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 Preface 1 Vectors and Curves 1.1 Tangent Vectors 1.2 Differentiable Maps 1.3 Curves in R3 1.3.1 Parametric Curves 1.3.2 Velocity 1.3.3 Frenet Frames 1.4 Fundamental Theorem of Curves 1.4.1 Isometries 1.4.2 Natural Equations 2 Differential Forms 2.1 One-Forms 2.2 Tensors 2.2.1 Tensor Products 2.2.2 Inner Product 2.2.3 Minkowski Space 2.2.4 Wedge Products and 2-Forms 2.2.5 Determinants 2.2.6 Vector Identities 2.2.7 n-Forms 2.3 Exterior Derivatives 2.3.1 Pull-back 2.3.2 Stokes' Theorem in Rn 2.4 The Hodge Operator 2.4.1 Dual Forms 2.4.2 Laplacian 2.4.3 Maxwell Equations 3 Connections 3.1 Frames 3.2 Curvilinear Coordinates 3.3 Covariant Derivative 3.4 Cartan Equations 4 Theory of Surfaces 4.1 Manifolds 4.2 The First Fundamental Form 4.3 The Second Fundamental Form 4.4 Curvature 4.4.1 Classical Formulation of Curvature 4.4.2 Covariant Derivative Formulation of Curvature 4.5 Fundamental Equations 4.5.1 Gauss-Weingarten Equations 4.5.2 Curvature Tensor, Gauss's Theorema Egregium 5 Geometry of Surfaces 5.1 Surfaces of Constant Curvature 5.1.1 Ruled and Developable Surfaces 5.1.2 Surfaces of Constant Positive Curvature 5.1.3 Surfaces of Constant Negative Curvature 5.1.4 Bäcklund Transforms 5.2 Minimal Surfaces 5.2.1 Minimal Area Property 5.2.2 Conformal Mappings 5.2.3 Isothermal Coordinates 5.2.4 Stereographic Projection 5.2.5 Minimal Surfaces by Conformal Maps 6 Riemannian Geometry 6.1 Riemannian Manifolds 6.2 Submanifolds 6.3 Sectional Curvature 6.4 Big D 6.4.1 Linear Connections 6.4.2 Affine Connections 6.4.3 Exterior Covariant Derivative 6.4.4 Parallelism 6.5 Lorentzian Manifolds 6.6 Geodesics 6.7 Geodesics in GR 6.8 Gauss-Bonnet Theorem 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. The Lecture Notes here is a short version of my book which only includes the chapters covered in our one-semester course in differential geometry. In the list above, this would be chapters 1-4 and chapter 6. Thank you all for supporting higher learning 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.