<i>Differential Geometry in Physics</i> is a treatment of the mathematical foundations of the theory of general relativity and gauge theory of quantum fields. The material is intended to help bridge the gap that often exists between theoretical physics and applied mathematics.<br><br>The approach is to carve an optimal path to learning this challenging field by appealing to the much more accessible theory of curves and surfaces. The transition from classical differential geometry as developed by Gauss, Riemann and other giants, to the modern approach, is facilitated by a very intuitive approach that sacrifices some mathematical rigor for the sake of understanding the physics. The book features numerous examples of beautiful curves and surfaces often reflected in nature, plus more advanced computations of trajectory of particles in black holes. Also embedded in the later chapters is a detailed description of the famous Dirac monopole and instantons.<br><br>Features of this book:<br>* Chapters 1-4 and chapter 5 comprise the content of a one-semester course taught by the author for many years.<br>* The material in the other chapters has served as the foundation for many master's thesis at University of North Carolina Wilmington for students seeking doctoral degrees.<br>* An open access ebook edition is available at Open UNC (https://openunc.org)<br>* The book contains over 80 illustrations, including a large array of surfaces related to the theory of soliton waves that does not commonly appear in standard mathematical texts on differential geometry.<br><br>
Author(s): Gabriel Lugo
Edition: 1
Publisher: University of North Carolina Wilmington William Madison Randall Library
Year: 2021
Language: English
Commentary: No attempt at file size reduction
Pages: 382
Tags: Differential Geometry; Mathematical Physics
Cover
Front Matter
Title Page
Copyright
Dedication
Contents
Preface
Preface
Vectors and Curves
Tangent Vectors
Differentiable Maps
Curves in R3
Parametric Curves
Velocity
Frenet Frames
Fundamental Theorem of Curves
Isometries
Natural Equations
Differential Forms
One-Forms
Tensors
Tensor Products
Inner Product
Minkowski Space
Wedge Products and 2-Forms
Determinants
Vector Identities
n-Forms
Exterior Derivatives
Pull-back
Stokes' Theorem in Rn
The Hodge Operator
Dual Forms
Laplacian
Maxwell Equations
Connections
Frames
Curvilinear Coordinates
Covariant Derivative
Cartan Equations
Theory of Surfaces
Manifolds
The First Fundamental Form
The Second Fundamental Form
Curvature
Classical Formulation of Curvature
Covariant Derivative Formulation of Curvature
Fundamental Equations
Gauss-Weingarten Equations
Curvature Tensor, Gauss's Theorema Egregium
Geometry of Surfaces
Surfaces of Constant Curvature
Ruled and Developable Surfaces
Surfaces of Constant Positive Curvature
Surfaces of Constant Negative Curvature
Bäcklund Transforms
Minimal Surfaces
Minimal Area Property
Conformal Mappings
Isothermal Coordinates
Stereographic Projection
Minimal Surfaces by Conformal Maps
Riemannian Geometry
Riemannian Manifolds
Submanifolds
Sectional Curvature
Big D
Linear Connections
Affine Connections
Exterior Covariant Derivative
Parallelism
Lorentzian Manifolds
Geodesics
Geodesics in GR
Gauss-Bonnet Theorem
Groups of Transformations
Lie Groups
One-Parameter Groups of Transformations
Lie Derivatives
Lie Algebras
The Exponential Map
The Adjoint Map
The Maurer-Cartan Form
Cartan Subalgebra
Transformation Groups
Classical Groups in Physics
Orthogonal Groups
Rotations in R2
Rotations in R3
SU(2)
Hopf Fibration
Angular Momentum
Lorentz Group
Infinitesimal Transformations
Spinors
N-P Formalism
The Kerr Metric
Eth Operator
SU(3)
Bundles and Applications
Fiber Bundles
Principal Fiber Bundles
Connections on PFB's
Ehresmann Connection
Horizontal Lift
Curvature Form
Gauge Fields
Electrodynamics
Dirac Monopole
BPST Instanton
References
Index
