra representations
Lie group and Lie algebra representations
Combining and decomposing representations
Other representations
Classification of Lie groups
Compact Lie groups
Simple Lie algebras
Classifying representations
Clifford groups
Classification of Clifford algebras
Isomorphisms
Representations and spinors
Chiral decomposition
Pauli and Dirac matrices
Clifford groups and representations
Reflections
Rotations
Lie group properties
Lorentz transformations
Representations in spacetime
Spacetime and spinors in geometric algebra
Riemannian manifolds
Introducing parallel transport of vectors
Change of frame
The parallel transporter
The covariant derivative
The connection
The covariant derivative in terms of the connection
The parallel transporter in terms of the connection
Geodesics and normal coordinates
Summary
Manifolds with connection
The covariant derivative on the tensor algebra
The exterior covariant derivative of vector-valued forms
The exterior covariant derivative of algebra-valued forms
Torsion
Curvature
First Bianchi identity
Second Bianchi identity
The holonomy group
Introducing lengths and angles
The Riemannian metric
The Levi-Civita connection
Independent quantities and dependencies
The divergence and conserved quantities
Ricci and sectional curvature
Curvature and geodesics
Jacobi Fields and Volumes
Summary
Fiber bundles
Gauge theory
Matter fields and gauges
The gauge potential and field strength
Spinor fields
Defining bundles
Fiber bundles
G-bundles
Principal bundles
Generalizing tangent spaces
Associated bundles
Vector bundles
Frame bundles
Gauge transformations on frame bundles
Smooth bundles and jets
Vertical tangents and horizontal equivariant forms
Generalizing connections
Connections on bundles
Parallel transport on the frame bundle
The exterior covariant derivative on bundles
Curvature on principal bundles
The tangent bundle and solder form
Torsion on the tangent frame bundle
Spinor bundles
Characterizing bundles
Universal bundles
Characteristic classes
Related constructions and facts
Appendix: Categories and functors
Generalizing sets and mappings
Mapping mappings
Appendix: The divergence, currents, and tensor densities
Coordinate and tensor divergences
Coordinate and tensor divergence theorems
Current forms and densities
Tensor densities
Conserved currents and quantities
References
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An Illustrated Handbook