Tensor Analysis for Engineers

Cover
Half-Title
Title
Copyright
Dedication
Contents
Preface
About the Author
Chapter 1: Introduction
	1.1 Index Notation—The Einstein Summation Convention
Chapter 2: Coordinate Systems Definition
Chapter 3: Basis Vectors and Scale  Factors
Chapter 4: Contravariant Components and Transformations
Chapter 5: Covariant Components and Transformations
Chapter 6: Physical Components and Transformations
Chapter 7: Tensors—Mixed and Metric
Chapter 8: Metric Tensor Operation on Tensor Indices
	8.1 Example: Cylindrical Coordinate Systems
	8.2 Example: Spherical Coordinate Systems
Chapter 9: Dot and Cross Products of Tensors
	9.1 Determinant of an N × N Matrix Using Permutation Symbols
Chapter 10: Gradient Vector Operator—Christoffel Symbols
	10.1 Covariant Derivatives of Vectors—Christoffel Symbols of the 2nd Kind
	10.2 Contravariant Derivatives of Vectors
	10.3 Covariant Derivatives of a Mixed Tensor
	10.4 Christoffel Symbol Relations and Properties—1st and 2nd Kinds
Chapter 11: Derivative Forms—Curl, Divergence, Laplacian
	11.1 Curl Operations on Tensors
	11.2 Physical Components of the Curl of Tensors—3D Orthogonal Systems
	11.3 Divergence Operation on Tensors
	11.4 Laplacian Operations on Tensors
	11.5 Biharmonic Operations on Tensors
	11.6 Physical Components of the Laplacian of a Vector—3D Orthogonal Systems
Chapter 12: Cartesian Tensor Transformation—Rotations
	12.1 Rotation Matrix
	12.2 Equivalent Single Rotation: Eigenvalues and Eigenvectors
Chapter 13: Coordinate Independent Governing Equations
	13.1 The Acceleration Vector—Contravariant Components
	13.2 The Acceleration Vector—Physical Components
	13.3 The Acceleration Vector in Orthogonal Systems—Physical Components
	13.4 Substantial Time Derivatives of Tensors
	13.5 Conservation Equations—Coordinate Independent Forms
Chapter 14: Collection of Relations for Selected Coordinate Systems
	14.1 Cartesian Coordinate System
	14.2 Cylindrical Coordinate Systems
	14.3 Spherical Coordinate Systems
	14.4 Parabolic Coordinate Systems
	14.5 Orthogonal Curvilinear Coordinate Systems
Chapter 15: Rigid Body Rotation: Euler Angles, Quaternions, and Rotation Matrix
	15.1 Active and Passive Rotations
	15.2 Euler Angles
	15.3 Categorizing Euler Angles
	15.4 Gimbal Lock-Euler Angles Limitation
	15.5 Quaternions-Applications for Rigid Body Rotation
	15.6 From a Given Quaternion to Rotation Matrix
	15.7 From a Given Rotation Matrix to Quaternion
	15.8 From Euler Angles to a Quaternion
	15.9 Putting It All Together
Chapter 16: Worked-out Examples
	16.1 Example: Einstein Summation Conventions
	16.2 Example: Conversion from Vector to Index Notations
	16.3 Example: Oblique Rectilinear Coordinate Systems
	16.4 Example: Quantities Related to Parabolic Coordinate System
	16.5 Example: Quantities Related to Bi-Polar Coordinate Systems
	16.6 Example: Application of Contravariant Metric Tensors
	16.7 Example: Dot and Cross Products in Cylindrical and Spherical Coordinates
	16.8 Example: Relation between Jacobian and Metric Tensor Determinants
	16.9 Example: Determinant of Metric Tensors Using Displacement Vectors
	16.10 Example: Determinant of a 4 × 4 Matrix Using Permutation Symbols
	16.11 Example: Time Derivatives of the Jacobian
	16.12 Example: Covariant Derivatives of a Constant Vector
	16.13 Example: Covariant Derivatives of Physical Components of a Vector
	16.14 Example: Continuity Equations in Several Coordinate Systems
	16.15 Example: 4D Spherical Coordinate Systems
	16.16 Example: Complex Double Dot-Cross Product Expressions
	16.17 Example: Covariant Derivatives of Metric Tensors
	16.18 Example: Active Rotation Using Single-Axis and Quaternions Methods
	16.19 Example: Passive Rotation Using Single-Axis and Quaternions Methods
	16.20 Example: Successive Rotations Using Quaternions Method
Chapter 17: Exercise Problems
References
Index