“Vector and Tensor Analysis” by Dr Nawazish Ali is a comprehensive textbook that provides a detailed introduction to the fundamental concepts of vector and tensor analysis. The book is designed for advanced undergraduate and graduate students in engineering, physics, and mathematics.
The book starts with a brief introduction to vectors, including their algebraic and geometric properties. It then covers vector calculus, including gradient, divergence, curl, and Laplacian. The author then provides a comprehensive overview of tensors, including their algebraic properties, tensor calculus, and applications to linear transformations.
One of the unique features of this book is its clear and concise presentation of the material. The author uses clear explanations, detailed examples, and numerous figures and illustrations to help students understand the concepts and techniques presented. The book is also well-organized, making it easy for students to navigate and find the information they need.
Table of Contents
CHAPTER 1: ALGEBRA OF VECTORS
- Introduction
- Scalars and vectors
- Geometric Represention of a Vector
- Fundamental Definaions Using Geometry Representation
- Parallel Vactor
- Rectangular Coordinate System in Space
- Unit Vectors i,j,k
- Components of a Vector
- Analytic Representation of a Vector
- Position Vector
- Fundamental Definitions Using Analytic Representation
- Properties of Vector Addition
- Properties of Scalar multiplication
- Dot or Scalar Product
- Dot Product of unit Vectors
- Dot Product in Terms of Components
- Condition for Orthogonality
- Properties of Dot Product
- Direction Cosines of a Vector
- Component and Projection of a Vectors
- Geometric Interpretation of Dot Product
- Application of Dot Product
- Cross or Vector Product
- Cross Product of unit Vectors
- Cross Product in Terms of Components
- Geometrical Interpretation of the Magnitude of Cross Product
- Condition for Parallemlism
- Properties of Cross Product
- Application of Cross Product
- Scalar Triple Product
- Scalar Triple Product in Terms of Components
- Scalar Triple Products of Unit Vectors
- Geometrical Interpretation of Magnitude of A.B, C
- Condition for Coplanarity
- Properties of Scalar Triple Product
- Vectors Triple Product
- Fundamental Identities for the Vector Triple Product
- Scalar and vector Products of Four Vectors
- Linear Combination of vectors
- Linear Dependence and Independence of Vectors
- Collinear Vectors
- Coplanar Vectors Reciprocal Vectors
- Solved Problems
- Exercise
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CHAPTER 2: GEOMETRY OF VECTORS
- Introduction
- Application to Geometry
- Vector Equations of a straight line
- Vector Equation of a straight line Through Two points
- Distance From a Point to a line in space
- Vector Equation of a plane
- Plane Through Three Points
- Distance From a point to a plane
- Angle Between Two Plane
- Angle Between a Line and a plane
- Vector Equation of a Sphere
- Solved Problems
- Exercise
CHAPTER 3: VECTOR DIFFERENTIATION AND INTEGRATION
- Introduction
- Scalar and Vector Functions of One Variable
- Domain and Range
- Limit and Continuity of a vector Function
- Ordinary Derivative of a Vector Function
- Geometrical Interpretation of Vector Derivative
- Velocity and Acceleration
- Differentiation Formulas
- Space Curves
- Vector Functions of more Then one variable
- Partial Derivatives of a vectors Functions
- Surface in space
- Vector Integration
- Solved Problems
- Exercise
CHAPTER 4: GRADIENT, DIVERGENCE, AND CURL
- Introduction
- Scalar and Vector Fields
- Level Surfaces
- The Operator Del
- Gradient of a Scalar Point Functional
- Properties of the Gradient
- Geometrical Interpretation of Gradient
- Directional Derivative
- Normal Derivative
- Alternative Definition of Gradient
- Divergence of a Vector Point Function
- Properties of the Divergence
- Physical Interpretation of the Divergence
- Laplacian
- Curl of a Vector point Function
- Properties of the Curl
- Geometrical Interpretational of the Curl
- Operations with
- Vector Identities
- Solved Problems
- Exercise
CHAPTER 5: LINE, SURFACE, AND VOLUME INTEGRALS AND RELATED INTEGRAL THEOREMS
- Introduction
- Tangential Line Integral
- Line Integral Dependent on path
- Line Integral Independent of path
- Theorems on line Interals Independent of Path
- Normal Surface integral
- Evaluation of the Surface Intergrals
- Volumes Integral
- simply and multiply Connected Regions
- Green,s Theorems in the Plane
- Green,s Theorems in the plane in Vector Notation
- Stokes Theorem
- Gauss Divergence Theorem
- The Gradient Theorem
- The Curl Theorem
- Green,s Identities
- Solid Angle
- Solved Problems
- Exercise
CHAPTER 6: CURVILINEAR COORDINATES
- Introduction
- Transformation of Coordinates
- Coordination Surfaces and Coordinate
- Units Vectors in Curvilinear Coordination System
- Orthogonal Curvilinear Coordinate System
- Expressions for Arc Length, Area, and Volume Elements in Orthogonal Curvilinear Coordinates
- Expression For Jacobian in Orthogonal Curvilinear Coordinate
- Gradient, Divergence, Curl, and Laplacian in Orthogonal Curvilinear Coordinate
- Expression For Arc Length, Area, and Volume Elements in rectangular Cartesian Coordinates
- Expression For Arc Length, Area, and Volume Elements in rectangular Cartesian Coordinates
- Expression for Jacobain In Rectangular Cartesian Coordinates
- 6.12 Expression for Gradient, Divergence, and laplacian in Rectangular Cartesian Coordinates
- Cylindrical polar Coordinates
- Cylindrical Coordinates in Terms of Cartesian Coordinates
- Unit Vectors In Cylindrical Coordinate System
- Orthogonality of Cylindrical Coordinate
- Relationships Among Unit Vectors in Cylindrical system
- Cartesian Unit Vectors in Terms of Cylindrical Unit Vectors
- Position Vector in Cylindrical Coordinate System
- Relationships between Cartesian and Cylindrical Components of a Vector
- Expressions for Arc Length, Area, and Volume Elements in Cylindrical Polar Coordinates
- Expressions for Jacobian in cylindrical polar Coordinates
- Expressions for ———–in Cylindrical Coordinates
- Expressions For Gradient, Divergence, Curl , and Laplacian in Cylindrical
- Polar Coordinates
- Alternative Method Using Transformation Equations
- Spherical Polar Coordinates
- Equations Expressing Spherical Coordinates in terms of Cartesian Coordinates
- Unit Vectors in Spherical Coordinates System
- Orthogonality of spherical Coordinate System
- Relationships Among Unit Vectors In Spherical System
- Cartesians Unit Vectors in Terms of Spherical Unit Vectors
- Position Vectors in Spherical Coordinate System
- Relationships Between Cartesian and Spherical Components of a Vectors
- Expressions for Arc Length, Area and Volume Elements in spherical polar Coordinates
- Expression for Jacbian in Spherical polar Coordinates
- Expression for———–in Spherical Coordinates
- Expressions for Gradient Divergence, Curl, and Laplacian in Spherical polar Coordinate
- Alternative Method Using Transformation Equations
- Transformation Equations Forms Cylindrical to Spherical Systems
- Relationships Between the unit Vectors of Cylindrical and Spherical Coordinate system
- Solved Problems
- Exercise
CHAPTER 7: CARTESIAN TENSORS
- Introduction
- Summation Convention
- Double Sums
- Substitutions
- Algebra and the Summation Convention
- The Kronecker Delta
- Rectangular Coordinate System
- Direction Cosines
- Orthogonal Rotation of Axes
- Proper and Improper Transformations
- Transformation equation
- Orthonormality Conditions
- Translation and Rotation
- Invariance with Respect to Rotation of Axes
- Scalar Invariant Operators
- The Alternating Symbol
- Tensors
- Algebra of Tensors
- Contraction of Tensors
- Inner Multiplication of Tensors
- Quotient Theorem
- Symmetric and Anti-Symmetric Tensors
- Invariance of Symmetric and Anti-Symmetric Tensors
- Fundamental Property of Tensor Equations
- Isotropic Tensors
- Tensor Calculus
- Application to Vector analysis
- Integral Theoram in Tensor From
- Eigenvalues and Eigenvector of a Second Order Tensor
- Eigenvalues and Eigenvector of a Second Order real Symmetric Tensor
- Principal axes and Principal Directions
- Invariants of a Tensor
- Deviators
- Solved Problems
- Exercise
CHAPTER 8: GENERAL TENSORS
- Introduction
- n-Dimensional Space
- Coordinate Transformations
- Einstein Summation Conventional
- Double Sums
- Substitutions
- The Kronecker Delta
- Tensor Form of the Transformation Equation Between Cartesian Coordinate and polar, Cylindrical and Spherical Coordinates
- Tensor Natation for Matrices
- The Jacobian
- Tensors
- Zeroth- Order Tensors
- First Order Tensor
- Second Order Tensor
- Tensors of Higher Order
- Tensor Fields
- Inverses Transformations Laws For the First and Second Order Tensors
- Partial Derivatives of First-Order Tensors
- Rank of a Tensor
- Algebra of Tensors
- Contraction
- Inner Multiplication
- Quotient Theorem
- Symmetric and skew –Symmetric Tensors
- In variance of Symmetric and Skew –Symmetric Properties of a Tensor
- Fundamental Property of Tensor Equations
- The Line Element and Metric Tensor
- Metric Tensor in Cylindrical and Spherical Polar Coordinates
- Conjugate or Reciprocal Tensor
- Conjugate Metric Tensor In Cylindrical and Spherical polar Coordinates
- Associated Tensors
- Christoffel Symbols
- Transformation Laws of Christoffel,s Symbols
- The formula for Second-Order Partial Derivatives
- Christoffel Symbols of the First Kind in Rectangular, Cylindrical, and Spherical Coordinates
- Christoffel Symbols of the Second Kind in Rectangular, Cylindrical, and Spherical Coordinates
- Converiant Derivatives
- Rules for Covariant Differentiation
- Magnitude or Length of a Vector
- Unit Vector
- Angle Between Two Vectors
- Divergence in Cylinderical and Spherical Coordinates
- Laplacian in Cylinderiacal And Spherical Coordinates
- Integral Theorems in Tensor Form
- Riemann- Christoffel Tensor
- Convariant Curvature Tensor
- Bianchi,s Second Identity
- Flat and Non_Flat Riemannian
- Euclidean and Non_Euclidean Metrics
- Ricci Tensor
- Scalar Curvature
- Cyclic Ricci Tensor
- Codazzi Type Tensor
- Einstein Space
- Space of Constant Curvature
- Elinstein Space
- The Intrinsic or Absolute Derivative
- Rules for Intrinsic Derivative
- Tensors form of Velocity and Acceleration
- Relative and Absolute Tensor
- Algebra of Relative Tensors
- Covariant Derivative of Relative Tensors
- Permutation Symbols and Permutation Tensors
- A Result From Calculus of Variations
- Geodesic, Geodesics Coordinates, and Geodesics Coordinate system
- Differential Equations For the Geodesic in Cylindrical and Spherical coordinates
- Bianchi,s Second Identity
- Solved Problems
- Exercise
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In addition, the book includes numerous exercises and problems at the end of each chapter, providing students with opportunities to practice and apply the concepts they have learned. The solutions to these problems are included in the book, allowing students to check their understanding of the material.
The book concludes with a discussion of the applications of vector and tensor analysis, including fluid mechanics, elasticity, and general relativity. The author also includes a discussion of the geometric and physical interpretations of vector and tensor fields, making the book a valuable resource for students who are interested in the applications of these fields.
Overall, “Vector and Tensor Analysis” by Dr. Nawazish Ali is an excellent textbook for students who are looking to gain a deep understanding of the fundamental concepts of vector and tensor analysis. The book’s clear and concise presentation, well-organized structure, and numerous exercises and problems make it an ideal resource for students who are looking to master these important topics.
In conclusion, if you are looking for a comprehensive and accessible textbook on vector and tensor analysis, “Vector and Tensor Analysis” by Dr. Nawazish Ali is an excellent choice. The book’s clear explanations, numerous examples, and well-designed exercises and problems make it an ideal resource for students who are looking to gain a deep understanding of these important concepts.
