A Historical Study of Vector Analysis
C.T. Tai
Radiation Laboratory
Department of Electrical Engineering and Computer Science
The University of Michigan
Ann Arbor, Michigan
48109-2122
Technical Report RL 915
THE UNIVERSITY OF MICHIGAN
Radiation Laboratory
Department of Electrical Engineering
and Computer Science
Ann Arbor, Michigan 48109-2122
USA
May, 1995

An Historical Study of Vector Analysis
Contents................................................... ii
1. Introduction................................................ 1
2. Notations and Operators...................................... 2
2.1 Past and Present Notations in Vector Analysis..................... 4
2.2 Quaternion Analysis.............................................. 6
2.3 Dyadic Analysis.......................................... 9
2.4. Operators.............................................. 12
3. The Pioneer Works of J. Willard Gibbs (1839 - 1903).............. 16
3.1 Two Pamphlets Printed in 1881 and 1884........................ 16
3.2 Divergence and Curl Operators and Their New Notations.............. 21
4. Book by Edwin Bidwell Wilson Founded upon the Lectures of J. Willard
Gibbs........................................................ 24
4.1 Gibbs' Lecture Notes................... 24
4.2 Wilson's Book.................................... 25
4.3 The Spread of the Formal Scalar Product (FSP) and Formal Vector Product
(FVP)............................. 31
5. V in the Hands of Oliver Heaviside (1850 - 1925)................... 35
6. Shilov's Formulation of Vector Analysis.......................... 38
7. Orthogonal Curvilinear Systems.............................. 39
7.1 Invariance of the Differential Operators V, V and V in Orthogonal Curvilinear
Systems.................................. 39
7.2 Two Examples from the Book by Moon and Spencer............... 42
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An Historical Study of Vector Analysis
7.3 A Search for the Divergence Operator in Orthogonal Curvilinear Coordinate
Systems............................................. 45
7.4 The Use of V to Derive Vector Identities....................... 46
8. The Method of Symbolic Vector............................... 50
9. General Curvilinear Coordinate Systems....................... 58
9.1 Unitary Vectors and Reciprocal Vectors....................... 58
9.2 Gradient, Divergence and Curl in General Curvilinear a System.......... 61
10. Retrospect....................... 70
Acknowledgement......................................... 73
References................................................. 74
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1. Introduction
Vector analysis is an indispensable tool in the teaching and learning of electromagnetics,
hydrodynamics, and mechanics. In a book on the history of vector analysis [], Michael J. Crowe
made a very thorough investigation of the decline of quaternion analysis and the evolution of
vector analysis during the 19th century until the beginning of this century. The topics covered are
mostly vector algebra and quaternion analysis. He did not comment much on the technical aspects
of the subject from the point of view of a mathematician or theoretical physicist. For example, the
difference between the presentations of Gibbs and Heaviside, considered to be two founders of
modern vector analysis, is not discussed in Crowe's book, and less attention is paid to the history
of vector differentiation and integration, and to the role played by the del operator, V.
A short history of vector analysis is also found in several other books. For example, in a book by
Burali-Forti and Marcolongo [2] published in 1920, there are four historical notes in the appendix
entitled: On the definition of abstraction, On vectors, On vector and scalar (interior) products, and
On grad, rot, div. In another book published in 1965 by Moon and Spencer [3] there is a brief but
very critical review of the history of vector analysis from a technical perspective. Many of the
assertions in that book will be discussed later. One important reason for these two authors to
present vector analysis by way of tensor analysis is stated very firmly in the introduction of that
book [3, p. 9]:
The present book differs from the customary textbook on vectors in stressing the
idea of invariance under groups of transformations. In other words, elementary
tensor technique is introduced, and in this way, the subject is placed on the firm,
logical foundation which vector textbooks have previously lacked.
In Appendix C of that book [3, p. 323] they make the following comment about the del operator:
In reading the foregoing book [referring to their book], one may wonder why
nothing has been said about the operator V, which is usually considered such an
important part of vector analysis. The truth is that V, though providing the subject
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with fluency, is an unreliable device because it often gives incorrect results For
this reason - and because it is not necessary - we have omitted it in the body of the
book. Here, however, we shall indicate briefly the use of the operator V
These two quotations are sufficient to indicate that after decades of application of vector analysis
there seems to be no systematic treatment of the subject that could be considered satisfactory
according to these two authors. This observation is also supported by the fact that we have so far
no standard notations in vector analysis. Many books on electromagnetics, for example, use the
linguistic notations for the gradient, divergence, and curl - namely, grad ui, div f, and curl f,
while many others prefer Gibbs' notations for these functions, namely V i, V f, and V x f. Is
there a good explanation to the students why we do not yet have a universally accepted standard
notation besides saying: "It is a matter of personal choice."? In regard to Moon and Spencer's
comments about the lack of a firm, logical foundation in previous books on vector analysis, there
has been no elaboration. They do give an example of an incorrect result from using V to find the
expression for divergence in an orthogonal curvilinear coordinate system, but no explanation was
given as to the cause of such a wrong result. In fact, the views expressed by these two authors are
also found in many books treating vector analysis. These will be reviewed and commented upon
later.
In writing this essay, we have in mind the reader who already has an acquaintance with the subject
matter of vector analysis, and who feels the need for a critical scrutiny of what he or she has
already learned. Many students must have felt such a need, because the conventional curriculum
avoids thorough critical examination of many topics. The primary objective of many schools of
physical and engineering sciences is to teach the students how to use particular tools (such as
vector analysis) to formulate and to solve problems. It is usually felt that students in the hard
sciences, particularly at the undergraduate level, do not have the luxury of time to deal with the
logic and many of the fine points of subject fundamentals. Many of these subtle details are
overlooked in favor of developing skills in applying results, sometimes bluntly. This essay
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attempts to point out those inadequate or illogical treatments of some basic aspects of the subject
which have arisen in the past, and offers a more logical and systematic alternative for the reader to
consider.
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2. Notations and Operators
2.1 Past and Present Notations in Vector Analysis
In a book on advanced vector analysis published in 1924, Weatherburn [4] compiled a table of
notations in vector analysis which had been used up to that time. The names of the authors in that
table are: Gibbs/Wilson, Heaviside, Abraham, Ignatowsky, Lorentz, and Burali-Forti/Marcolongo.
In Moon and Spencer's Vectors (quoted in the previous chapter), published in 1965, there is also a
table of notations. The names of the authors in that table are: Maxwell, Gibbs, Gibbs/Wilson,
Heaviside, Gans, Lagally, Burali-Forti, Marcolongo, Phillips, and Moon/Spencer. Among these
authors, Gibbs, Wilson, Phillips, Moon, and Spencer are American. Maxwell and Heaviside
belong to the English schools. Abraham, Ignatowsky, Gans, and Lagally belong to the German
schools. Lorentz was a Dutch physicist and Burali-Forti and Marcolongo were Italians.
Ignatowsky was a native of Russia but was trained in Germany. For our study, we prepare
another list which contains several contemporary authors and some more notations in Table I. The
dyadic notation is added because we need it to characterize the gradient of a vector, which is a
dyadic function. A rudimentary introduction to dyadic analysis will be given after we present the
list of notations given below. In looking at this list, most readers will recognize the linguistic
notations grad u, div a, curl a, or rota for the three key functions. They are probably
accustomed to Gibbs' notations Vu, V.a, and V x a except that the period'.' in V.a is now
replaced by a dot'*' as in Wilson's notations, and his Greek letters for vectors are now commonly
replaced by boldface, Clarendon or equivalent fonts while the linguistic notations are used by
many authors in Europe and a few in the U. S. A. There is no doubt that Gibbs' notations have
been adopted in many books published in the U. S. A. We quote here two very well known books
in electromagnetic theory, one by Stratton, and another by Jackson Their treatises are well known
to many electrical engineers as well as physicists.
Historically, vector analysis was developed a few years after Maxwell formulated his monumental
work in electromagnetic theory. When he wrote his treatise on electricity and magnetism [5] in
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Table I: Notations
Author(s) Vectors Scalar Product Vector Product Dyadic Tensor
in 3-space in 3-space
Maxwell [5] a,p S p Vcap
/,~
Gibbs [6] a,8/ a. a x f afp
Wilson [7] a,b a b axb ab
Heaviside [8] a,b ab Vab
Gans [9] W ( s {[A
Burati-Forti/ a,b ax b aAb -
Marcolongo [2]
Stratton[10] a,b a.b axb - Ti
Jackson [11] a,b a.b axb T T7
Moon/ a,b a'b ax b - T
Spencer [12]
Author(s) gradient of gradient of divergence of curl or rot Laplacian of Laplacian of
a scalar a vector a vector of a vector a scalar a vector
Maxwell [5] Vz - -SVp VVp V2u
Gibbs[6] Vu Va V.a Vxa V.Vu V.Va
Wilson [7] Vuz Va V.a Vxa V.Vu V.Va
Heaviside [8] Vu V.a Va; div a VVa; curl a V2u V2a
Gans [9] Vu; grad z - V a; div a V x a; rot a A u Aa
Burati-Forti/ gradu - div a rot a A2u A' a
Marcolongo [2]
Stratton[10] VIJ Va V-a Vxa V2u V2a
Jackson[11] Vu Va V a Vxa V2u Va
Moon/ grad u - div a curl a V2u *a
Spencer [12]
* Upper case script symbols are used here in place of capital German letters originally used by Maxwell and Gans.
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1873, vector analysis was not yet available. Its forerunner, quaternion analysis, developed by
Hamilton (1805 - 1865) in 1843, was then advocated by many of Hamilton's followers. It is
probably for this reason that Maxwell wrote an article in his book (Article 618) entitled
"Quaternion Expressions for the Electromagnetic Equations." Maxwell's notations on our list are
based on this document. Actually, he used very little of these notations in the entire book and in
his papers published elsewhere.
The notation used by Heaviside is not conventional from the present point of view. His notation
for the scalar product and the divergence does not have a dot and his notation for the curl is of
quaternion form like Maxwell's. The notations used by Burati-Forti and Marcolongo are obsolete
now. Occasionally we still see the notation a A b for the cross product in European books. As a
whole, we now have basically two sets of notations in current use: the linguistic notation and
Gibbs' notation. The names of Moon and Spencer are included on our list primarily because these
two authors considered the use of V to be unreliable and they frequently emphasize their view
that the rigorous method of formulating vector analysis is to follow ihe route of tensor analysis. In
addition, their new notation for the Laplacian of a vector function will be a subject of detailed
examination in the chapter on orthogonal curvilinear systems.
2.2 Quaternion Analysis
The rise of vector analysis as a distinct branch of applied mathematics has its origin in quaternion
analysis. It is therefore necessary to review briefly the laws of quaternion analysis to show its
influence upon the development of vector analysis and also explain the notations in the previous
list. Quaternions are complex numbers of the form
q = w + ix + jy + kz (2.1)
where w, x, y, and z are real numbers, and i, j, and k are unit vectors, directed along the x, y, and z
axes respectively. These unit vectors obey the following laws of multiplication:
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k = A, jk = i, ki = j
ji =-k, kj= -i, ik=- - (2.2)
ii= -jj kk= -
We must not at this stage associate the above relations with our current laws in vector analysis.
We consider the subject as a new algebra, which is indeed' the case. The product of the
multiplication of two quaternions a and p in which the scalar parts it' and it"' are zero is obtained
as follows:
We let
ac= iD + jD + kD,
p=iX + jY + kZ
then
op = -(D, X + DY + D3Z)
-( - 3Z)
+ i(D.Z - D3Y)
(2.3)
+ j(D3X- D,Z)
+k(DY- DX)
The resultant quaternion, ap, has two parts, one scalar and one vector. In Hamilton's original
notation they are:
S. p = -(DX + D2Y + D3Z) (2.4)
V. ap= i(D2Z - D3Y)
+j(D3X-D Z) (2.5)
+ k(DY - DX)
The period between S or V and up can be omitted without causing any ambiguity. When one
identifies a as V, Hamilton's del operator, that is:
= V= i-+ + k (2.6)
then
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SVP-=- -4 - (27)
V {' &J
C( C C7) \(2.8)
+J
SYVp is the quaternion notation used by Maxwell for the negative of the divergence of p and he
called it the convergence of p. He used the quaternion notation VVp for the curl of p. The term'curl' was his creation, and it is now a standard name. According to Crowe [1, p. 142] the term'divergence' was originally due to William Kingdom Clifford (1845 - 1879) who was also the first
person to define the modern notations for the scalar and vector products. However, his original
definition of the scalar product is the negative of the modem scalar product. In the list of
notations, we notice that Heaviside used the quaternion notation for the curl even though he was
opposed to quaternion analysis. In one of his writings [8, p. 35] he concurred with Gibbs'
treatment of vector analysis but criticized Gibbs' notations without offering a reason. Heaviside's
remark will be quoted and discussed in Sec. 3.1.
We should mention that the long controversy between quaternionists led by Tait and the
proponents of the then new vector analysis led by Gibbs was covered in great detail by Crowe [1].
Such stories are very educational to young scientists and engineers. The other topic which needs
to be reviewed deals with the dyadic notation which was used by Gibbs, Wilson, and Jackson but
not the other authors in the previous list. An understanding of this notation is necessary in order
to explain the Laplacian of a vector function, particularly in the general curvilinear coordinate
system. Additionally, dyadic analysis is a natural extension of vector analysis. Problems
formulated using tensor analysis in a three-dimensional Euclidean space can be handled by dyadic
analysis in a relatively simpler format.
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2.3 Dyadic Analysis
For the time being, we will provide the basic formulas in dyadic analysis to be used in our
investigation of the past works. In vector algebra, a vector function denoted by PF is represented
in a Cartesian coordinate system by:
F- FZIi, i=1-2,3 (2.9)
where x,, with i = 1,2,3 denotes the unit vector in the x,y,z direction and FI corresponds to the
components of F in the x,y, and z direction. It is understood that the summation goes from i= to
3. A dyadic function or a dyadic for short, denoted by F is defined in the same Cartesian
coordinate system by
F = -Fjj, j = 1,2,3 (2.10)
where
Fj = EIx,, i=1,2,3 (2.11)
i
denotes three independent or distinct vector functions. The relative position of Fj and x3 in
(2.10) must be maintained in that order, and one is not supposed to interchange the ordering of
these two vectors. In other words, the commutative rule does not apply to (2.10). When (2.11) is
substituted into (2.10), we obtain:
F =ZZ Fixixj (2.12)
i j
Equations (2.10) - (2.12) contain the definition of a dyadic in a Cartesian coordinate system.
There are nine scalar components of F. The doublets xixj, juxtaposed together i and X j with
= (1,2,3), are called dyads, and there are nine of them too. The dyads are not commutative,
that is:
Xi Xj.x,, IlJ
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_ - _~= T
The transpose of a dyadic F, denoted by F is defined by
T
J
= FZi,j,- i (2.13)
i J
= ZZF,,~,j
i j
There are two scalar products between a dyadic and a vector. The anterior scalar product
between a vector a and a dyadic F, denoted by a F, is defined by
a.F-Z(a F j)j - ZZ i' (2.14)
j i j
which is a vector. The posterior scalar product between a and F, denoted by F a, is defined by
F=.a =ZFr(xj a)- ZZ= ajF (2.15)
j i j
which is also a vector. In general,
a. F F'a
The two products are equal when F is a symmetric dyadic characterized by Fj = F.T. There are
two vector products possible between a and F. The anterior vector product is defined by
a x= (a x F )xEI = F (a xxi. ) (2.16)
i i j
which is a dyadic. The posterior vector product is defined by
Fxa= i(xj ~xa)= F "xi(xj xa) (2.17)
j i j
which is another dyadic.
The gradient of a vector function in a Cartesian system, denoted by VF, is defined by
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V/- = E> r — l = /> >x — c
JF C(2-. 18)
= L —,x X
I J I I
which is a dyadic. We will derive the expression for the gradient of F in general curvilinear
systems later. This introduction of dyadic analysis is merely presented to show that the gradient of
a vector function as tabulated in the list of notations in Table I is a dyadic. It should be mentioned
that a dyadic can also be written in the form
F=A B (2.19)
Knowing F and with a specified A we can find B as follows:
~2A.F=A AB A - 2B
Hence,
B=- A F (2.20)
where A F is the anterior scalar product between A and F. If B is specified, we can find A as
follows:
F-B=A B.B=AB2
hence
A=-2 FB (2.21)
B
where F B is the posterior scalar product between B and F. In the list of notations, Gibbs and
Wilson use the A B form for the dyadics. The next topic to be reviewed deals with operators.
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2.4 Operators
For our convenience we would like to discuss in sufficient detail, the classification and the
characteristics of a number of operators appearing in this study. We will focus on unary and
binary operators and will consider such operators in cascade or compound arrangements as the
complexity of the case at hand requires.
A unary operator involves only one operand. A binary operator needs two operands, one anterior,
and another posterior. A cascade operator could be unary or binary. As an example we consider
the derivative symbol to be a unary operator. When it operates on an operand P, it produces
the derivative, -. In some writings, the operator is denoted by D,. The operand under
consideration can be a scalar function of x and other independent variables or a vector function,
or a dyadic function, that is,
dP a da dF
a Be' Be' e
are all valid applications of the unary differential operator.
The partial derivative of a dyadic function in a Cartesian system is defined by
dF dF,.
(2.22)
8F..
i j
We list in Table II below, several commonly used unary operators and their possible operands.
The function a in the weighted differential operator a- is assumed to be a scalar function. A
vector operator such as a- can operate on a dyadic that would yield a'tridic' - a quantity which
is not included in this study. The last operator in the table is the del operator or the gradient
operator. It can be applied to an operand which is either a scalar or a vector.
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Table II: Valid Application of Some Unary Differential Operators
/C'- C^"~ _ — 8 _^
Operator a a- a = ( 2.3)
Type of Operand b, b, b b, b, b 3, b h, h
cb 8b 8h h b c b b h -b h - b
Result,, a,a a — a, a-, Vb, Vh
8x 8x ax x x ex Afx x c x
A binary operator requires two operands. In arithmetic and algebra we have four binary operators:. (addition), - (subtraction), x (multiplication), and - (division). In these cases, we need two
operands, one anterior and another posterior, as in 2+3, 4-3, 5 x 3, and 6 3. It should be
remarked that the symbols + and - are also used to denote'plus' and'minus' signs. For example,
-a = 1a' when a is negative. In this case the minus sign is not considered to be a binary operator in
our classification, but rather as a unary'sign change' operator. The two binary operators involved
frequently in our work are the dot (*) and the cross (x). They appear in Gibbs' notations for the
scalar and vector products, that is, a b and a x b. We consider the dot and the cross as two
binary operators, and their operands, one anterior and one posterior, must be vectors, that is
A.B and AxB
The dot operator is not the same as the multiplication operator in arithmetic and the cross
operator is not the same as the multiplication operator, although we use the same symbol.
According to the definitions of the scalar and vector products,
A.B=B. A = 1ABcos (2.24)
AxB=-BxA= A lBsins0c (2.25)
where 0 is the angle measured from A to B in the plane containing these two vectors and c is the
unit vector I to both A and B and is pointed in the right-screw advancing direction when A turns
into B. The dot and the cross can also be applied to operands where one of them or both are
dyadics. Thus, we have
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A B, B.A,
ABB_ _=__(2.26)
Ax B,Bx A,A B,B A
The first two entities are vectors and the remaining four are dyadics.
The last group of operators are called cascade or compound operators. Of particular concern in
this study is the proper treatment of a pair of operators of different types, which are applied
sequentially. When one of the operators is a scalar differential unary operator, and the other is a
vector binary operator, there arise a number of hazards in their application which, if not properly
treated, could lead to invalid results. Several commonly used cascade operators are of the form:
V, x-, xV (2.27)
ey Oy
These operators also require two operands; the anterior operand must be a vector and the
posterior operand must be compatible with the part in front. Thus we can have
- dB - dB
A —, A.
A.- Vu AVB; (2.28)
- dB dB
Ax-, Ax --
AxVu, AxVB
In (2.27) the unary operator and V, and the binary operators* and x are not commutative;
dy
hence, the following combinations or assemblies are not valid cascade operators.
d (
V., — x Vx (2.29)
8y y'
These assemblies are formed by interchanging the positions of the symbols in (2.27). They are not
operators in the sense that we cannot find an operand to form a meaningful entity. For example,
AA, -B; V.A, VB;
~ ^ _ ^ (2.30)
- xA, - xB; VxA, VxB
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do not have any meaningful interpretation.
The reader has probably noticed that there are two assemblies, V A and V x 4 in (2.30) which
correspond to Gibbs' notation for the divergence and curl This is very true, but that does not
mean that V A is a scalar product between V and A, nor is V x A a vector product between V
and A. In fact, this is a central issue in this study to be examined very critically in the following
chapters. We now have the necessary tools to investigate many of the past presentations of vector
analysis.
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3. The Pioneer Works of J. Willard Gibbs (1839 - 1903)
3.1 Two Pamphlets Printed in 1881 and 1884
Gibbs' original works on vector analysis are found in two pamphlets entitled Elements of V'ector
Analysis [6], privately printed in New Haven. The first consists of 33 pages published in 1881 and
the second of 40 pages published in 1884. These pamphlets were distributed to his students at
Yale University and also to many scientists and mathematicians including Heaviside, Helmholtz,
Kirchhoff, Lorentz, Rayleigh (Lord), Stokes, Tait, and Thomson (J. J.) [12, Appendix IV]. The
contents are divided into five chapters and a note on bivectors:
Chapter I. Concerning the algebra of vectors
Chapter II. Concerning the differential and integral calculus of vectors
Chapter III. Concerning linear vector functions
Chapter IV. Concerning the differential and integral calculus of vectors (Supplement to
Chapter II)
Chapter V. Concerning transcendental functions of dyadics
A Note on bivector analysis
The most important formulations for our immediate discussions are covered in Articles 50 - 54
and 68 - 71 which are reproduced below:
Functions of Positions in Space
50. Def - If u is any scalar function of position in space (i.e., any scalar quantity having
continuously varying values in space), Vu is the vector function of position in space which
has everywhere the direction of the most rapid increase of u, and a magnitude equal to the
rate of that increase per unit of length. Vu may be called the derivative of u, and u, the
primitive of Vu.
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We may also take any one of the Nos. 51, 52, 53 for the definition of Vu.
51. If p is the vector defining the position of a point in space,
du = Vu dp
52. V du du du
52. VI/= i-+j-+-k- (3.1)
dx dy dz
du du du=
53. -= i VT, -= j V-, -= k Vu
dx dy dz
54. Def - If c is a vector having continuously varying values in space,.d do dc
V. = i.-+.-+ k.- (3.2)
dx dy dz
do dco do
Vxco=ix +jx +kx — (3.3)
dx dy dz
V co is called the divergence of co and V x c its curl.
If we set
o =Xi+ Yj +Zk,
we obtain by substitution the equation
dX dY dZ
V-c= —-d - +.+ (3.4)
dx dy dz
and
(dZ dY' (dX dZ\ (dY dX\
Vx o=/ - - - +j -- - - + k\ - - -- (3.5)
{ dy dz) i dz dx) dx dy )
which may also be regarded as defining V co and V x o.
Combinations of the Operators V, V, and V x
68. If c is any vector function of space, V - V x c = 0. This may be deduced directly from
the definition of No. 54.
The converse of this proposition will be proved hereafter.
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69. If t is any scalar function of position in space, we have by Nos. 52 and 54
d2 hc d:
V.Vu=d. + d + 2 j (3.6)
70. Def - If aw is any vector function of position in space, we may define VV y t by the
equation
V Vo- Va ^ +: + j' 0w (3.7)
a&C -dy dz
the expression V V being regarded, for the present at least, as a single operator when
applied to a vector. (It will be remembered that no meaning has been attributed to V
before a vector.) It should be noticed that if
= iX + jY + kZ,
V. V = iV VX+ jV.VY+kV VZ, (3.8)
that is, the operator V * V applied to a vector affects separately its scalar components.
71. From the above definition with those of Nos. 52 and 54 we may easily obtain
VVo= VV.o-VxVx w (3.9)
The effect of the operator V - V is therefore independent of the direction of the axes used
in its definition.
In quoting these sections we have changed Gibbs' original notation for the divergence from V.wc
to V-o, i.e., the period has been replaced by a dot. In addition, some equation numbers have been
added for our reference later on.
After Gibbs revealed his new work on vector analysis he was attacked fiercely by Tait, a chief
advocate of the quaternion analysis, who stated [13, Preface]:
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Even Prof Willard Gibbs must be ranked as one of the retarders of quaternion
progress, in virtue of his pamphlet on vector analysis, a sort of hermaphrodite
monster, compounded by the notations of Hamilton and Grassman.
This infamous statement has been quoted by many authors in the-past. Gibbs' gentlemanly but firm
response to Tait's attack was [14]:
The merit or demerits of a pamphlet printed for private distribution a good many
years ago do not constitute a subject of any great importance, but the assumption
implied in the sentence quoted are suggestive of certain reflections and inquiries
which are of broad interest; and seem not untimely at a period when the methods
and results of the various forms of multiple algebra are attracting so much
attention. It seems to be assumed that a departure from quaternionic usage in the
treatment of vectors is an enormity. If this assumption is true, it is an important
truth; if not, it would be unfortunate if it should remain unchallenged, especially
when supported by so high an authority. The criticism relates particularly to
notations, but I believe that there is a deeper question of notions underlying that of
notations. Indeed, if my offense had been solely in the matter of notation, it would
have been less accurate to describe my production as a monstrosity, than to
characterize its dress as uncouth.
Gibbs then continued on to explain the advantage of his treatment of vector analysis in
comparison to quaternion analysis. In the final part of that paper he stated:
The particular form of signs we adopt is a matter of minor consequence. In order
to keep within the resources of an ordinary printing office, I have used a dot and a
cross, which are already associated with multiplication, which is best denoted by
the simple juxtaposition of factors. I have no special predilection for these
particular signs. The use of the dot is indeed liable to the objection that it interferes
with its use as a separatrix, or instead of a parenthesis.
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An Histoncal Study of Vector Analysis page 20
Although Gibbs considered his choice of the signs or notations a matter of minor importance,
actually it had a tremendous consequence as will be shown in this study. Before we discuss it, a
comment from Heaviside, generally considered by the scientific community as a co-founder with
Gibbs of the modern vector analysis, should be quoted. During the peak of the controversy
between Tait and Gibbs, Heaviside made the following remark [8, p. 35]:
Prof W. Gibbs is well able to take care of himself. I may, however, remark that the
modifications referred to are evidence of modifications felt to be needed, and that
Prof. Gibbs' pamphlet (Not published, New Haven, 1881-4, p.83), is not a
quaternionic treatise, but an able and in some respects original little treatise on
vector analysis, though too condensed and also too advanced for learners' use, and
that Prof Gibbs, being no doubt a little touched by Prof Tait's condemnation, has
recently (in the pages of Nature) made a powerful defense of his position. He has
by a long way the best of the argument, unless Prof Tait's rejoinder has still to
appear. Prof Gibbs clearly separates the quaternionic question from the question
of a suitable notation, and argues strongly against the quaternionic establishment of
vector analysis. I am able (and am happy) to express a general concurrence of
opinion with him about the quaternion and its comparative uselessness in practical
vector analysis. As regards his notation, however, I do not like it. Mine is Tait's,
but simplified, and made to harmonize with Cartesians.
There are two implications in Heaviside's remark which are of interest to us. When he considered
Gibbs' pamphlet to be too condensed it implies that some of the treatments may not have been
obvious to him (or may not even have been comprehended by him). Secondly, he stated dislike for
Gibbs' notations but without giving his reason(s). The fact that Heaviside. used some of Tait's
quaternionic notations seems to indicate that he did not approve of Gibbs' notations at all. We
now believe that many workers, including Heaviside, did not appreciate the most eloquent and
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An Historical Study of Vector Analysis page 21
complete theory of vector analysis formulated by Gibbs. For this reason, we would like to offer a
digest of Gibbs' work so we can have a clear understanding of his formulation.
3.2 Divergence and Curl Operators and Their New Notations
The basic definitions of the gradient, divergence, and the curl formulated by Gibbs are given by
(3.1), (3.2), and (3.3). For convenience, we will make some changes in symbols to allow the
convenience of using the summation sign. These changes are:
x, Y, zto x, x2, x3
i, j, k, toxl, x2, x-3
The old total derivative symbols will be replaced by partial derivatives and the Greek letters for
vectors by boldface letters. Thus, Eqs. (3.1) - (3.3) become:
Ol
V2= x - (3.10)
V. F=x.- - (3.11); dxi
dF
i Xi
VxF=Zk xx (3.12)
It is understood that the summation goes from i = 1 to 3
The most important information passed to us by Gibbs concerns the nomenclature for the
notations in these expressions. In the title preceding Article 68 quoted previously, he designated
V, V and V x as operators. If we examine the expressions given by (3.10), (3.11), and (3.12) it
is quite obvious that the gradient operator or the del operator is unmistakably given by
V = ^i- (3.13)
i oxi
For the divergence, Gibbs used two symbols, a del followed by a dot, to denote his divergence
operator. For the curl, he used a del followed by a cross to denote the curl operator. If we
examine the expressions for the divergence and the curl defined by (3.11) and (3.12) it is clear
that his two notations mean:
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An Historcal Study of Vector Analysis page 22
(V').)G Z v (3.14)
-, -cx
C X
(V x)G -E.', (3.15)
We emphasize this point by labeling his two notations with a subscript'(;', and we use an arrow
instead of an equal sign to denote'a notation for.'
According to our classification of the operators in Chapter 2, Gibbs' (V.) and (V x)G are not
compound operators; they are assemblies used by Gibbs as the notations for the divergence and
curl. On the other hand, the terms at the right side of (3.14) and (3.15) are indeed compound
operators according to our classification. Since these operators are distinct from the gradient
operator we will introduce two notations for them. They are:
V-=Xi - (3.16)
cxi
v ~= E xi X^x (3.17)
They are called, respectively, the divergence operator and the curl operator. Although these
operators are so far defined in the Cartesian coordinate system we will demonstrate later that they
are invariant to the choice of coordinate system. One important feature of V and V is that both
these operators are independent of the gradient operator V. In other words, V is not a constituent
of the divergence operator nor of the curl operator. These two symbols are suggested by the
appearance of the dot or the cross in between the unit vectors xi and the partial derivatives
Oxi
of the V operator as defined by (3.13). In Gibbs' notations, (V.)G and (V x)G, V is a part of his
notations for the divergence and the curl that leads to a very serious misinterpretation by many
later users and which is a key issue in our study. With the introduction of these two new
notations, Eqs. (3.1) to (3.9) become:
8u
Vu = i. (3.18)
d 8xi
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An Historical Study of Vector Analysis page 23
VF=Y-i6F (3 19)
V F = (3.20)
i F'X,
V F ^-', X (3.21)
CXi. X Jdx. Xk) (3.22)
(i,j, k)= (1,2,3) in cyclic order
Vt- = 2 (3.23)
VVF = 2 (3.24)
xi
VVF= i VVF (3.25)
VVF =VVF- VVF (3.26)
In these formulas the del operator only enters in the gradient of a scalar, (3.18), or of a vector,
(3.24) - (3.26). Except for the notations for the divergence and the curl, we have not changed the
content of Gibbs' work at all. These equations will be used later in our study of other people's
presentations.
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An Historical Study of Vector Analysis page 24
4. Book by Edwin Bidwell Wilson Founded upon the Lectures of J. Willard
Gibbs
4.1 Gibbs' Lecture Notes
In 1901 the first book on vector analysis by an American author was published. The book was
written by Wilson [7], then an instructor at Yale University, and founded upon the lectures of
Gibbs. According to the general preface of that book, the greater part of the material has been
taken from the course of lectures on Vector Analysis delivered annually at Yale University by
Professor Gibbs. There is one historical document well-kept at the Sterling Memorial Library of
Yale University which is the record of the lectures [15]. It is a cloth-bound book of notes, handwritten in ink on 8-1/2" by 11" ruled paper, consisting of fifteen chapters covering 289 - plus
pages. The title page and the table of contents are:
Lectures Delivered upon
Vector Analysis
and its
Applications to Geometry and Physics
by
Professor J. Willard Gibbs 1899-90
reported by Mr. E. B. Wilson
Table of contents page
Ch. 1 Fundamental Notions and Operators 1
Ch. 2 Geometrical Applications of Vector Analysis 11
Ch. 3 Products of Vectors 25
Ch. 4 Geometrical Applications of Products 50
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An Historical Study of Vector Analysis page 25
Ch. 5 Crystallography 62
Ch. 6 Scalar Differentiation of Vectors 72
Ch. 7 Differentiating and Integrating Operations 83
Ch. 8 Potentials, Newtonians, Laplacians, Maxwellians 110
Ch. 9 Theory of Parabolic Orbits 125
Ch. 10 Linear Vector Functions 164
Ch. 11 Rotations and Strains 200
Ch. 12 Quadratic Surfaces 223
Ch. 13 Curvature of Curved Surfaces 234
Ch. 14 Dynamics of a Solid Body 261
Ch. 15 Hydrodynamics 276
4.2 Wilson's Book
Presumably, Wilson's book (436 pages) is mainly based on these notes. It was mentioned in the
preface of Wilson's book that some use, however, has been made of the chapters on vector
analysis in Heaviside's Electromagnetic Theory (1893) and in Foppl's lectures on Maxwell's
Theory of Electricity (1894). Apparently, Gibbs himself was not involved in the preparation of
this book. We quote here two paragraphs in the preface by Professor Gibbs:
I was very glad to have one of the hearers of my course on Vector Analysis in the
year 1899-1900 undertake the preparation of a text-book on the subject.
I have not desired that Dr. Wilson should aim simply at the reproduction of my
lectures, but rather that he should use his own judgment in all respects for the
production of a text-book in which the subject should be so illustrated by an
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adequate number of examples as to meet the wants of students of geometry and
physics.
In the general preface, Wilson stated:
When I undertook to adapt the lectures of Professor Gibbs on Vector Analysis for
publication in the Yale Bicentennial Series, Professor Gibbs himself was already so
fully engaged in his work to appear in the same series, Elementary. Principles in
Statistical Mechanics, that it was understood no material assistance in the
composition of this book could be expected from him. For this reason he wished
me to feel entirely free to use my own discretion alike in the selection of the topics
to be treated and in the mode of treatment. It has been my endeavor to use the
freedom thus granted only in so far as was necessary for presenting his method in
text-book form.
One very important remark by Wilson is found in the preface:
It has been the aim here to give also an exposition of scalar and vector products of
the operator V, of divergence and curl which have gained such universal
recognition since the appearance of Maxwell's Treatise on Electricity and
Magnletism, slope, potential, linear vector functions, etc. such as shall be adequate
for the needs of students of physics at the present day and adapted to them.
We would like to point out here that in Gibbs' pamphlets and in the lecture notes reported by
Wilson, there is no mention of the scalar and vector products of the operator V. We believe this
concept or interpretation was created by Wilson and unfortunately, it has had a tremendously
detrimental effect upon the learning of vector analysis within the framework of Gibbs' original
contributions.
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In explaining the meaning of the divergence of a vector function Wilson misinterpreted Gibbs'
notation for this function, namely V F After definingl the V operator for the gradient in a
Cartesian system as
C C C
V=-i-+ j +k (4.1)
c x )cy cz
he stated in Sec. 70, p. 150 of Wilson's book [7]:
Although the operation VV has not been defined and cannot be at present, tcwo
formal combinations of the vector operator V and a vector function V mav be
treated. These are the (formal) scalar product and the (formal) vector product of V
into V. They are:
V~V= - k+j- +k.V (4.2)
VxV= i-+j +k- xV (4.3)
Ox d'a OZ
The differentiations-,, -, being scalar operators, pass by the dot and the
t6x ffy fZ.
cross, that is
V V e V 8V)
V V= -..+j k (4.4)
e x y dz )
VxV= i x d +j d V+kx d (4.5)
They may be expressed in terms of the components V, YV, 1V of V
We have identified the equations with our own numbers. In order to compare these expressions
with Gibbs' expressions now described by (3.19) to (3.22), we again, will change the notations for
V; x, y, z; i,j, k; to F; x, x, x,3;, 2, 3; and V-Vand VxV to VF and VF. Eq. (4.1) to
(4.5) become:
V=-"i (4.6)
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An Historical Study of Vector Analysis page 28
VF= Z, cj JF (4.7)
-, F= Z X XF (4.8)
vF=X.-X F (4.9)
Fxi
v F X=.'i x — (4.10)
i dxi
Equations (4.6), (4.9), and (4.10) are identical to Gibbs' (3.13), (3.19), and (3.21). However,
(4.7) and (4.8) are not found in Gibbs' works. Wilson obtained or derived (4.9) and (4.10) from
(4.7) and (4.8). The derivation involves two crucial steps or assumptions. First, he considers
Gibbs' notations V F and V x F as'formal' scalar and vector products between V and F. In the
following we will refer to this model as the FSP (formal scalar product) and FVP (formal vector
product). He did not explain the meaning of the word'formal'. Secondly, after he formed the FSP
and FVP he let the differentiation pass by the dot and the cross with the argument that the
Oxi
differentiations, (i = 1,2,3) are scalar operators. The statement appears to be quite firm. But
Oxi
standard books on mathematical analysis do not have such a theorem. Later on, [7, p. 152]
Wilson attempts to soften his attitude by saying:
From some standpoints objections may be brought forward against treating V as a
symbolic vector and introducing V V and V x V as the symbolic scalar and vector
products of V into V respectively. These objections may be avoided by simply
laying down the definition that the symbol V. and Vx, which may be looked upon
as entirely new operators quite distinct from V, shall be
e9V. 9V 8V
V.V = -/-+ j.+ +k- (4.11)
Ax ay dz
and
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An Historical Study of Vector Analysis page 29
VxV/ix - +J -X +kx - (4.12)
CX C t' C Z
But for practical purposes and for remembering formulas, it seems by all means
advisable to regard
V = + —+ J kdx cy' z
as a symbolic vector differentiator. This symbol obeys the same laws as a vector
just in so far as the differentiations,, -, obey the same laws as ordinary
ox 8dy z
scalar quantities.
The contradictions between the above statement and the FSP and FVP assertion seems quite
evident. Equations (4.11) and (4.12), of course, are the same as Gibbs' (3.1 1) and (3.12) with V
replaced by F and x, y, z; and i, j, k; by x,, x,, x3 and x,,:, x3. The difference is that Gibbs
never spoke of a FSP and FVP but Wilson introduced these concepts to derive the expressions for
div F and curl F by imposing some non-valid manipulations. What is the consequence? Many
later authors followed his practice and encountered difficulties when the same treatment was
applied to orthogonal curvilinear coordinate systems. Before we discuss this topic, Heaviside's
treatment of vector analysis, particularly his handling of V should be reviewed and commented
upon.
We have pointed out that Gibbs' pamphlets were communicated to Heaviside. On the other hand,
Wilson also mentioned some use of Heaviside's treatment of vector analysis in his book
Electronmaginetic Theory (1893) in his preface. The exchange between Heaviside and Wilson was
therefore, mutual. However, Heaviside goes his own way in presenting the same topics. Before
we turn to the next chapter, Wilson's FSP and FVP model will be analytically examined.
If we start with one of Gibbs' definitions of divergence, without using his notation but rather by
using the linguistic notation, i.e.,
div F =y - (4.13)
9xd
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An Historical Study of Vector Analysis page 30
then by substituting F, = -,F into (4.13) we find
J(_i - F)
div F = Ccx
(4.14)
Zeta CeF C^v
C Xi CXi
Since 0 (4.14) reduces to
Since = 0 (4.14) reduces to
exi
div F = - (4.15)
which is obviously not equal to
8axi- F
or V F. This is a proof of the lack of validity of the FSP. A similar proof can be executed with
respect to the FVP. Another demonstration of the fallacy of a FSP is to consider a'twisted'
differential operator of the form
V, = 2 - + X3-+ X - (4.16)
-xl 6Ox, aX3
and a'twisted' vector function defined by
F = 2F; + 3F=2 + X1F3 (4.17)
If the FSP were a valid product then by following Wilson's pass-by procedure we obtain
aF aFF aF3
Vt.Ft =, x + + (4.18)
dxl OX2 dX3
In other words, div F is now treated as the formal scalar product between Vt and Ft. The result
is the same as Wilson's FSP between V and F. Such a manipulation is, of course, not a valid
mathematical procedure. We have now refuted Wilson's treatment of div F and curl F based on
the FSP and FVP. The legitimate compound differential operators for the divergence and the curl
are, respectively, V and V defined by (3.16) and (3.17). (V )G and (V x)G are merely Gibbs'
notations suggested for the divergence and the curl. They are not operators.
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4.3 The Spread of the Formal Scalar Product (FSP) and Formal Vector Product (FNVP)
Being the first book on vector analysis published in 1901 in the U S. A., Wilson's book became
very popular. The 8th reprinting was made in 1943 and a paperback reprint by Dover Publications
appeared in 1960. Many later authors freely adopted Wilson's presentation using the FSP and
FVP to derive the expressions for divergence and curl in the Cartesian coordinate system. We
have found over fifty books [16] containing such a treatment. We now quote herein a few
examples to show Wilson's influence.
1.) In the book Advanced Vector Analysis by Weatherburn [4] published in 1924, we find the
following statement:
To justify the notation V-, we have only to expand the formal products according
to the distributive law, then
v.f -= f ai -f= =divf
We shall remark here that any distributive law in mathematics should be proved. In this case, there
is no distributive law to speak of because the author is dealing with an assembly of mathematical
symbols and V. is not a compound operator. Incidentally, Weatherburn's book appears to be the
first book published in England wherein Gibbs' notations, but not Heaviside's, have been used in
addition to the linguistic notations, namely, grad 1u, div f, and curl f.
2) A German book by Lagally [17] published in 1928 contains the following statement on p. 123:
The rotation (curl) of f is denoted by the vector product between V with field
function f... and
divgradf= V.V f = i j ji ~}xd =V2f
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An Historical Study of Vector Analysis page 32
The above is an English translation of the original text in German. It is seen that a term like
x- l.IN - is an assembly of symbols. It is not a compound operator.
CXl CX1
3.) In a book by Mason and Weaver [18, p. 336] we find the following statement:
The differential operator V can be considered formally as a vector of components
-,-,-, so that its scalar and vector products with another vector may be
&x ay 5z
taken.
In comparison with Wilson's treatment Mason and Weaver have used the word'formally' to be
associated with V and then speak of scalar and vector products with vector functions.
4.) In a book Applied Mathematics by Schelkunoff [19, p. 126], the author first derived the
differential expression for the divergence based on the flux model; then he added:
In Sec. 6 the vector operator del was introduced. If we treat it as a vector and
multiply it by a vector F, we find
V F Xi jF div F
fd to secy t Ca ten ordxia
For this reason V. may be used as an alternative for div; however, the notation is
tied too specifically to Cartesian coordinates.
There are two messages in this statement: the first one is his acceptance of the FSP as a valid
entity. The second one is his implication that FSP only applies to the Cartesian system. Actually,
the divergence operator, V, is invariant with respect to the choice of the coordinate system, a
property to be demonstrated later, but V.' is an assembly, not an operator. Only by means of an
illegitimate manipulation does it yield the differential expression for the divergence in the
Cartesian coordinate system.
5.) From a well-known book by Feynman, Leighton and Sands [20, p. 2-7] we find the following
statement:
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An Historical Study of Vector Analysis page 33
Let us try the dot product between V with a vector field that we know, say f we
write
V.f=Vjx+V f,+Vf
v. -f = +vf + fz
v.yf= +
cx Jy 8z
The authors remarked on the same page before the above statement:
With operators we must always keep the sequence right, so that the operations
make the proper sense...
This remark is very important. Our discussion and use of the operators in chapter 2, particularly
that related to the compound operators, closely adheres to this principle. In the case of Gibbs'
notation, V -, we are faced with a dot symbol after V, so that the differentiation cannot be
applied to f, it is blocked by a dot in the assembly. Thus, the authors seem to have violated their
own rule by trying to form a dot product or FSP.
6.) In the English translation of a Russian book by Borisenko and Tarapov [21, p. 157] we find
the following statement:
The expression (4.29) V = Eik dx for the operator V implies the following
representation for the divergence of A:
div A= Ak =i- = V.A
dXk JXk
A coordinate-free symbolic representation of the operator V is
V(...) Lim n(..)dS (4.19)
V-0 s,
where (..) is some expression (possibly preceded by a dot or a cross) on which the
given operator acts. In fact, according to (4.31) and (4.29),
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An Historcal Study of Vector Analysis page 34
grad o= Lirn- iS, (4.20),' —o (r /d
div A - Limn A.ida (4. 1)
From the above statement we see that the two authors believe the validity of the FSP. Their
(4. 19) also implies that they consider V as a constituent of the divergence and the curl in addition
to comprising the gradient operator. The formula described by (4.19) appeared earlier in the book
by Gans [22, p. 49, Sixth Edition] who used both Gibbs' notations and the linguistic notations in
this edition.
There are several authors presenting V as defined by (J/dxi))i instead of E ki(/J xi), and
the Laplacian, defined by div grad, is often treated as the scalar product between two nablas,
presumably because Gibbs used V V as the notation for this compound operator. These
practices, including the use of a FSP and FVP, go beyond the boundary of the U. S. A. and
continental Europe. There are books in Chinese and Japanese doing the same.
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5. V in the Hands of Oliver Heaviside (1850 - 1925)
Although we have traced the concept of the FSP and FVP as due to Wilson, the same practice is
found in the works of Heaviside. In Vol. I of his book Elecitrolmagetiwc 7Teory [8. ~ 127]
published in 1893, Heavside stated:
When the operand of V is a vector, say I), we have both the scalar product and
the vector product to consider. Taking the formula along first, we have
divD = VD, + VD + V3D3.
This function of D is called the divergence and is a very important function in
physical mathematics.
He then considered the curl of a vector function as the vector product between V and that vector.
At the time of his writing he was already aware of Gibbs' pamphlets on vector analysis but
Wilson's book was not yet published. It seems, therefore, that Heaviside and Wilson
independently introduced the misleading concept for the scalar and vector products between V
and a vector function. Both were, perhaps, induced by Gibbs' notations for the divergence and the
curl. Heaviside did not even include the word'formal' in his description of the products. We
should mention that Heaviside's notations for these two products and the gradient are not the
same as Gibbs' (See the table of notations in Sec. 2.1). His notation for the divergence of f is
Vf and his notation for the curl of f is VVf (a quaternion notation) while his notation for the
gradient of a scalar function f is V.f. Having treated V f and V x f (Gibbs' notations for the
divergence and the curl) as two'products', Heaviside simply considered V as a vector in deriving
various differential identities. One of them was presented as follows [8, ~ 132]:
The examples relate principally to the modification introduced by the
differentiating functions of V.
(a) We have the parallelopiped property
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An Historical Study of Vector Analysis page 36
N I'VE = VI'E N ='NV (176)
where V is a common vector. The equations remain true when V is vex, provided
we consistently employ the differentiating power in the three forms. Thus, the first
form, expressing N component of curl E, is not open to misconception. But in
the second form, expressing the divergence of IE N, since N follows V, we must
understand that N is supposed to remain constant. In the third form, again, the
operand E precedes the differentiator; we must either, then, assume that V acts
backwards, or else, which is preferable, change the third form to VNV.E, the
scalar product of VNV and E, or (VNV)E if that is plainer.
(b) Suppose, however, that both vectors in the vector product are variable. Thus,
required the divergence of VEH, expanded vectorially. We have
VVE H = E VHV = H VVE, (177)
where the first form alone is entirely unambiguous. But we may use either of the
others, provided that the differentiating power of V is made to act on both E and
H. But if we keep to the plainer and more usual convention that the operand is to
follow the operator, then the third term, in which E alone is differentiated, gives
one part of the result, whilst the second form, or rather its equivalent, -EVVH,
wherein H alone is differentiated, gives the rest. So we have, complete, and
without ambiguity
div VE H = HCurl E - E Curl H (178)
a very important transformation.
First of all, in terms of Gibbs' notations, Heaviside's Eqs. (176), (177), and (178) would be written
in the form
N.VxE=V.(ExN)=E.(NxV) (5.1)
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An Historical Study of Vector Analysis page 37
V(E X H)= E(H xV)= H(VxE) (5.2)
V (Ex H)= H V x E- E V x (5.3)
And in terms of the new notations for the divergence operator and the curl operator, Heaviside's
three equations will be written in the form
N E =V(ExN)= E (N V) (5.4)
(E x H) = E.(H x V)= H.(VE) (5.5)
v(EXH)=H-vE-E-vH (5.6)
According to the established mathematical rules, Heaviside's logic in arriving at his (178) or our
(5. 1) or (5.4) is entirely unnacceptable; in particular, present day students would never write an
equation (177) or (5.2) or (5.5) with V being the V operator. The second term in (5.2) or (5.5) is
a weighted operator while the first and the third are functions and they are not equal to each
other. His Eq. (178) or (5.3) in Gibbs' notation or (5.6) in our notation is a valid vector identity
but his derivation of this identity is not based on established mathematical rules. It is obtained by a
manipulation of mathematical symbols and selecting the desired terms. The most important
message passed to us is his practice of considering V f and V x f as two legitimate products,
the same as Wilson's FSP and FVP. Heaviside's'equations' will be examined again in a later
section and will be cast in proper form in terms of the symbolic vector and/or a partial symbolic
vector.
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6. Shilov's Formulation of Vector Analysis
A book in Russian on vector analysis was written by Shilov [23] in 1954, who advocated a nesw
formulation with the intent of providing a rather broader treatment of vector analysis. Shilov's
work was adopted by Fang [24] who studied in the U. S. S. R. We were informed of Shilov's
work through Fang. After a careful examination of the English translation of the two key chapters
in that book we found the contradictions as described below:
Shilov defined an'expression' for V denoted by T(V) as:
(V) - T(i) + T(j)+ T(k) (6.1)
8x Oy az
where i,j,kdenote the Cartesian unit vectors and V (nabla) is identified as the Hamilton
differential operator, that is,
V=i — +j+k
ex Oy 0z
Equation (6.1) is the same as Shilov's Eq. (18) on p. 18 of [24]. We want to emphatically call
attention to the fact that the only meaningful expression for T(V) involving V is Vf, the gradient
off. In that case, (6.1) is an identity because the right side of (6.1) yields
i f df df
(if)+ (if)+ (kf) =i + J +kOx ay 0: Ox y 8z
which is Vf.
The most serious contradiction in Shilov's work is his derivation of the expression for the
divergence and the curl by letting T(V) equal to V f and V x f respectively. We have pointed
out before that these two products do not exist. Shilov is defining a meaningless assembly to
make it meaningful. It is like defining 2 + x3 to be equal to 2 x +3 (= +6).
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7. Orthogonal Curvilinear Systems
After having revealed a number of'historical' confusions and contradictions in vector analysis so
far presented in the Cartesian system, we now examine several presentations in curvilinear
coordinate systems. We will show even more clearly the sources of the various
misrepresentations.
In this section we limit ourselves to problems in orthogonal curvilinear coordinate systems,
leaving the discussion of non-orthogonal curvilinear systems to a later section. In an orthogonal
curvilinear system with coordinate variables vi, i=1,2,3, the total differential of the position
vector r of a point in 3-dimensional space can be written in the form
8r r r
dr = dv+ +- dv-, +- dvl3 (7.1)
If we denote the metric coefficients in an orthogonal curvilinear system by hk with i = (1,2,3),
then
Or
r = hiu., i = (1,2,3) (7.2)
where ^ii denote the unit vectors in that system. An orthogonal system is characterized by
ui,. i j (7.3)
and ui xuj =uk, it jk (7.4)
with (i,j,k) in cyclic order of (1,2,3). Thus ^,,i2, and ti3 form a right-hand system.
For our discussion, a review of the invariance of the three differential operators is in order.
7.1 Invariance of the Differential Operators V, V and V in Orthogonal Curvilinear
Systems
When (7.2) is substituted into (7.1) we obtain
dr h,a dv, (7.5)
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It is understood that the summation in (7.5) goes from i = I to 3. The total differential of a scalar
function is defined by
df =- -d' (7.6)
i Clvi
which can be written in the form
df = of hidvi
df=C-Xh,Ohd
(7.7)
hi 4fi
The last line of(7.7) represents the scalar product of the gradient off and dr given by (7.5). Thus
df = Vf dr (7.8)
where
Vf =ii d=f (7.9)
hi &vi
If we write dr = dr in (7.8), then
Of=r3 Vf (7.10)
Jr
The name for the gradient used by Maxwell is space-variation, presumably because of the
relationship described by (7.10). In view of (7.9), the gradient operator in an orthogonal
curvilinear coordinate system is therefore given by
hV-'Ii 8v (7.11)
hi Ovi
For the special case of the Cartesian coordinate system, h, = 1, u^ = xi, v = xi, we recover the
del operator of Hamilton. Although our derivation of (7. 11) is independent of the choice of the
coordinate system, it is desirable to show analytically that the gradient operator is indeed invariant
to that choice. If we have another orthogonal curvilinear coordinate system with coordinate
variables l', unit vectors i< and metric coefficients hk' and we denote the gradient operator in that
system by V', we want to show that
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V=V' (7.12)
Since
dr= Zhidv, =- Zhujd,' (7.13)
i J
hence
hdvj'hi u i U dv (7.14)
Thus,
l,,;.
hj g J = hi u j i
d vi
or;;,. ih' dvj
uui=' (7.15)
hi dvi
Now
u =(Ut, u,) (7.16)
Substituting (7.15) into (7.16), we obtain
h. dv
U. = diUi
Odvi
or
h=j dv i (7.17)
h. h, dvi
By definition,
V' =,u j (7.18)
j h dv
and by the chain rule of differentiation, we have
V'P= hi dvk d (7.19)
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Eliminating u' I /h between (7. 17) and (7.19) we obtain
V' = Ej k E C I'c l A. c
j kI ~~1 (7.20)
Since
dv,; i'V cv, i0, i k
eq. (7.20) reduces to
V'= i ^- =V (7.21)
i hi Ovi
This completes our proof By following the same procedure it is not difficult to show that the
divergence operator and the curl operator are also invariant with respect to the choice of the
orthogonal curvilinear coordinate system, i.e.,
v-"' d'.=yu. =V, (7.22)
i hid h dvJ
and
i dv, i h dvF
With these expressions at our disposal, let us look at some of the treatments of the FSP and FVP
in orthogonal curvilinear coordinate systems and some of the presentations of vector identities
involving the del operator.
7.2 Two Examples from the Book by Moon and Spencer
In the book by Moon and Spencer [3, p. 325] they stated:
Let me apply the definition, Eq. (1.4), (of V in the orthogonal curvilinear system,
our 7.20) to divergence. By the usual definition of a scalar product,
1V.^ V + ( + 1 (9V,3 (7.24)
(g1)2- x1 (g22) OX (g33)~ Ox 3
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But this is not divergence, which is found to be...
Similar inconsistencies are obtained with other applications of Eq. (14).
In (7.24), their (g,,) correspond to our metric coefficients /, and their x' to our variables i,
In the first place, they have now applied the FSP to V and V[in an orthogonal curvilinear
coordinate system without realizing that the FSP is a non-valid entity in any coordinate system
including the Cartesian system. After obtaining a wrong formula for the divergence, (7.24), they
did not offer an explanation of the reason for the failure. It is seen in our discussion of the
invariance of V that Vf is equal to V'f' and Vf or V'f' is not a scalar product between V
and f nor V' and f'.
In discussing the Laplacian of a vector function, the two authors stated [3, p. 235]:
Section (7.08) showed that there are three meaningful combinations of differential
operators: div grad, grad div, and curl curl. Of these, the first is the scalar
Laplacian, V2. It is convenient to combine the other two operators to form the
vector Laplacian, >:
= grad div - curl curl (7.25)
Evidently the vector Laplacian can operate only on a vector, so
*E = grad div E - curl curl E (7.26)
Since the quantities on the right are vectors, WE transforms as a univalent tensor
or vector.
As noted in Table 1.01 (their table of notations on page 10), the scalar and vector
Laplacians are often represented by the same symbol. This is poor practice,
however, since the two are basically quite different:
V2 = div grad (7.27)
-- grad div - curl curl (7.28)
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This difference is evident also when the expression for the vector Laplacian is
expanded....
Analytically it can be proved [25, pp. 124 - 126] that in any orthogonal curvilinear system,
div gradf = grad divf - curl curlf
or
VVf= VVf - f (7.29)
where Vf denotes the gradient of a vector function that is a dyadic function. The divergence of a
dyadic function is a vector function. The use of V to denote the Laplacian is an old practice, but
the use of V V is preferred because it shows the structure of the Laplacian when it is applied to
either a scalar function or a vector function. By treating (7.25) as the definition for the Laplacian
applied to a vector function, the two authors have probably been influenced by a remark made by
Stratton [10, p. 50]:
The vector V VF may now be obtained by subtraction of (85) [an expansion of
V x V x F in an orthogonal curvilinear system] from the expansion of VV F, and
the result differs from that which follows a direct application of the Laplacian to
the curvilinear components of F.
As shown in our proof [25, pp. 124 - 126] VFis a dyadic, where the gradient operator must
apply to the entire vector function containing both the components and the unit vectors. When
this is done, we find that (7.29) is indeed an identity. In view of our analysis, it is clear that a
special symbol for the Laplacian is not necessary when it is operating on a vector function. The
same remark holds true for the two different notations for the Laplacian introduced by BuratiForti and Marcolongo as shown in Table I.
These two examples also show why Moon and Spencer thought that V is an unreliable device.
The past history of vector analysis seems to have led them to make such a conclusion. V is a
reliable device when it is used in the gradient of a scalar or vector function, but not in any other
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application. For the divergence and the curl, the divergence operator, V, and the curl operator,
V, are the proper operators They are distinctly different from V.
7.3 A Search for the Divergence Operator in Orthogonal Curvilinear Coordinate
Systems
In a very well known book on the methods of theoretical physics [26, p. 44] the authors try to
find the differential operators for the three key functions in an orthogonal curvilinear coordinate
system. They state:
The vector operator must have different forms for its different uses:
V = ~ i - for the gradient
i hi dvi
ui v- for the divergence
and no form which can be written for the curl.
We have used Q to represent Ih,h3 and have changed their coordinate variables si to vl and their
symbols ai to ti,. It is obvious that the'operator' introduced by these two authors for the
divergence can produce the correct expression for the divergence only if the operation is
interpreted as
[* i - < —- +J] — Z -- -.f (7.25)
Q O dvi(hi) Q. dvi \hi )
Such an interpretation is quite arbitrary, and it does not follow the accepted rule of a differential
operator because the first term within the bracket is a function so the entire expression represents
the scalar product of [" ] and f. One is not supposed to move the unit vector ui to the right side
of / h and then combine Ut with.f as shown in the right term of (7.25). It is a matter of
creating a desired expression by arbitrarily rearranging the terms in a function and the position of
the dot operator. A reader must recognize now that V can never be a part of the divergence
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operator nor the curl operator. The proper operators for the divergence and the curl are V and V
respectively. We could have used any two symbols for that matter such as I) and C.
7.4 The Use of V to Derive Vector Identities
There are many authors who have tried to apply identities in vector algebra to'derive' vector
identities involving the differential functions Vf, Vf, and \Vf We quote here two examples. The
first example is from the book by Borisenko and Tarapov [21, p. 180] where a problem is posed
and'solved':
Prob. 7. Find V(A B)
Solution. Clearly
V(A B)= V(A. B) + V(A BC) (7.26)
where the subscript c has the same meaning as on p. 170 (the subscript c denotes
that the quantity to which it is attached is momentarily being held fixed).
According to formula (1.30)
c(a b) = (a c)b - a x (b x c) (7.27)
Hence, setting
a=Ac, b=B, c=V,
we have
V(Ac. B) = (Ac V)B + A x (V x B) (7.28)
and similarly,
V(A Bc)= V(B A) = (B V)A + BC x (V x A) (7.29)
Thus, finally,
V(A B) =(A V)B +(B. V)A + A x curl B +B x curl A (7.30)
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As far as the final result, (7.30), is concerned, they have indeed obtained a correct answer But
what is the justification of applying (7.27) with c replaced by V and why cannot the same
formula be applied directly to V(A B-)? There is no way to provide the answers to these
questions. A reader has to accept blindly such a treatment.
The second example is found in the book by Panofsky and Phillips [27, p. 470]. They wrote:
V x (A x B) = (V B)A -(V A)B
=(. Bc)A +(. B)A, -(V.A.) — (v.) A7.
where the subscript c indicates that the function is constant and may be permuted
with the vector operator, with due regard to sign changes if such changes are
indicated by the ordinary vector relations.
It is seen that their (V B)A in the first line is not (div B)A. Rather it is equal to
(V B)A +(V B)AC. Secondly, if BC is constant, the established rule in differential calculus
would consider their V Bc (i.e., div Bc) = 0. The use of algebraic identities to derive differential
identities by replacing a vector by V has no foundation - the first line of (7.31). For the exercise in
consideration, one way to find the identity is to prove first that
(A x B) =(BA-AB)
or
Curl (A x B)= div (BA - AB) (7.32)
where AB is a dyadic and BA its transpose (see Sec. 2.3). Then by means of dyadic analysis one
finds
V(BA)=(VB)A + B VA (7.33)
V(AB) =( A)B + A VB (7.34)
Hence
V(AxB)=(VB)A+B VA
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-(VA)B-A VB (7.35)
where VA and V1B are two dyadic functions. A simpler method of deriving (7.35) will be shown
in later section. It should be emphasized that one cannot legitimately write
Vx(A x B)=(V B)A -(V.A)B
as the two authors did and then change (V B)A to V (BA), and similarly for (V. A)B in order
to create a desired identity.
A general comment on the analogy and no analogy between algebraic vector identities and
differential vector identities was made by Milne [28]. It was stated on p. 77:
The above examples [referring to nine differential vector identities expressed in
linguistic notations such as [grad (X. Y) = (grad X). Y + (grad Y) X etc.] whilst
exhibiting the relations between the symbols in vector or tensor form, conceal the
nature of the identities. A little gain of insight is obtained occasionally if the symbol
V is employed. E.g., Example (9) [curl curl X = grad divX -V2X] may be
written
V x (V x X)= V(V. X)- V2X (7.36)
which bears an obvious analogy to
Q x(Q x X) = Q(Q X)- Q2X (7.37)
where Q denotes a vector function.
On the other hand Example (5)
[Curl (X x Y) = Y grad X -X grad Y + X div Y - Y divX]
may be written
Vx( x Y)= Y.VX -X VY+X(V Y)- Y(V X) (7.38)
which bears no obvious analogy to
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Q) x(X x Y) = X(Q) Y)- Y(Q.) ) (7.39)
To obtain a better analogy, one would have to write
Q x (X x Y) = Q (YX - XY) (7.40)
and replace Q by V.
We do not understand why (7.40) is a better analogy than (7.39) because as algebraic vector
identities, they are equivalent. There is only one interpretation of (7.40), namely,
Q x (X x Y)=(() Y)X -(Q X)Y (7.41)
By replacing Q by V in (7.40), and treating the resultant expression as the divergence of the
dyadic XY - YX the manipulation is identical to the one used by Panofsky and Phillips. This
short paragraph on the role played by del in an authoritative book on vectorial mechanics shows
the consequence of treating Gibbs' notations for the divergence and the curl as two products, one
scalar and and one vector.
We have now shown the failures by several prominent authors in trying to invoke V as an
operator, not only for the gradient but also for the divergence and the curl. The role is now filled
in by the symbolic vector to be discussed in the next section. Many of the ambiguities which have
occurred in the past presentations covered in this paper will be recast correctly and
unambiguously by our new method utilizing the symbolic vector. In fact, the entire subject of
vector analysis can be developed from one single defining equation including the derivation of
vector identities and integral theorems. The method can also be extended to non-orthogonal
curvilinear systems.
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8. The Method of Symbolic Vector
The method of symbolic vector was first disclosed in an article published in 1991 [29]. The entire
subject was then expanded and compiled in a book [25] published in 1992. We are not going to
give any derivations nor proofs of the theorems to be used here. Rather, we will just use some
important formulas in that method in order to clarify the historical presentations covered in the
previous sections.
In the first place, a symbolic expression, denoted by T(V), involves a symbolic vector, or a
dummy vector denoted by V. The symbolic expression is defined by
JT(n)dS
T( )= n (8.1)
AV-0 AV(8.1)
where n denotes an outward unit normal vector of a surface, S, enclosing the volume AV The
symbolic expression is created by replacing a vector in a well-defined vector expression by V
Some of the well-defined vector expressions are:
ad, a d, axd, a(d xb), d(a b), d (ab), d x (ab), and d x(a x b) (8.2)
where the dot and the cross represent the two binary operators, namely, the scalar product
operator and the vector product operator.
When the vector d in (8.2) is replaced by V, the symbolic vector or the dummy vector, we obtain
the following symbolic expressions:
aV,, aV, x, a(Vxb), V(a b), V.(ab), Vx(ab), and Vx(axb) (8.3)
To be more specific, if T(V) contains only one function besides V, like the first three expressions
in (8.3), we sometimes would use the notation T( V,a) where a may be a scalar or a vector.
When T(V) contains two functions like the last five in (8.3), the notation T( V,a, b) will be used
for clarity if necessary, where a and b may be scalars or vectors or one of each. As far as the
notation is concerned, we may use any other symbol to denote the symbolic vector such as S
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(symbolic) or D (dummy). We adopt the symbol V because by a proper choice of the symbolic
expression, we can produce the three key functions. Vf, V F, and V F that makes V the creator
of the operators V, V, and V.
The definition of T(V) given by (8.1) is the most important formula in our new method. We
would like to consider three simple symbolic expressions to derive the expressions for the
gradient, the divergence, and the curl. Let us consider an expression 7(V) given by
T(V)=a V (8.4)
T(ni) is then given by
T(ni)=a n (8.5)
which is a function of both a and n. By substituting (8.4) and (8.5) into (8.1), we obtain
a. nidS
a *V= Linm s (8.6)
Av-O A V
For simplicity, let us evaluate the limit of the integral-differential expression in (8.6) in the
Cartesian coordinate system; we obtain:
a* = i Oa (8.7)
di xi
which is the expression for the divergence of a now denoted by V a. The chain of events is
recapitulated in the following line:
a.' dS
a.V= Linm s = X, n
AF-+O AV i xi
= V a (divergence of a) (8.8)
If we start with a T(V) represented by V -a we would obtain the same result because n' a = a n,
i.e.,
V.a= = a.V=Va (8.9)
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We also want to remark that the dot in a -V or V a is a sign or a binary operator for the scalar
product, but the product is executed in n a or a i. This is one of the most important and delicate
concepts in the method of symbolic vector. A reader must grasp firmly this concept in order to
understand and to use this method freely without uncertainty.
The symbolic vector V and the function a in the symbolic expression V-a is therefore
commutative. At this stage we request the reader to leave aside Gibbs' notations for the
divergence and the curl. The use of Gibbs' notation now would bring lots of confusion into
understanding the method of symbolic vector. On the other hand, if we had used S as the notation
for the symbolic vector we would obtain
S a=a S=Va (8.10)
that does not change the result at all.
We consider now a symbolic expression given by f V or V f. Then, an application of (8.1) yields
the expression for the gradient, i.e.,
fV = Vf=Vf (8.11)
Finally, if we let T(V) equal to V xf or -f x V we obtain the expression for the curl:
V xf = -f x V= f (8.12)
The three sample examples show very clearly that by means of the definition of T(V) we can
readily derive the differential expressions for the three key functions in vector analysis. In general,
the integral/differential limit for any T(V) defined by (8.1) can be evaluated in an orthogonal
curvilinear system that yields
( ) ZV vi^Thl17 (8.13)
where h, and i, denote, respectively, the metric coefficients and the unit vectors in that system,
and Q = /hh3. The derivation of(8.13) is found in [25, p. 38].
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There are two lemmas associated with the definition of T(V):
Lemma 1: For any symbolic expression T(V), which is generated from a valid vector expression,
we can treat V in that expression as a vector, and all of the algebraic identities in vector algebra
are applicable. Thus we have
aV=V a (8.14)
Va-a =a. V
V la=aV (8.15)
V xa =-axV (8.16)
b.(a xV)= V.(b x )=- (Vxb) (8.17)
V x(a x b) =( Vb)a- ( V a)b (8.18)
When T(V) contains more than one function besides V, say, two functions a and b which could
be both scalars or vectors or one of each, we will specifically use the form T( V,a,b) to write
such a symbolic expression. In this case we have
Lemma 2: For a symbolic expression containing two functions, the following relation holds true:
T(V,a, b)= T( Va,a,b) +T (Vb,a,b) (8.19)
where a and Vb denote two partial symbolic vectors.
A symbolic expression containing a partial symbolic vector is defined by
[T(in,a,b)dS
T( V a, b)= Lim s - (8.20)
a AV —O AV
In an orthogonal curvilinear coordinate system, the differential form of T( Va,a, b) is given by
T( Vaa,b) = T(uabb) (8.21):o
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Similarly for T( V^a,b). Lemma I also applies to expressions containing a partial symbolic
vector. There is no need for us to consider symbolic expressions containing more than two
functions. For example, to determine an expression like
(a x ).(bxc)
we can treat (a x V) d first. After we obtain the result we let d = b x c and simplify the resultant.
With (8.13) - (8.21) and the two lemmas at our disposal, we can re-examine the presentations by
these authors discussed previously.
To find the expressions for the three key functions in orthogonal curvilinear systems, we let T(V)
equal to V f or f V, f V or Vf, and V xf or -f x V respectively in (8.13), one finds
Vf =fV=i = Vf(gradient off) (8.22)
kh, 5v
f =f.= I -t = Vf(divergence off) (8.23)
h. Ov,
Vxf=-xV=f x' h =f (curloff)- (8.24)
i 1. Ov
Equations (8.23) and (8.24) can be converted into the form
f- dI iT v fi (8.25)
f=- h (f) ( h f) (8.26)
In (8.26) the summation is taken in the cyclic order of i,j,k = (1,2,3). In obtaining these results
we made use of the following identities [25, p. 13 - 15]:'O f2 " = (
E, d(u.)=O (8.27)
i ( 1 vh, - 1 0h \, — -\- h k I + —-U (8.28)
8dv, kh~dv h,9 kvk
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cut ch:-=_'JQ (i j) (8.29)
Cl' 7 C\'
In (8.28) i,j,k = (1,2,3) is taken in cyclic order. It is very clear that the method of symbolic
vector yields directly the operational expressions for the divergence and the curl as well as the
gradient. Morse and Feshbach's failure to find the divergence operator and the curl operator is
remedied in this analysis. The curl operator evolves just as readily as the divergence operator
From the structure of (8.23) we see clearly that Vf is not the scalar product between V and f,
an assumption made by Moon and Spencer for the orthogonal curvilinear system.
In regard to Heaviside's treatment the proper substitutes for his'equation' (5. 1) to (5.3) or (5.4) to
(5.6) are:
N V xE (E x N)=E.(NxV) (8.30)
V.(E x H)=E(H x )=H xE) (8.31)
V(E x H)= VE(E x H)- V.(H x E)
=H.(VEE)-E-(VHxH) (8.32)
Equation (8.32) yields
V(E x H)= H VE-E VH (8.33)
Although our equations have the same form as Heaviside's except that his V has been replaced by
V, the symbolic vector, yet there is a vast difference in meaning of the two sets. For example, his
H Vx E in (5.2) is interpreted as H CurlE but our H VxE is the same as V (E xH)
because of Lemma 2 and it is equal to V (E x H).
Every term in (8.30) to (8.33) is well defined. Both Lemma 1 and Lemma 2 are used to obtain the
vector identity stated by (8.33).
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Return now to the problems posed by Borinsenko and Parapov we start with the symbolic
expression v(A- B) for V(A- B); then by applying Lemma 2 we have
V(A B) =)V 4(A B)+V (A B) (8.34)
Applying Lemma 1 we have
V4(A B) = (B V )A- B x(A x V ) (8.35)
and
V (A B) =(A B)B - A x(B x B) (8.36)
hence,
V (A B) = B VA + B x A (8.37)
and
B(A B) = A VB + Ax v B (8.38)
Thus,
V(A B) = A. VB+B VA + A x + B x VA (8.39)
Our derivation of (8.39) appears to be similar to the derivation by Borisenko and Tarapov in
form, but the use of the FSP and FVP in their formulation and the treatment of (7.26) as an
algebraic identity is entirely unacceptable while each of our steps are supported by the basic
principle in the method of symbolic vector, particularly the two Lemmas therein.
The exercise posed by Panofsky and Phillips can be formulated correctly by our new method. The
steps are outlined below:
We start with V x(A x B) which is the symbolic expression of (A x B); then by means of
Lemma 2
V x(A x B) = V Ax(A x B)+ V Bx(A x B) (8.40)
By means of Lemma 1, we have
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V x(A BW) = (B V 4)A-( V.A)B
= B.VA-BVA
Similarly
VBx(A x B) = (B B)A-(A.VB)B
=AVB-A VB
hence,'(Ax B)= AVB-AVB
-BVA+B.VA (8.41)
which is the same as (7.35) obtained previously in Sec. 7 by a more complicated analysis. The
convenience and the simplicity of the method of symbolic vector to derive vector identities
hopefully has been demonstrated very clearly in the last two examples. All commonly used vector
identities have been derived in this way as shown in [25, pp. 52 - 54]. The method of symbolic
vector has so far been applied only to orthogonal curvilinear systems. The method can be applied
equally well to general or non-orthogonal curvilinear systems. The formulation is shown in the
next section.
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9. General Curvilinear Coordinate Systems
Expressions for the three key functions in vector analysis have been derived previously by
Stratton [10, pp. 38 - 47] in the general curvilinear coordinate system or non-orthogonal
coordinate system. The expression for the gradient is obtained by means of the relation
do - V -T dr using the contravarient components of the displacement vector dr. The expression
for the divergence is found by applying Gauss's theorem to an infinitesimal region and the
expression for the curl is obtained by applying Stokes' theorem to an infinitesimal contour. We
will derive these expressions by the method of symbolic vector by applying the basic definition of
T( V) to all three functions. To carry out the analysis it is necessary to reveal the concept of
unitary vectors and reciprocal vectors. However, we will not follow the usual treatment based on
tensor analysis and notations. Rather we will treat the subject entirely within the framework of
vector analysis except to share some technical nomenclature commonly used in tensor analysis.
Vector analysis in general curvilinear coordinate systems is not covered in [25]; the material to be
presented is new.
9.1 Unitary Vectors and Reciprocal Vectors
In the general curvilinear coordinate system, henceforth to be abbreviated as GCS, the total
differential of a displacement vector will be written in the form
dr a, dv, (9.1)
where a, with i = 1,2, 3, are called unitary vectors and vi the coordinate variables. The unitary
vectors are not necessarily of unit length, nor of the dimension of length. The three base vectors
define a differential volume given by
dV = a,.(a2 x a3)dv,dv2dv3
= Advldv2dv3
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where
A = (a x ) = a, a (a. x a,)=( a.(a, x a)
a, a,, and a3 are in general, not orthogonal to each other. Three reciprocal vectors, denoted by
b with j= 1,2,3, are defined by
1 1 I
b = -a x a3, b = -a x a (9 3)
IA 3 A' - A a
They are called reciprocal vectors because
]1, i = j
b.a, - i, (9.4)
The unitary vectors can be expressed in terms of the reciprocal vectors in the form:
a, = A(b2 x b), a2 = A(b x b,), a, = A(b, x b2) (9.4)*
The total differential of the same displacement vector defined in (9.1) can be written in the form
dr= bjdwi (9.5)
where w,, with = 1,2,3, denote the coordinate variables measured along the reciprocal base
vectors, so
dr.W = bj (9.6)
while
dr
-= a (9.7)
dv,
It can be readily shown that
b, (b2 xb)=b2 (b xb,)=b3 (b, xb)= 1 (9.8)
* Stratton [10, p. 39] had inadvertently written, in our notation, al = A-(b x b), etc.
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A Hisbrical Study od Vecor Analysis pae) 6
A vector function F can be expressed either in terms of the unitary vectors or their reciprocal
vectors. We write:
F =fa, =gjb (9.9)
i i
The notation Fi will be reserved to denote the components of F in an orthogonal curvilinear
system. Our fA and gi correspond, respectively, to the contravariant and covariant components of
F designated by Stratton [26], two names commonly used in tensor analysis. We merely call g,
the components of F in the unitary (vector) system, or the unitary components, and fi the
components of F in the reciprocal (vector) system, or the reciprocal components.
If we denote
aij = ai aj = aji (9.10)
and
Pij = bi bj = Jji (9.11)
then the relations between fi and g, are:
fi = Pijgj (9.12)
gi aijfj (9.13)
We have purposely avoided the superscript notations for gi and fij commonly used in tensor
analysis for these quantities, mainly to show that vector analysis can be treated properly without
the aid of tensor analysis.
On account of the orthogonal relations between a, and bi as stated by (9.4) the following relations
can be derived from (9.9):
fi=F b, (9.14)
gj = F.a (9.15)
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An Historical Study of Vector Analysis page 61
hence
F=-(F-b,), = (F.an)b1 (9 16)' ~ ~~;~~J
In the language of dyadic analysis
ba, = a jb = (9.17)
t J
where I is called the idemfactor such that
F I=I F=F (9.18)
We have now sufficient materials to apply the method of symbolic vector to the general
curvilinear system.
9.2 Gradient, Divergence and Curl in a General Curvilinear System
In a GCS, the differential length along the coordinates vi in the direction of the unitary vector a,
is:
ds, = anidv,, i = 1,2,3 (9. 19)
hence,dsi = dsi =(aai ) dv(9.20)
= (aii) dv
A differential area bounded by ds2 and ds3 is given by
dS1 = ds2 x ds3
1S 12 3 (9.21)
= a2 x a3dv2dv3
In general,
dS, = dsj x dsk
i x (9.22)
=(aj X a)dvjdv,
where i, j, k are carried out in cyclic order of (1, 2,3).
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An Historical Study of Vector Analysis page 62
Then
dSi = [(j x ak) (a x ak)]'jdv
[(1j.aj)(ak a)-(j.ak)(ak- aJ)],'jdtdl (9.23)
= ( axa/, - a ) dddvkjs
where the a,'s are defined in (9.10).
The differential volume is given by
dV = a1 (al x a3)dv\,dv,dv3 = Adv\ldvdv3 (9.24)
As shown by Stratton, [10, p. 43]
Ca1 a12 a3
A= a21 a2 a23 (9.25)
a31 a32 a33
Because our notations are different from Stratton's we have repeated some of his presentations
mainly for the readers to get accustomed to our notations. We must mention that much of the
basic works on unitary and reciprocal vectors are the original contributions of Gibbs found in his
first pamphlet [6, Vol. 1, Chapter 1].
In the subsequent analysis we need a theorem which states:
E x (a xa )=O (9.26)
To prove this theorem we have, in view of (9.7),
&a Ya3- (9.27)
Yvj dv,
hence,
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An Historical Study of Vector Analysis page 63
z c ( a c).6,L - ^ a, a, -- - ax a,
ca, (ac\
-+ a3.'( ( a i (9.28)
ca, c a
+ X - + x a,
C'3 C'
Since
dva3 3 a a, Ca d a ca,
G V %3 V _ 8c c- _cV.
the six terms in (9.28) cancel each other so (9.26) holds true. The geometrical interpretation of
this theorem is that the total vector area of a closed surface vanishes. The theorem for an
orthogonal curvilinear system was proved in [26, p. 15]. The proof therein appears more
complicated than the present proof using unitary vectors.
We consider the symbolic expression introduced previously by (8. 1) but now write it in the form
T(ni )ASi
T( ) Lima i (9.29)
vv-eo AV
with
niASi = AS = (aj x ak )AvjAVk
with
AV = AAv^Av:Av3.
Because of the linearity of T(n,) with respect to ii we find that the differential expression for
T( V ) in a general curvilinear coordinate system is given by
T(V)=Z T(a xa ) (9.30)
A -Ov.
where i, j,k = (1,2,3) taken in cyclic order.
To find the expression for the gradient, we let
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An Histoncal Study of Vector Analysis page 64
T(V)=Vf = fV
so
T(a, x ak )=(a X ak )f
that yields
Vf =fV=I afxak = Vf = gradf (9.31)
where we have made use of (9.26) to eliminate the sum of the derivatives of the cross products. In
terms of the reciprocal vectors b,
a1 x ak = Ab,;
hence
Of
Vf= bi (9.32)
The gradient operator in GCS is therefore represented by
V= b,-i (9.33)
i vi
To find the expression for the divergence, we let
T(V)= V.F=F.V
so
T(aj xak )=(aj xak,)F=F.(aj xak).
Substituting them into (9.30) we obtain
VF=ZF. 1 A d [(a xak)-F]
- (zaj xa, d. F (9.34)
= bi'd = VF=divF
i 9vi
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Equation (9.34) shows that the divergence operator in the general curvilinear system is represented
by
d (9.35)
V=' b, i - (divergence operator)
d v
and'-' is a compound operator which is now applied to a vector as the posterior operand. By
changing aj x ak to Ab, in the first line of (9.34) we obtain
v 1 9 (b (9.36)
F=- - (Ab,. F)
A
Since
bi F =fi
according to (9.14) where fi denotes the reciprocal components of F or the contravariant
components of F, (9.36) can be written in the form
V 1 d (9.37)
F= 1 r9(Afi)
A dv,
which gives directly, the differential form of V F without the need to evaluate the derivatives of
the unitary vectors or the reciprocal vectors.
By letting
T(V)=VxF =-FxV (9.38)
we can obtain the operational form and the direct differential form of v F. They are given by
V F = b x d- (operational form) (9.39)
i dv1
with'=- b x - (Curl operator) (9.40)
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A Hisorical Study of Vector Analysis page 66
and
F =-a. a g - a g (differential form) (9.41)
A d c vj d vk
where gi denotes the unitary components or the covariant components of F.
Equations (9.32), (9.37), and (9.41) are the same as Stratton's (49), (55), and (63). on pp. 44 -
47 of his book [10]. His formulation does not yield directly our equations (9.34) and (9.39)
although they can be obtained from (9.37) and (9.41), or Stratton's (55) and (63) by a proper
transformation of variables and with the aid of (9.26). However, it is a rather complicated exercise.
Presumably it is for this reason that the proper forms of the divergence operator and the curl
operator in general curvilinear coordinate systems are not treated in the literature, including
orthoginal curvilinear systems except the Cartesian system as found in Gibbs' classic.
In comparing the present method of deriving (9.37) and (9.42) with that of Stratton, we see that
Stratton applies the divergence theorem to obtain the differential expression for the divergence and
Stokes' theorem to obtain the differential expression for the curl while our method is based upon
one single expression, namely (9.30), from which the expressions for the three key functions can
be derived.
Before we apply (9.32), (9.36), and (9.41) to the orthogonal curvilinear systems as special cases,
it is desirable to show the invariance of the three operatorsin the general curvilinear system. We
consider first the gradient operator:
V=^,Cb di (9.42)
In a primed system with coordinate variables v and reciprocal vectors b the gradient operator will
be denoted by V' and given by
V'=C; f - a@,(9.43)
We want to show that V = V'.
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An Historical Study of Vector Analysis page 67
Let
bh I cjb; (9.44)
By taking the scalar product of (9.44) with ia, a typical unitary vector in the primed sysstem, we
obtain,k- = bi a
or ci- bi a (9.45)
Thus
b- =E(b.)b'j (9.46)
By definition,
dr = idar, = la'd (9.47)
i j
By taking the scalar product of (9.47) with bk, a typical reciprocal vectorin the unprimed system,
we obtain
dvk = (bk-.a)dv;
or dv, = (b,.a)dv; (9.48)
Hence, =bi, a (9.49)
8v-.
Substituting (9.49) into (9.46), and in view of (9.42), we have
i = Avj' i
L- J Ovf, 01v
i6'~ v~j'~ ~(9.50)
= lbj'., = V'
6
Equation (9.50) describes the invariance property of the gradient operator in any two GCS's.
Similar proofs apply to the invariance of V and V.
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To obtain the formulas for the three key functions in an orthogonal curvilinear system which is a
special case of the GCS, we let
a, = h,,, i = 1,2,3 (9.51)
where the h'.s denote the metric coefficients, and
o, ij /
U,i XU. (9.52)
J u, i j k
with i,j, k = (1,2,3) in cyclic order. Then
A = Q = h/h, (9.53)
bi = ui,/h, (9.54)
In an orthogonal system the components of F have been denoted by F,, i.e.,
F= F u, (9.55)
Thus, in view of (9.14), (9.15), (9.51), and (9.54) we have
F, - b, = F,/h, (9.56)
g, = F, a,= h,F, (9.57)
Equations (9.32) to (9.37) and (9.39) to (9.41) become
Vf/= Z -_f (9.58)
VF= ZF (9.59)
t_ Ui v
1 0 f_' _
VF= h j (9.60)
VF=-. udvi F (9.61)
1 00 / QRV i (962)
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An Historical Study of Vector Analysis page 69
F= = ESx (9.63)
h, C*',
v UX (9 64)
1 Y~(hF:)
VF= jhi"^,aF,> C-I (9.65)
As shown before, these formulas can be derived by using Gibbs' formulas for the three key
functions in the Cartesian system and invoking the invariance of the three distinct differential
operators V, V, and V. We list these expressions at the end of the main body of this essay to
point out that all these expressions are now derived from one single defining equation, namely the
symbolic expression T( V) given by (8.1).
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10. Retrospect
In this work we have examined critically some practices of presenting vector analysis in several
early works and in a few contemporary writings. It should be pointed out emphatically that the
whole subject of vector analysis was formulated by the great American scientist J. Willard Gibbs
in a very precise and elegant fashion. Although his original works are confined to formulations in
a Cartesian coordinate system, they can be extended to curvilinear systems as a result of the
invariance of the differential operators, as reviewed in this paper, without the necessity of
resorting to the aid of tensor analysis.
In spite of the richness of Gibbs' theory of vector analysis, his notations for the divergence and the
curl, in the opinion of this author, have induced several later workers, including one of his
students, Wilson, to make some inappropriate interpretations. The adoption of these
interpretations is world-wide. We have selected a few examples from the works of several
seasoned scientists and engineers to illustrate the prevalence of the improper use of V.
Many authors in the past have considered Heaviside to be a co-founder with Gibbs of the modern
vector analysis. We do not share this view. In Heaviside's treatment of vector analysis, he spoke
freely of the scalar product and the vector product between V and a vector function F and he
used V as a vector in deriving algebraic vector identities which incorporate differential entities. In
view of these mathematically insupportable treatments, Heaviside's status as a pioneer in vector
analysis is not of the same level as Gibbs'. In the historical introduction of a 1950 edition of
Heaviside's book on Electromagnetic Theory [8], Ernst Weber stated:
Chap. III of the Electromagnetic Theory dealing with "The Elements of Vectorial
Algebra and Analysis" is practically the model of modern treatises on vector
analysis. Considerable moral assistance came from a pamphlet by J. W. Gibbs who
independently developed vector analysis during 1881-4 in Heaviside sense - but
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An Histoncal Study of Vector Analysis page 71
using the less attractive notation of Tait, however, Gibbs deferred publication until
1901.
The above statement contains, unfortunately, several misleading messages. In the first place, in
view of our detailed study of Heaviside's works, his treatment would be a poor model if it were
used to teach vector calculus. Secondly, if Heaviside truly received moral assistance from Gibbs'
pamphlet, he would not have committed himself to the improper use of V, and would have
restricted his use of it to the expression for the gradient. Most important of all, Gibbs did not
develop his theory in the Heaviside sense. His development is completely different from that of
Heaviside. Finally, the book published in 1901 was written by Wilson, not by Gibbs himself Even
though it was founded upon the lectures of Gibbs, it contained some of Wilson's own
interpretations which are not found in Gibbs' original pamphlets nor in his lecture notes reported
by Wilson. The two prefaces, one by Gibbs and another by Wilson, which we quote in Sec. 1, are
proofs of our assertion. We were reluctant to criticize a scientist of Heaviside's status and the
opinion expressed by Prof. Weber. After all, Heaviside had contributed very much to
electromagnetic theory and had been recognized as a rare genius. However, in the field of vector
analysis we must set the record straight and call attention to the outstanding contribution of Gibbs
who stood above all his contemporaries in the last century. For the sake of future generations of
students, we have the obligation to remove unsound arguments and arbitrary manipulations in an
otherwise precise branch of mathematical science.
The recently published symbolic method of treating vector analysis, which has been introduced
briefly in this paper, shows that some of the treatments can be remedied by using the technique in
this relatively new method. It also shows that the entire subject of vector calculus can be
conveniently developed based on one single defining expression that includes all the integral
theorems which are not mentioned in this paper, but can be found in [25]. As far as notations are
concerned, in addition to the long established notations of Gibbs and the linguistic notations, we
have presented new symbols for two distinct differential operators for the divergence and the curl
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An Historical Study of Vector Analysis page 72
to accompany the existing gradient operator. For a decision on whether these new notations will
be of interest and use to students, we leave the matter in the hands of future generations.
We have examined a history covering a period of over one century. It represents a very interesting
period in the development of the mathematical foundations of electromagnetic theory. However,
in view of the long-entrenched and widespread mis-use of the gradient operator, V, as a
component of the divergence and curl operators, the obligation of sharing the insight presented
here with many of our colleagues in this field has been a labour fraught with frustration.
We hope that this presentation is clear enough that the issue(s) will be understood by the serious
workers in this subject, and that future students will not have to ponder over contradictions and
misrepresentations.
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An Historical Study of Vector Analysis page 73
Acknowledgement
I am grateful to Prof Philip H. Alexander of the University of Windsor, Canada, currently a
visiting scholar at the University of Michigan, for his most valuable suggestions and assistance
during my preparation of this manuscript. Dr. John Bryant has given me constant advice and
encouragement ever since this research was started. Prof W. Jack Cunningham of Yale University
was very kind in helping me to search for the Lecture Notes of Prof Gibbs recorded by Dr.
Wilson, currently stored at Sterling Library of Yale University. It is a very valuable document in
my study. The support which I have received from Dr. Thomas Phipps, Jr. is very much
appreciated. I want to thank Prof. Fawwaz Ulaby, Director of the Radiation Laboratory at the
University of Michigan for his encouragement, and Dr. Shou-Zhong Wang for his technical
assistance.
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An Historical Study of Vector Analysis - References page 74
References
[1] Crowe, Michael J., A History of Vector Analysis, Dover Publications, N. Y., 1985, an
unabridged and corrected republication of the work first published by the University of
Notre Dame Press in 1967.
[2] Burati-Forti, C., and Marcolongo, R., Elements de Calcul I'ectoriel, French edition of the
original in Italian by S. Lattes, Librairie Scientifique, A. Hermann et Fils, Paris, 1910.
[3] Moon, Parry and Spencer, Domina Eberle, Vectors, D. Van Nostrand, N. J., 1965.
[4] Weatherburn, Advanced Vector Analysis, G. Bell and Sons, London, 1924.
~,
[5] Maxwell, J. C., A Treatise on Electricity and Magnetism, Oxford University Press, Oxford,
1873.
[6] Gibbs, J. W., Elements of Vector Analysis, privately printed, New Haven (first part in 1881,
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An Historical Study of Vector Analysis - References page 75
[13] Tait, P. G., An Elenmentary Treatise on Quaternions. Cambridge University Press, 1890
[14] Gibbs, J. W., "On the role of quaternions in the algebra of vectors," Nature. Vol. XLIII, pp.
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An Historical Study of Vector Analysis - ReferencesERST OF MICHIGAN page 76
3 9015 03695 6574
Chapter 2, and pp. 85 - 92 of Chapter 8 containing the essential formulas of Shilov's
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Electronics.
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China, 1986.
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date: 95/05/23