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

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

Published $\text {1909}$, John Wiley & Sons

### Contents

Preface (New York, April 9, 1909)

CHAPTER $\text {I}$. Elementary Operations of Vector Analysis.
$1$. 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
Exercises and Problems

CHAPTER $\text {II}$. Scalar and Vector Products of Two Vectors.
$15$. 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
Exercises and Problems

CHAPTER $\text {III}$. Vector and Scalar Products of Three Vectors.
$24$. 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
Exercises and Problems

CHAPTER $\text {IV}$. Differentiation of Vectors.
$35$. 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 $\nabla$
Exercises and Problems

CHAPTER $\text {V}$. The Differential Operators. $\nabla \equiv \mathbf i \dfrac \partial {\partial x} + \mathbf j \dfrac \partial {\partial y} + \mathbf k \dfrac \partial {\partial z}$
$44$. 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 $\nabla^2$
Exercises and Problems

CHAPTER $\text {VI}$. Applications to Electrical Theory.
$65$. 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 $\operatorname {Pot} = \ds \iiint_\infty \dfrac {\paren {\ } d v} r$
$70$. 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
Exercises and Problems

CHAPTER $\text {VII}$. Applications to Dynamics, Mechanics and Hydrodynamics.
$85$. 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
Exercises and Problems

APPENDIX.
Notation and Formulæ.
Various Notations in Use
Hamilton
Heaviside
Grassmann
Gibbs
Comparison of Formulæ in Different Notations
Notation of this Book
Formulæ.
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