Preface
Notation
1 Introduction
1.1 Vector (linear) spaces
1.2 The scalar product
1.3 Complex numbers
1.4 Quaternions
1.5 The cross product
1.6 The outer product
1.7 Notes
1.8 Exercises
2 Geometric algebra in two and three dimensions
2.1 A new product for vectors
2.2 An outline of geometric algebra
2.3 Geometric algebra of the plane
2.4 The geometric algebra of space
2.5 Conventions
2.6 Reflections
2.7 Rotations
2.8 Notes
2.9 Exercises
3 Classical mechanics
3.1 Elementary principles
3.2 Two-body central force interactions
3.3 Celestial mechanics and perturbations
3.4 Rotating systems and rigid-body motion
3.5 Notes
3.6 Exercises
4 Foundations of geometric algebra
4.1 Axiomatic development
4.2 Rotations and reflections
4.3 Bases, frames and components
4.4 Linear algebra
4.5 Tensors and components
4.6 Notes
4.7 Exercises
5 Relativity and spacetime
5.1 An algebra for spacetime
5.2 Observers, trajectories and frames
5.3 Lorentz transformations
5.4 The Lorentz group
5.5 Spacetime dynamics
5.6 Notes
5.7 Exercises
6 Geometric calculus
6.1 The vector derivative
6.2 Curvilinear coordinates
6.3 Analytic functions
6.4 Directed integration theory
6.5 Embedded surfaces and vector manifolds
6.6 Elasticity
6.7 Notes
6.8 Exercises
7 Classical electrodynamics
7.1 Maxwell's equations
7.2 Integral and conservation theorems
7.3 The electromagnetic field of a point charge
7.4 Electromagnetic waves
7.5 Scattering and diffraction
7.6 Scattering
7.7 Notes
7.8 Exercises
8 Quantum theory and spinors
8.1 Non-relativistic quantum spin
8.2 Relativistic quantum states
8.3 The Dirac equation
8.4 Central potentials
8.5 Scattering theory
8.6 Notes
8.7 Exercises
9 Multiparticle states and quantum entanglement
9.1 Many-body quantum theory
9.2 Multiparticle spacetime algebra
9.3 Systems of two particles
9.4 Relativistic states and operators
9.5 Two-spinor calculus
9.6 Notes
9.7 Exercises
10 Geometry
10.1 Projective geometry
10.2 Conformed geometry
10.3 Conformed transformations
10.4 Geometric primitives in conformed space
10.5 Intersection and reflection in conformal space
10.6 Non-Euclidean geometry
10.7 Spacetime conformed geometry
10.8 Notes
10.9 Exercises
11 Further topics in calculus and group theory
11.1 Multivector calculus
11.2 Grassmann calculus
11.3 Lie groups
11.4 Complex structures and unitary groups
11.5 The general linear group
11.6 Notes
11.7 Exercises
12 Lagrangian and Hamiltonian techniques
12.1 The Euler-Lagrange equations
12.2 Classical models for spin-1/2 particles
12.3 Hamiltonian techniques
12.4 Lagrangian field theory
12.5 Notes
12.6 Exercises
13 Symmetry and gauge theory
13.1 Conservation laws in field theory
13.2 Electromagnetism
13.3 Dirac theory
13.4 Gauge principles for gravitation
13.5 The gravitational field equations
13.6 The structure of the Riemann tensor
13.7 Notes
13.8 Exercises
14 Gravitation
14.1 Solving the field equations
14.2 Spherically-symmetric systems
14.3 Schwarzschild black holes
14.4 Quantum mechanics in a black hole background
14.5 Cosmology
14.6 Cylindrical systems
14.7 Axially-symmetric systems
14.8 Notes
14.9 Exercises
Bibliography
Index
· · · · · · (收起)