Invariance

Chapters

1. Index notation and Cartesian tensors
   • A bar that fails on a slant
   • The summation convention and the Kronecker delta
   • The Levi-Civita symbol and vector identities
   • The rotation matrix and passive changes of axes
   • What makes a Cartesian tensor
   • Contraction, outer products and the quotient theorem
   • Symmetric and antisymmetric parts of a tensor
   • Principal axes and the inertia tensor
   • Pseudotensors and mirror images
   • Isotropic tensors and isotropic materials
   • What is settled for Cartesian tensors
   • Problems
2. General bases, the metric and tensors in spacetime
   • Two sets of components for one vector
   • The dual basis and the reciprocal lattice
   • Contravariant and covariant components
   • The metric tensor raises and lowers indices
   • Coordinates and the general tensor transformation law
   • The tensor density and the volume element
   • The Minkowski metric and four-vectors with indices
   • The field-strength tensor and its invariants
   • What is settled for tensor algebra
   • Problems
3. Differential forms and the exterior derivative
   • Heat, work and the paths between states
   • One-forms and line integrals without a metric
   • The wedge product and oriented areas
   • Forms of degree k
   • The exterior derivative
   • Closed and exact forms, and entropy
   • Pullback and change of variables
   • The Hodge star and the Laplacian
   • What is settled for local forms
   • Problems
4. Integrating forms: Stokes's theorem, holes and Maxwell's equations
   • A field with no curl and a circulation
   • Orientability and the Möbius strip
   • The integral of a form by pullback
   • The generalised Stokes theorem
   • The Poincaré lemma
   • The winding number of a closed form
   • Maxwell's equations as forms
   • Gauge freedom and charge conservation
   • What is settled for forms in flat space
   • Problems
5. Manifolds, charts and tangent spaces
   • Flat charts of a round Earth
   • Coordinate charts and the stereographic projection
   • Atlases and the smooth manifold
   • Tangent vectors as derivatives along curves
   • The cotangent space and forms on a manifold
   • The pushforward of a vector
   • A vector field and its flow
   • The Lie bracket of vector fields
   • What is settled for manifolds
   • Problems
6. The metric, the connection and geodesics
   • The route that does not keep its heading
   • A Riemannian metric on the sphere
   • The covariant derivative repairs the partial derivative
   • The Levi-Civita connection and Christoffel symbols
   • The covariant divergence in any coordinates
   • Parallel transport on the sphere
   • The Foucault pendulum as parallel transport
   • Geodesics and great circles
   • What is settled for the connection
   • Problems
7. Curvature
   • A spear carried round a triangle
   • Holonomy around a loop
   • The Riemann curvature tensor
   • Ricci tensor and scalar curvature
   • Gaussian curvature and the theorema egregium
   • The Gauss-Bonnet theorem and the survey triangle
   • Geodesic deviation and the tidal field
   • The hyperbolic plane and the space of velocities
   • What is settled for Riemannian geometry
   • Problems
8. Groups: the algebra of symmetry operations
   • Two operations in two orders
   • Symmetry operations and their composition
   • The group axioms
   • Subgroups, cosets and Lagrange's theorem
   • Conjugacy classes
   • Homomorphisms, kernels and quotients
   • Permutation groups and Cayley's theorem
   • Direct products
   • The orbit-stabiliser theorem
   • What is settled for finite groups
   • Problems
9. Point groups and crystallographic symmetry
   • A diffraction pattern with tenfold symmetry
   • Symmetry elements and their matrices
   • Point groups of molecules
   • Polarity and chirality rules from the point group
   • The crystallographic restriction
   • Plane lattices and the Bravais lattice classification
   • Neumann's principle and crystal properties
   • Quasicrystals and the limits of the crystallographic restriction
   • What is settled for point groups and crystals
   • Problems
10. Representations, Schur's lemma and characters
    • Frequencies that come in pairs
    • A group representation on three coordinates
    • Equivalent and unitary representations
    • Reducible and irreducible representations
    • Commuting matrices and Schur's lemma
    • The great orthogonality theorem
    • Group characters and their orthogonality
    • Building the character table of C3v​
    • The degeneracy theorem
    • What is settled for representations
    • Problems
11. Character tables at work: vibrations, selection rules and splitting
    • Nine vibrations and two infrared bands
    • The displacement representation
    • Vibrational symmetry species, from water to carbon dioxide
    • The projection operator and symmetry-adapted coordinates
    • Direct product representations
    • The vanishing integral theorem
    • Infrared and Raman selection rules
    • Descent in symmetry and crystal-field splitting
    • What is settled for character tables in use
    • Problems
12. Lie groups and Lie algebras: rotations, SU(2) and the double cover
    • A neutron that needs two turns
    • The Lie group as a manifold
    • The one-parameter subgroup and generator
    • The Lie algebra of rotations
    • The exponential map
    • The SU(2) double cover of SO(3)
    • The adjoint representation
    • The Lorentz algebra and non-commuting boosts
    • What is settled for Lie groups
    • Problems
13. Representations of Lie algebras: spin, products and SU(3)
    • Nine states split as one, three and five
    • The highest-weight representation of su(2)
    • The Casimir operator labels a representation
    • The character of a rotation
    • Tensor product decomposition and the Clebsch-Gordan series
    • Crystal fields from the character of a rotation
    • The SU(3) weight diagram
    • Counting multiplets on the SU(3) weight diagram
    • What is settled and what comes next
    • Problems