Introduction to Differential Geometry with Tensor Applications

Description

INTRODUCTION TO DIFFERENTIAL GEOMETRY WITH TENSOR APPLICATIONS This is the only volume of its kind to explain, in precise and easy-to-understand language, the fundamentals of tensors and their applications in differential geometry and analytical mechanics with examples for practical applications and questions for use in a course setting. Introduction to Differential Geometry with Tensor Applications discusses the theory of tensors, curves and surfaces and their applications in Newtonian mechanics. Since tensor analysis deals with entities and properties that are independent of the choice of reference frames, it forms an ideal tool for the study of differential geometry and also of classical and celestial mechanics. This book provides a profound introduction to the basic theory of differential geometry: curves and surfaces and analytical mechanics with tensor applications. The author has tried to keep the treatment of the advanced material as lucid and comprehensive as possible, mainly by including utmost detailed calculations, numerous illustrative examples, and a wealth of complementing exercises with complete solutions making the book easily accessible even to beginners in the field. Groundbreaking and thought-provoking, this volume is an outstanding primer for modern differential geometry and is a basic source for a profound introductory course or as a valuable reference. It can even be used for self-study, by students or by practicing engineers interested in the subject. Whether for the student or the veteran engineer or scientist, Introduction to Differential Geometry with Tensor Applications is a must-have for any library. This outstanding new volume: Presents a unique perspective on the theories in the field not available anywhere elseExplains the basic concepts of tensors and matrices and their applications in differential geometry and analytical mechanicsIs filled with hundreds of examples and unworked problems, useful not just for the student, but also for the engineer in the fieldIs a valuable reference for the professional engineer or a textbook for the engineering student Cover Half-Title Page Series Page Title Page Copyright Page Dedication Contents Preface About the Book Introduction Part I: TENSOR THEORY 1 Preliminaries 1.1 Introduction 1.2 Systems of Different Orders 1.3 Summation Convention Certain Index 1.3.1 Dummy Index 1.3.2 Free Index 1.4 Kronecker Symbols 1.5 Linear Equations 1.6 Results on Matrices and Determinants of Systems 1.7 Differentiation of a Determinant 1.8 Examples 1.9 Exercises 2 Tensor Algebra 2.1 Introduction 2.2 Scope of Tensor Analysis 2.2.1 n-Dimensional Space 2.3 Transformation of Coordinates in Sn 2.3.1 Properties of Admissible Transformation of Coordinates 2.4 Transformation by Invariance 2.5 Transformation by Covariant Tensor and Contravariant Tensor 2.6 The Tensor Concept: Contravariant and Covariant Tensors 2.6.1 Covariant Tensors 2.6.2 Contravariant Vectors 2.6.3 Tensor of Higher Order 2.6.3.1 Contravariant Tensors of Order Two 2.6.3.2 Covariant Tensor of Order Two 2.6.3.3 Mixed Tensors of Order Two 2.7 Algebra of Tensors 2.7.1 Equality of Two Tensors of Same Type 2.8 Symmetric and Skew-Symmetric Tensors 2.8.1 Symmetric Tensors 2.8.2 Skew-Symmetric Tensors 2.9 Outer Multiplication and Contraction 2.9.1 Outer Multiplication 2.9.2 Contraction of a Tensor 2.9.3 Inner Product of Two Tensors 2.10 Quotient Law of Tensors 2.11 Reciprocal Tensor of a Tensor 2.12 Relative Tensor, Cartesian Tensor, Affine Tensor, and Isotropic Tensors 2.12.1 Relative Tensors 2.12.2 Cartesian Tensors 2.12.3 Affine Tensor 2.12.4 Isotropic Tensor 2.12.5 Pseudo-Tensor 2.13 Examples 2.14 Exercises 3 Riemannian Metric 3.1 Introduction 3.2 The Metric Tensor 3.3 Conjugate Tensor 3.4 Associated Tensors 3.5 Length of a Vector 3.5.1 Length of Vector 3.5.2 Unit Vector 3.5.3 Null Vector 3.6 Angle Between Two Vectors 3.6.1 Orthogonality of Two Vectors 3.7 Hypersurface 3.8 Angle Between Two Coordinate Hypersurfaces 3.9 Exercises 4 Tensor Calculus 4.1 Introduction 4.2 Christoffel Symbols 4.2.1 Properties of Christoffel Symbols 4.3 Transformation of Christoffel Symbols 4.3.1 Law of Transformation of Christoffel Symbols of 1st Kind 4.3.2 Law of Transformation of Christoffel Symbols of 2nd Kind 4.4 Covariant Differentiation of Tensor 4.4.1 Covariant Derivative of Covariant Tensor 4.4.2 Covariant Derivative of Contravariant Tensor 4.4.3 Covariant Derivative of Tensors of Type (0,2) 4.4.4 Covariant Derivative of Tensors of Type (2,0) 4.4.5 Covariant Derivative of Mixed Tensor of Type (s, r) 4.4.6 Covariant Derivatives of Fundamental Tensors and the Kronecker Delta 4.4.7 Formulas for Covariant Differentiation 4.4.8 Covariant Differentiation of Relative Tensors 4.5 Gradient, Divergence, and Curl 4.5.1 Gradient 4.5.2 Divergence 4.5.2.1 Divergence of a Mixed Tensor (1,1) 4.5.3 Laplacian of an Invariant 4.5.4 Curl of a Covariant Vector 4.6 Exercises 5 Riemannian Geometry 5.1 Introduction 5.2 Riemannian-Christoffel Tensor 5.3 Properties of Riemann-Christoffel Tensors 5.3.1 Space of Constant Curvature 5.4 Ricci Tensor, Bianchi Identities, Einstein Tensors 5.4.1 Ricci Tensor 5.4.2 Bianchi Identity 5.4.3 Einstein Tensor 5.5 Einstein Space 5.6 Riemannian and Euclidean Spaces 5.6.1 Riemannian Spaces 5.6.2 Euclidean Spaces 5.7 Exercises 6 The e-Systems and the Generalized Kronecker Deltas 6.1 Introduction 6.2 e-Systems 6.3 Generalized Kronecker Delta 6.4 Contraction of δijkαβϒ 6.5 Application of e-Systems to Determinants and Tensor Characters of Generalized Kronecker Deltas 6.5.1 Curl of Covariant Vector 6.5.2 Vector Product of Two Covariant Vectors 6.6 Exercises Part II: DIFFERENTIAL GEOMETRY 7 Curvilinear Coordinates in Space 7.1 Introduction 7.2 Length of Arc 7.3 Curvilinear Coordinates in E₃ 7.3.1 Coordinate Surfaces 7.3.2 Coordinate Curves 7.3.3 Line Element 7.3.4 Length of a Vector 7.3.5 Angle Between Two Vectors 7.4 Reciprocal Base Systems 7.5 Partial Derivative 7.6 Exercises 8 Curves in Space 8.1 Introduction 8.2 Intrinsic Differentiation 8.3 Parallel Vector Fields 8.4 Geometry of Space Curves 8.4.1 Plane 8.5 Serret-Frenet Formula 8.5.1 Bertrand Curves 8.6 Equations of a Straight Line 8.7 Helix 8.7.1 Cylindrical Helix 8.7.2 Circular Helix 8.8 Exercises 9 Intrinsic Geometry of Surfaces 9.1 Introduction 9.2 Curvilinear Coordinates on a Surface 9.3 Intrinsic Geometry: First Fundamental Quadratic Form 9.3.1 Contravariant Metric Tensor 9.4 Angle Between Two Intersecting Curves on a Surface 9.4.1 Pictorial Interpretation 9.5 Geodesic in Rn 9.6 Geodesic Coordinates 9.7 Parallel Vectors on a Surface 9.8 Isometric Surface 9.8.1 Developable 9.9 The Riemannian–Christoffel Tensor and Gaussian Curvature 9.9.1 Einstein Curvature 9.10 The Geodesic Curvature 9.11 Exercises 10 Surfaces in Space 10.1 Introduction 10.2 The Tangent Vector 10.3 The Normal Line to the Surface 10.4 Tensor Derivatives 10.5 Second Fundamental Form of a Surface 10.5.1 Equivalence of Definition of Tensor bαβ 10.6 The Integrability Condition 10.7 Formulas of Weingarten 10.7.1 Third Fundamental Form 10.8 Equations of Gauss and Codazzi 10.9 Mean and Total Curvatures of a Surface 10.10 Exercises 11 Curves on a Surface 11.1 Introduction 11.2 Curve on a Surface: Theorem of Meusnier 11.2.1 Theorem of Meusnier 11.3 The Principal Curvatures of a Surface 11.3.1 Umbillic Point 11.3.2 Lines of Curvature 11.3.3 Asymptotic Lines 11.4 Rodrigue’s Formula 11.5 Exercises 12 Curvature of Surface 12.1 Introduction 12.2 Surface of Positive and Negative Curvature 12.3 Parallel Surfaces 12.3.1 Computation of āαβ and bαβ 12.4 The Gauss-Bonnet Theorem 12.5 The n-Dimensional Manifolds 12.6 Hypersurfaces 12.7 Exercises Part III: ANALYTICAL MECHANICS 13 Classical Mechanics 13.1 Introduction 13.2 Newtonian Laws of Motion 13.3 Equations of Motion of Particles 13.4 Conservative Force Field 13.5 Lagrangean Equations of Motion 13.6 Applications of Lagrangean Equations 13.7 Himilton’s Principle 13.8 Principle of Least Action 13.9 Generalized Coordinates 13.10 Lagrangean Equations in Generalized Coordinates 13.11 Divergence Theorem, Green’s Theorem, Laplacian Operator, and Stoke’s Theorem in Tensor Notation 13.12 Hamilton’s Canonical Equations 13.12.1 Generalized Momenta 13.13 Exercises 14 Newtonian Law of Gravitations 14.1 Introduction 14.2 Newtonian Laws of Gravitation 14.3 Theorem of Gauss 14.4 Poisson’s Equation 14.5 Solution of Poisson’s Equation 14.6 The Problem of Two Bodies 14.7 The Problem of Three Bodies 14.8 Exercises Appendix A: Answers to Even-Numbered Exercises References Index Also of Interest EULA