An Introduction to Vectors, Vector Operators and Vector Analysis

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