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

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## Joseph George Coffin:

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

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

### Subject Matter

### 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

- Notation and Formulæ.

- 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

- Formulæ.

## Further Editions

- 1911: Joseph George Coffin:
*Vector Analysis: An Introduction to Vector-methods and their Various Applications to Physics and Mathematics*(2nd ed.)

## Cited by

## Sources

- 1921: C.E. Weatherburn:
*Elementary Vector Analysis*... (previous) ... (next): Preface