Introduction to Differential Geometry with Tensor Applications

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