Book:Joseph George Coffin/Vector Analysis: An Introduction to Vector-methods and their Various Applications to Physics and Mathematics

Subject Matter

 * Vector Analysis

Contents

 * Preface (New York, April 9, 1909)


 * CHAPTER $\text {I}$..


 * $1$. Definitions — Vector — Scalar
 * $2$. Graphical Representation of a Vector
 * $3$. Equality of Vectors — Negative Vector — Unit Vector — Reciprocal Vector
 * $4$. Composition of Vectors — Addition and Subtraction — Vector Sum as an Integration
 * $5$. Scalar and Vector Fields — Point-Function — Definition of Lamé — Continuity of Scalar and Vector Functions
 * $6$. Decomposition of Vectors
 * $7$. The Unit Vectors $\mathbf {i \, j \, k}$
 * $8$. Vector Equations — Equations of Straight Line and Plane
 * $9$. Condition that Three Vectors Terminate in Same Straight Line — Examples
 * $10$. Equation of a Plane
 * $11$. Plane Passing through Ends of Three Given Vectors
 * $12$. Condition that Four Vectors Terminate in Same Plane
 * $13$. To Divide a Line in a Given Ratio — Centroid
 * $14$. Relations Independent of the Origin — General Condition




 * CHAPTER $\text {II}$..


 * $15$. Scalar or Dot Product — Laws of the Scalar Product
 * $16$. Line-Integral of a Vector
 * $17$. Surface-Integral of a Vector
 * $18$. Vector or Cross Product — Definition
 * $19$. Distributive Law of Vector Products — Physical Proof
 * $20$. Cartesian Expansion of the Vector Product
 * $21$. Applications to Mechanics — Moment
 * $22$. Motion of a Rigid Body
 * $23$. Composition of Angular Velocities




 * CHAPTER $\text {III}$..


 * $24$. Possible Combinations of Three Vectors
 * $25$. Triple Scalar Product $V = \mathbf a \cdot \paren {\mathbf b \times \mathbf c}$
 * $26$. Condition that Three Vectors lie in a Plane — Manipulation of Scalar Magnitudes of Vectors
 * $27$. Triple Vector Product $\mathbf q = \mathbf a \times \paren {\mathbf b \times \mathbf c}$ — Expansion and Proof
 * $28$. Demonstration by Cartesian Expansion
 * $29$. Third Proof
 * $30$. Products of More than Three Vectors
 * $31$. Reciprocal System of Vectors
 * $32$. Plane Normal to $\mathbf a$ and Passing through End of $\mathbf b$ — Plane through Ends of Three Given Vectors — Vector Perpendicular from Origin to a Plane
 * $33$. Line through End of $\mathbf b$ Parallel to $\mathbf a$
 * $34$. Circle and Sphere
 * $34 \text a$. Resolution of System of Forces Acting on a Rigid Body — Central Axis — Minimum Couple




 * CHAPTER $\text {IV}$..
 * $35$. Two Ways in which a Vector may Vary — Differentiation with Respect to Scalar Variables
 * $36$. Differentiation of Scalar and Vector Products
 * $37$. Applications to Geometry — Tangent and Normal
 * $38$. Curvature — Osculating Plane — Tortuosity — Geodetic Lines on a Surface
 * $39$. Equations of Surfaces — Curvilinear Coordinates — Orthogonal System
 * $40$. Applications to Kinematics of a Particle — Hodographs — Equations of Hodographs
 * $41$. Integration with Respect to a Scalar Variable — Orbit of a Planet — Harmonic Motion — Ellipse
 * $42$. Hodograph and Orbit under Newtonian Forces
 * $43$. Partial Differentiation — Origin of the Operator $\nabla$




 * CHAPTER $\text {V}$. . $\nabla \equiv \mathbf i \dfrac \partial {\partial x} + \mathbf j \dfrac \partial {\partial y} + \mathbf k \dfrac \partial {\partial z}$


 * $44$. Scalar and Vector Fields
 * $45$. Scalar and Vector Functions of Position — Mathematical and Physical Discontinuities
 * $46$. Potential — Level or Equipotential Surfaces — Relation between Force and Potential
 * $47$. $\nabla$ applied to a Scalar Function — Gradient — Independence of Axes — Fourier's Law
 * $48$. $\nabla$ applied to Scalar Functions — Effect of $\nabla$ on Scalar Product
 * $49$. The Operator $\mathbf s_1 \cdot \nabla$, or Directional Derivative — Total Derivative
 * $50$. Directional Derivative of a Vector — $\nabla$ applied to a Vector Point-Function
 * $51$. Divergence — The Operator $\nabla \cdot$
 * $52$. The Divergence Theorem — Examples — Equation of Flow of Heat
 * $53$. Equation of Continuity — Solenoidal Distribution of a Vector
 * $54$. Curl — The Operator $\nabla \times$ — Example of Curl
 * $55$. Motion of Rotation without Curl — Irrotational Motion
 * $56$. $\nabla$, $\nabla \cdot$, $\nabla \times$ applied to Various Functions — Proofs of Formulæ
 * $57$. Expansion Analogous to Taylor's Theorem
 * $58$. Stokes' Theorem
 * $59$. Condition for Vanishing of the Curl — Conservative System of Forces
 * $60$. Condition for a Perfect Differential
 * $61$. Expression for Taylor's Theorem — The Operator $\map {e^{\epsilon \cdot \nabla} } {\ }$
 * $62$. Euler's Theorem on Homogeneous Functions
 * $63$. Operators Involving $\nabla$ Twice — Possible Combinations — The Operator $\nabla^2 = \nabla \cdot \nabla$
 * $64$. Differentiation of $r^m$ by $\nabla^2$




 * CHAPTER $\text {VI}$..


 * $65$. Gauss's Theorem — Solid Angle — Gauss's Theorem for the Plane — Second Proof
 * $66$. The Potential Function — Poisson's and Laplace's Equations — Harmonic Function
 * $67$. Green's Theorems
 * $68$. Green's Formulæ — Green's Function
 * $69$. Solution of Poisson's Equation — The Integrating Operator $\operatorname {Pot} = \displaystyle \iiint_\infty \dfrac {\paren {\ } d v} r$
 * $70$. Vector-Potential
 * $71$. Separation of a Vector-Function into Solenoidal and Lamellar Components — Other Systems of Units
 * $72$. Energy in Terms of Potential
 * $73$. Energy in Terms of Field Intensity
 * $74$. Surface and Volume Density in Terms of Polarization
 * $75$. Electro-Magnetic Field — Maxwell's Equations
 * $76$. Equation of Propagation of Electro-Magnetic Waves
 * $77$. Poynting's Theorem — Radiant Vector
 * $78$. Magnetic Field due to a Current
 * $79$. Mechanical Force on an Element of Current
 * $80$. Theorem on Line Integral of the Normal Component of a Vector Function
 * $81$. Electric Field at any Point due to a Current
 * $82$. Mutual Energy . of Circuits — Inductance — Neumann's Integral
 * $83$. Vector-Potential of a Current — Mutual Energy of Systems of Conductors — Integration Theorem
 * $84$. Mutual and Self-Energies of Two Circuits




 * CHAPTER $\text {VII}$..


 * $85$. Equations of Motion of a Rigid Body — D'Alembert's Equation — Equations of Translation — Motion of Center of Mass
 * $86$. Equations of Rotation — Kinetic Energy of Rotation — Moment of Inertia
 * $87$. Linear Vector-Function — Instantaneous Axis
 * $88$. Motion of Rotation under No Forces — Poinsot Ellipsoid — Moments and Products of Inertia — Coördinates of a Linear Vector-Function — Principal Moments of Inertia — Principal Axes
 * $89$. Geometrical Representation of the Motion — Invariable Plane — Invariable Axis
 * $90$. Polhode and Herpolhode Curves — Permanent Axes — Equations of Polhode and Herpolhode
 * $91$. Moving Axes and Relative Motion — Theorem of Coriolis
 * $92$. Transformation of Equations of Motion — Centrifugal Couple — Gyroscope
 * $93$. Euler's Equations of Motion
 * $94$. Analytical Solution of Euler's Equations under No Impressed Forces
 * $95$. Hamilton's Principle — Lagrangian Function
 * $96$. Extension of Vector to More than Three Dimensions — Definitions
 * $97$. Lagrange's Generalized Equations of Motion — The Operator $\overline {\nabla L} = 0$ Contains the Whole of Mechanics
 * $98$. Hydrodynamics — Fundamental Equations — Equation of Continuity — Euler's Equations of Motion of a Fluid
 * $99$. Transformations of the Equations of Motion
 * $100$. Steady Motion — Practical Application
 * $101$. Vortex Motion — Non-creatable in a Frictionless System — Helmholtz's Equations
 * $102$. Circulation — Definition
 * $103$. Velocity-Potential — Circulation Invariable in a Frictionless Fluid




 * APPENDIX.


 * Various Notations in Use
 * Hamilton
 * Heaviside
 * Grassmann
 * Gibbs
 * Comparison of Formulæ in Different Notations
 * Notation of this Book
 * Notation of this Book


 * Résumé of the Principal Formulæ of Vector Analysis
 * Vectors
 * Vector and Scalar Products — Products of Two Vectors
 * Products of Three Vectors
 * Differentiation of Vectors
 * The Operator $\nabla$, del
 * Linear Vector Function
 * Index
 * Index