Tensor analysis is used in engineering and science fields. This new edition provides engineers and applied scientists the tools and techniques of tensor analysis for applications in practical problem solving and analysis activities. The geometry is limited to the Euclidean space/geometry, where the Pythagorean Theorem applies, with well-defined Cartesian coordinate systems as the reference. Quantities defined in curvilinear coordinate systems, like cylindrical, spherical, parabolic, etc. are discussed and several examples and coordinates sketches with related calculations are presented. In addition, the book has several worked-out examples for helping readers with mastering the topics provided in the prior sections.
Author(s): Mehrzad Tabatabaian
Edition: 2
Publisher: Mercury Learning and Information
Year: 2020
Language: English
Pages: 197
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
