Differential Geometry in Physics

Differential Geometry in Physics 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. 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. Features of this book: * Chapters 1-4 and chapter 5 comprise the content of a one-semester course taught by the author for many years. * 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. * An open access ebook edition is available at Open UNC (https://openunc.org) * 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. 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 __Differential Geometry in Physics__ 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. 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. Features of this book: \* Chapters 1-4 and chapter 5 comprise the content of a one-semester course taught by the author for many years. \* 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. \* An open access ebook edition is available at Open UNC (https://openunc.org) \* 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. Differential Geometry In Physics 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. 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. Features Of This Book: * Chapters 1-4 And Chapter 5 Comprise The Content Of A One-semester Course Taught By The Author For Many Years. * 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. * An Open Access Ebook Edition Is Available At Open Unc (https: //openunc.org) * 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. Presents 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.