A Geometrical Approach to Physics

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Foreword
CHAPTER 1: Differential forms
	1.1. MOTIVATION
	1.2. ASPECTS OF THERMODYNAMICS IN TERMS OF DIFFERENTIAL FORMS
	1.3. THE RULES OF EXTERIOR CALCULUS
CHAPTER 2: Vector fields and their relationship with differential forms
	2.1. INTRODUCTION
	2.2. DIFFERENTIAL 1-FORMS AS COTANGENT VECTOR FIELDS
		2.2.1. Tangent vector fields
		2.2.2. Heat capacity from a geometrical perspective
		2.2.3. Heat capacity and the first law of thermodynamics
	2.3. INTERIOR PRODUCT ON DIFFERENTIAL FORMS
		2.3.1. A relationship between thermodynamic properties of a gas
	2.4. CONSOLIDATION
		2.4.1. Differential forms
		2.4.2. The exterior derivative
		2.4.3. Vector fields
		2.4.4. The interior product
CHAPTER 3: Aspects of integration
	3.1. INTRODUCTION
	3.2. A UNIFIED PERSPECTIVE ON MULTIVARIABLE INTEGRATION
		3.2.1. The emergence of 2-forms in the context of magnetostatics
		3.2.2. The relationship between integrals of differential forms and ordinary integrals
		3.2.3. The generalised Stokes theorem
	3.3. AN ALTERNATIVE PERSPECTIVE ON TWO WELL-KNOWN IDENTITIES IN VECTOR CALCULUS
CHAPTER 4: The metric tensor
	4.1. INTRODUCING THE METRIC TENSOR
	4.2. ORTHONORMAL FRAMES
	4.3. MEASUREMENTS
		4.3.1. Lengths
		4.3.2. Angles and rapidities
		4.3.3. Volumes
	4.4. EXAMPLES
		4.4.1. Hyperbolic space
		4.4.2. Cosmological spacetimes
		4.4.3. Rindler spacetime
	4.5. DUALITIES
		4.5.1. Metric dual
		4.5.2. Hodge map
	4.6. CONFORMAL STRUCTURE
	4.7. SYMMETRIES AND KILLING VECTORS
CHAPTER 5: Maxwell’s equations in terms of differential forms
	5.1. INTRODUCTION
	5.2. THE IMPORTANCE OF THE METRIC
	5.3. THE VACUUM MAXWELL EQUATIONS FROM A FOUR-DIMENSIONAL PERSPECTIVE
	5.4. ELECTROMAGNETIC WAVES FROM A SPACETIME PERSPECTIVE
	5.5. OBSERVERS AND THE FIELDS THEY PERCEIVE
	5.6. ELECTRIC CHARGE AND ELECTRIC CURRENT FROM A FOUR-DIMENSIONAL PERSPECTIVE
		5.6.1. Charge conservation
	5.7. POLARISATION AND MAGNETISATION
CHAPTER 6: Classical mechanics
	6.1. THE TANGENT BUNDLE
	6.2. LAGRANGIAN MECHANICS
		6.2.1. Example: Free particle
		6.2.2. Example: Harmonic oscillator
		6.2.3. Example: Particle in a magnetic field
	6.3. THE COTANGENT BUNDLE
	6.4. HAMILTONIAN MECHANICS
	6.5. POISSON BRACKETS
	6.6. CONSERVED QUANTITIES
	6.7. SINGULAR LAGRANGIANS
	6.8. ALMOST SYMPLECTIC GEOMETRY
CHAPTER 7: Connections
	7.1. INTRODUCTION
	7.2. COVARIANT DIFFERENTIATION
	7.3. PARALLEL TRANSPORT
	7.4. GEODESICS AND THE LEVI-CIVITA CONNECTION
	7.5. TORSION AND NONMETRICITY
		7.5.1. Torsion
		7.5.2. Non-metricity
		7.5.3. Reconstructing the connection
	7.6. CURVATURE
	7.7. TENSOR-VALUED FORMS AND CARTAN’S STRUCTURE EQUATIONS
	7.8. FORCES AND ACCELERATION
		7.8.1. Forces due to scalar fields
		7.8.2. Electromagnetic forces
	7.9. THE FERMI-WALKER DERIVATIVE
		7.9.1. Gyroscopes and the concept of rotation
		7.9.2. Classical behaviour of a particle with spin
CHAPTER 8: Generalised functions from a geometric perspective
	8.1. INTRODUCTION
	8.2. THE EXTERIOR CALCULUS OF LINEAR DISTRIBUTIONS
		8.2.1. The relationship with the Dirac delta function
	8.3. EDGE CURRENTS FROM A GEOMETRICAL PERSPECTIVE
	8.4. BOUNDARY CONDITIONS IN ELECTROMAGNETISM
		8.4.1. Reflection of a pulse from an accelerating mirror
APPENDIX A: Solutions and hints to the exercises
	A.1. DIFFERENTIAL FORMS
	A.2. VECTOR FIELDS AND THEIR RELATIONSHIP WITH DIFFERENTIAL FORMS
	A.3. ASPECTS OF INTEGRATION
	A.4. THE METRIC TENSOR
	A.5. MAXWELL’S EQUATIONS IN TERMS OF DIFFERENTIAL FORMS
	A.6. CLASSICAL MECHANICS
	A.7. CONNECTIONS
	A.8. GENERALISED FUNCTIONS FROM A GEOMETRIC PERSPECTIVE
Suggestions for further reading
Index