Tensor Analysis for Engineers : Transformations - Mathematics - Applications.

Format:

Book

Author/Creator:

Tabatabaian, Mehrzad.

Language:

English

Subjects (All):

Calculus of residues.

Physical Description:

1 online resource (251 pages)

Edition:

3rd ed.

Place of Publication:

Bloomfield : Mercury Learning & Information, 2023.

Summary:

Tensor analysis is used in engineering and science fields. This new edition provides engineers and applied scientists with the tools and techniques of tensor analysis for applications in practical problem solving and analysis activities. It includes expanded content on the application of mechanical stress transformation. 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 the readers with mastering the topics provided in the prior sections.

Contents:

Cover

Halftitle

Title

Copyright

Dedication

Contents

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: Mechanical Stress Transformation: Analytical and Mohr's Circle Methods

16.1 Plane Stress Condition

16.2 Principal Stresses and Directions: Eigenvalues and Eigenvectors

16.3 Analysis of Transformed Stresses: Mohr's Circle Graphical Method

16.4 3D Stress Transformation and Analysis

16.5 Principal Directions: Eigenvectors

16.6 Octahedral Stresses in Principal Coordinate System

16.7 Octahedral Stresses and Deviatoric Stresses

16.8 von Mises Yield Criterion vs Octahedral Shear Stress

Chapter 17: The Worked Examples

17.1 Example: Einstein Summation Conventions

17.2 Example: Conversion from Vector to Index Notations

17.3 Example: Oblique Rectilinear Coordinate Systems

17.4 Example: Quantities Related to Parabolic Coordinate System

17.5 Example: Quantities Related to Bi-Polar Coordinate Systems

17.6 Example: Application of Contravariant Metric Tensors

17.7 Example: Dot and Cross Products in Cylindrical and Spherical Coordinates

17.8 Example: Relation between Jacobian and Metric Tensor Determinants

17.9 Example: Determinant of Metric Tensors Using Displacement Vectors.

17.10 Example: Determinant of a 4 × 4 Matrix Using Permutation Symbols

17.11 Example: Time Derivatives of the Jacobian

17.12 Example: Covariant Derivatives of a Constant Vector

17.13 Example: Covariant Derivatives of Physical Components of a Vector

17.14 Example: Continuity Equations in Several Coordinate Systems

17.15 Example: 4D Spherical Coordinate Systems

17.16 Example: Complex Double Dot-Cross Product Expressions

17.17 Example: Covariant Derivatives of Metric Tensors

17.18 Example: Active Rotation Using Single-Axis and Quaternions Methods

17.19 Example: Passive Rotation Using Single-Axis and Quaternions Methods

17.20 Example: Successive Rotations Using Quaternions Method

Chapter 18: Exercises

References

Index.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

9781683929628

1683929624

9781683929635

1683929632

OCLC:

1389611716