Table Of ContentHandbook of
VECTOR and POLYADIC
ANALYSIS
THOMAS B. DREW
Professor of Chemical Engineering
Columbia University
New York, N. Y.
REINHOLD YUl3LISHlIl6:-47KYOKA"1 ItliN
NEW YORK, N. Y.
CHAPMAN & HALL, LTD., LONDON
Copyright Q 1961 by
REINHOLD PUBLISHING CORP.
All rights reserved
Library of Congress Catalog Card Number: 61-18276
Printed in the United States of America
Preface
This book results from occasional sets of notes prepared over the
past twenty years for the use of engineering graduate students in fluid
mechanics, heat transmission, and the theory of diffusional processes.
The primary service of mathematical analysis in teaching such subjects
is, in my view, linpial rather than computational. The use of tensor
analysis in the form of Gibbsian polyadics clears away the fog of mul-
tiple scalar partial differential equations in which the student otherwise
finds himself without requiring him to learn a secret code. the symbols
and ideas are only slightly different from those met in undergraduate
calculus and the mathematical models mirror the physical models
clearly. The apace given the several topics reflects my own teaching
experience; there could be no pretense of completeness in a book of
this length. The admission of complex vectors and polyadice has proved
useful in developing the notion of function-vectors in Fourier analysis
and in the treatment of integral equations.
I wish to acknowledge my debt to H. B. Phillips who first led me
to appreciate the value of vector analysis using electromagnetic theory
as the vehicle. Thanks are due my daughter Sarah who typed much of
this manuscript and the staff of the Waverly Press, Inc., for their help
in devising printable symbols that suggest the intended meaning.
Thomas B. Drew
Temple, New Hampshire
September, 1961
Contents
1. INTRODUCTION
Object & Scheme
1
Scalars vs. Vectors
1
Notation 3
II. VECTOR ALGEBRA
Definitions and Elementary Operations 5
Scalar Product 7
Vector Product 9
Position Vectors 10
Linear Vector Functions of Vectors
11
Dyadics, Tensors, and Linear Vector Operators 14
Dyadic Representation of Linear Vector Operators 17
Multiple Scalar Products: Higher Linear Operators 21
Definitions 21
Polydot Products of Polyads 21
Unit Polyads 22
Polyadics 24
Transposes and Adjoints of Polyadics 27
Idemfactors, Deviation Factors, Transposers, and related func-
tions 31
Linear Polyadic Functions of Polyadics 45
Use of Linear Operators with Arguments of Various Ranks 52
Polycross Products 57
Versors: Rotation Operators 58
Invariants of Vectors and Polyadics 64
III. VECTOR and POLYADIC CALCULUS
Differentiation and Integration with respect to Scalars 69
Differentiation with respect to Vectors and Polyadics 69
The Divergence and Gauss's Theorem 74
The Circulation, the Curl, Stokes Theorem 75
Operations with V and dW 78
Mixed Derivative Operators 82
Maclaurin's and Taylor's Series 83
vu
viii CONTENTS
APPENDICES
Appendix A: Summary of Formulas and Notation
Appendix B : Ternary Numeration
Appendix C : Comparison of Notation
Appendix D: Miscellaneous Exercises
References on Vector Analysis
I. Introduction
1.1 Object and Scheme
This book constitutes a brief exposition of the vector and tensor
concepts and methods convenient in studying the theory of diffusion,
fluid dynamics, and related topics. There is no intent to provide a text
or treatise but rather to give the reader with a knowledge of calculus a
handbook sufficiently interlineated to be intelligible to one familiar
with the elementary vector usage in college physics and engineering
mechanics. No prior formal study of vector or tensor analysis is assumed.
Willard Gibbs' system of vector analysis is used. It has the advantage
for the uninitiated that many of the notations of scalar calculus carry
over with analogous meanings into the calculus of vectors and tensors;
moreover, the modifications required to include complex vectors (the
"bi-vectors" of Gibbs) parallel those required in scalar analysis to
admit complex scalars. Because diffusion theory involves tensors of
rank higher than two, Gibbs' theory of dyadics has been extended to
that of "polyadics": linear vector functions of variable vectors are thus
extended to linear polyadic functions of polyadics. The presentation is
largely in terms of three-dimensional real vectors but in general the
formulae are so written that they are valid for complex vectors, and any
changes necessary for application to N-dimensions are pointed out. In
the first few sections, the geometric (or vector) method and the alge-
braic (or matrix) method are used in parallel so that the essential
equivalence of vector and matrix analysis may be seen and, further, so
that matrix interpretation may be available for numerical computation
when convenient. Polyadic analysis is tensor analysis restricted to
"Cartesian tensors" but written in terms of the usual symbols of calculus
rather than in a shorthand wholly new to the student.
1.2 Scalars vs. Vectors
The size of a physical quantity is commonly specified with reference
to an agreed scale by a real number. For this reason properties such as
mass, volume, and temperature, of which size on an agreed scale is the
primary analytical characteristic, are called scalar properties or, more
briefly, scalars. The numbers used to express size, and by extension all
algebraic numbers, are themselves often called scalars. This is the usage
1
2 VECTOR AND POLYADIC ANALYSIS
herein. Some writers restrict the word scalar to mean what are here
termed scalar invariants (v. infra).
Certain physical quantities, force and velocity for example, require
for their descriptions the specification of their directions as well as their
sizes; these are vector quantities. The fundamental idea of vector
analysis is that the notions of magnitude and direction may be regarded
as a composite property and represented jointly by a single letter symbol.
The dual entity is a vector. In the same sense in which the abstract
number 3 belongs to all classes of threes so do all quantities of a given
size and direction possess the same vector.
The geometrical representation of a real three-dimensional vector as
an arrow is familiar from elementary mechanics. The arrow flies with
the vector and its length measures the magnitude. All parallel arrows
of equal length, wherever located in space, are equally good representa-
tions of the same vector. The notion of position is not part of the vector
concept.
To state the numerical value of a vector, select any convenient system
of non-coplanar rectilinear axes and consider the particular representa-
tive arrow which rises from the origin. The coordinates of the arrow's
tip, stated in x, y, z order, fully determine the vector if the system of
axes is known; when reference to the vector is intended they are called
the coordinates of the vector with respect to that system of axes. A
system of rectilinear axes may be chosen in infinitely many orientations,
hence there belong to each vector infinitely many ordered sets of three
numbers, each associated with a particular frame of reference. Alge-
braically speaking, it is usual to take the collection of such sets asso-
ciated with right-handed rectangular Cartesian frames of reference to
be the vector. Every set of this restricted collection has the property
that the sum of the squares of its members is the same number: the
square of the magnitude of the vector. An ordered set of quantities is a
matrix;* algebraically speaking, therefore, a vector is a restricted col-
Typically, a matrix is a set ordered in two senses; one sense is represented by
position in rows of a rectangular array; the other, by position in columns. Thus
the matrix
a, b, c,
a, b, c,
as b, cc,
a b+ .
might be a tabulation in order of the coefficients of x, y, and z in the first, sec-
ond, third, and fourth of a set of linear equations. This is a "4 X 3 matrix"; that
INTRODUCTION 3
lection of matrices. Thus, following Murnaghan ("Applied Mathe-
matics", Wiley, 1948) :
A three-dimensional vector is a collection of 3 X 1 matrices such
that
(a) the sum of the squares of the absolute values of the elements
(coordinates) is a given number, a scalar invariant of the collection,
called the square of the magnitude of the vector, and
(b) the corresponding coordinates of any pair of the matrices, say
and
Y
1 1
z11
are related by a change of variables of the type
x' = 1,x + m,y + n,z
y' = 12x + m2y + n2z
z' = lax + may + n3z
where the determinant
1, m, nx
% = + 1
l2 m2
l3 ma n3
This definition is valid for vectors of any number of dimensions with
real or complex coordinates; in N-dimensions, N X I matrices replace
the 3 X 1 matrices and so on.
1.3 Notation
Vectors are denoted by symbols in bold face type; scalars, by letters
in ordinary italics. When in the same discussion symbols are needed
both for a vector and for its magnitude (a scalar), the same letter is
used when the typeface makes the interpretation clear. When neces-
sary for clarity J A I is used to denote the magnitude of the vector A;
there will be no confusion if I a I is used for the absolute value (or modu-
lus) of the scalar a.
The common practice of indicating the complex conjugate of a com-
plex number by an overscore is extended to complex vectors; thus, if,
is, it contains four rows and three columns. If a set is ordered in only one sense
as is true of the set of coordinates of a vector, it is simply a matrix of one column
or, if we prefer, of one row.
4 VECTOR AND POLYADIC ANALYSIS
with x and y real, w = x + iy, then iV = x - iy whether w, x, y repre-
sent scalars, vectors or polyadics.
The coordinates of a vector A with respect to the set of rectangular
axes x, y, z in use will often be denoted by A., A , A..
When a single letter is used to designate a matrix, it will be enclosed
by double bars; thus, 11 A 11. A curled overscore will be used to indicate
the transposed matrix. For example, if
all au als 1
IIA11 a- an an azs
an an an
then
all a21 a0i
an an an
11 1111
an ass an
that is, the columns of (I A II are the rows of II 111.
Tensors and polyadics (cf. ¢2.5, 2.8) will be denoted by Gothic letters,
e.g., P.
