An Introduction to Vectors, Vector Operators and Vector Analysis

✍ Scribed by Pramod S. Joag

• Publisher: Cambridge University Press
• Year: 2016
• Tongue: English
• Leaves: 546
• Category: Library

✦ Table of Contents

Cover
Copyright
Dedication
Contents
Figures
Tables
Preface
Nomenclature
Part I Basic Formulation
1 Getting Concepts and Gathering Tools
1.1 Vectors and Scalars
1.2 Space and Direction
1.3 Representing Vectors in Space
1.4 Addition and its Properties
1.4.1 Decomposition and resolution of vectors
1.4.2 Examples of vector addition
1.5 Coordinate Systems
1.5.1 Right-handed (dextral) and left-handed coordinate systems
1.6 Linear Independence, Basis
1.7 Scalar and Vector Products
1.7.1 Scalar product
1.7.2 Physical applications of the scalar product
1.7.3 Vector product
1.7.4 Generalizing the geometric interpretation of the vector product
1.7.5 Physical applications of the vector product
1.8 Products of Three or More Vectors
1.8.1 The scalar triple product
1.8.2 Physical applications of the scalar triple product
1.8.3 The vector triple product
1.9 Homomorphism and Isomorphism
1.10 Isomorphism with R3
1.11 A New Notation: Levi-Civita Symbols
1.12 Vector Identities
1.13 Vector Equations
1.14 Coordinate Systems Revisited: Curvilinear Coordinates
1.14.1 Spherical polar coordinates
1.14.2 Parabolic coordinates
1.15 Vector Fields
1.16 Orientation of a Triplet of Non-coplanar Vectors
1.16.1 Orientation of a plane
2 Vectors and Analytic Geometry
2.1 Straight Lines
2.2 Planes
2.3 Spheres
2.4 Conic Sections
3 Planar Vectors and Complex Numbers
3.1 Planar Curves on the Complex Plane
3.2 Comparison of Angles Between Vectors
3.3 Anharmonic Ratio: Parametric Equation to a Circle
3.4 Conformal Transforms, Inversion
3.5 Circle: Constant Angle and Constant Power Theorems
3.6 General Circle Formula
3.7 Circuit Impedance and Admittance
3.8 The Circle Transformation
Part II Vector Operators
4 Linear Operators
4.1 Linear Operators on E3
4.1.1 Adjoint operators
4.1.2 Inverse of an operator
4.1.3 Determinant of an invertible linear operator
4.1.4 Non-singular operators
4.1.5 Examples
4.2 Frames and Reciprocal Frames
4.3 Symmetric and Skewsymmetric Operators
4.3.1 Vector product as a skewsymmetric operator
4.4 Linear Operators and Matrices
4.5 An Equivalence Between Algebras
4.6 Change of Basis
5 Eigenvalues and Eigenvectors
5.1 Eigenvalues and Eigenvectors of a Linear Operator
5.1.1 Examples
5.2 Spectrum of a Symmetric Operator
5.3 Mohr’s Algorithm
5.3.1 Examples
5.4 Spectrum of a 2x2 Symmetric Matrix
5.5 Spectrum of Sn
6 Rotations and Reflections
6.1 Orthogonal Transformations: Rotations and Reflections
6.1.1 The canonical form of the orthogonal operator for reflection
6.1.2 Hamilton’s theorem
6.2 Canonical Form for Linear Operators
6.2.1 Examples
6.3 Rotations
6.3.1 Matrices representing rotations
6.4 Active and Passive Transformations: Symmetries
6.5 Euler Angles
6.6 Euler’s Theorem
7 Transformation Groups
7.1 Definition and Examples
7.2 The Rotation Group O +(3)
7.3 The Group of Isometries and the Euclidean Group
7.3.1 Chasles theorem
7.4 Similarities and Collineations
Part III Vector Analysis
8 Preliminaries
8.1 Fundamental Notions
8.2 Sets and Mappings
8.3 Convergence of a Sequence
8.4 Continuous Functions
9 Vector Valued Functions of a Scalar Variable
9.1 Continuity and Differentiation
9.2 Geometry and Kinematics: Space Curves and Frenet–Seret Formulae
9.2.1 Normal, rectifying and osculating planes
9.2.2 Order of contact
9.2.3 The osculating circle
9.2.4 Natural equations of a space curve
9.2.5 Evolutes and involutes
9.3 Plane Curves
9.3.1 Three different parameterizations of an ellipse
9.3.2 Cycloids, epicycloids and trochoids
9.3.3 Orientation of curves
9.4 Chain Rule
9.5 Scalar Integration
9.6 Taylor Series
10 Functions with Vector Arguments
10.1 Need for the Directional Derivative
10.2 Partial Derivatives
10.3 Chain Rule
10.4 Directional Derivative and the Grad Operator
10.5 Taylor Series
10.6 The Differential
10.7 Variation on a Curve
10.8 Gradient of a Potential
10.9 Inverse Maps and Implicit Functions
10.9.1 Inverse mapping theorem
10.9.2 Implicit function theorem
10.9.3 Algorithm to construct the inverse of a map
10.10 Differentiating Inverse Functions
10.11 Jacobian for the Composition of Maps
10.12 Surfaces
10.13 The Divergence and the Curl of a Vector Field
10.14 Differential Operators in Curvilinear Coordinates
11 Vector Integration
11.1 Line Integrals and Potential Functions
11.1.1 Curl of a vector field and the line integral
11.2 Applications of the Potential Functions
11.3 Area Integral
11.4 Multiple Integrals
11.4.1 Area of a planar region: Jordan measure
11.4.2 Double integral
11.4.3 Integral estimates
11.4.4 Triple integrals
11.4.5 Multiple integrals as successive single integrals
11.4.6 Changing variables of integration
11.4.7 Geometrical applications
11.4.8 Physical applications of multiple integrals
11.5 Integral Theorems of Gauss and Stokes in Twodimensions
11.5.1 Integration by parts in two dimensions: Green’s theorem
11.6 Applications to Twodimensional
11.7 Orientation of a Surface
11.8 Surface Integrals
11.8.1 Divergence of a vector field and the surface integral
11.9 Diveregence Theorem in Threedimensions
11.10 Applications of the Gauss’s Theorem
11.10.1 Exercises on the divergence theorem
11.11 Integration by Parts and Green’s Theorem in Threedimensions
11.11.1 Transformation of
∆U to spherical coordinates
11.12 Helmoltz Theorem
11.13 Stokes Theorem in Threedimensions
11.13.1 Physical interpretation of Stokes theorem
11.13.2 Exercises on Stoke’s theorem
12 Odds and Ends
12.1 Rotational Velocity of a Rigid Body
12.2 3-D Harmonic Oscillator
12.2.1 Anisotropic oscillator
12.3 Projectiles and Terestrial Effects
12.3.1 Optimum initial conditions for netting a basket ball
12.3.2 Optimum angle of striking a golf ball
12.3.3 Effects of Coriolis force on a projectile
12.4 Satellites and Orbits
12.4.1 Geometry and dynamics: Circular motion
12.4.2 Hodograph of an orbit
12.4.3 Orbit after an impulse
12.5 A Charged Particle in Uniform Electric and Magnetic Fields
12.5.1 Uniform magnetic field
12.5.2 Uniform electric and magnetic fields
12.6 Two-dimensional Steady and Irrotational Flow of an Incompressible Fluid
Appendices
A Matrices and Determinants
A.1 Matrices and Operations on them
A.2 Square Matrices, Inverse of a Matrix, Orthogonal Matrices
A.3 Linear and Multilinear Forms of Vectors
A.4 Alternating Multilinear Forms: Determinants
A.5 Principal Properties of Determinants
A.5.1 Determinants and systems of linear equations
A.5.2 Geometrical interpretation of determinants
B Dirac Delta Function
Bibliography
Index