'A 4 4 A SURVEY OF THE ERRORS IN VECTOR ANALYSIS C. T. Tai Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, Michigan 48109-2122 December, 1993 Technical Report No.906 RL-906 = RL-906

i A Survey of the Errors in Vector Analysis C. T. Tai Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, Michigan 48109-2122 January 14, 1994

1 Introduction In this report we have made a survey of the errors in vector analysis from the books available in the libraries of the University of Michigan and some personal libraries. There are probably more books on this subject, too many for us to search. The intent of this research is not to embarrass the authors whose works are quoted herein. Rather it is to show a mysterious event which has been propagated so long in the history of applied mathematics. It is hoped that after this revelation the teaching of vector analysis in the future will be more logical without any more confusion. The use of the new notations for the divergence and the curl would eliminate the possibility of the false product error. The linguistic notations can also be replaced by the new notations without any fear due to the misinterpretation of Gibbs's notations. 2 Books in Calculus, Applied Mathematics, and Vector analysis 1. Arflcen, George, Mathematical methods for Physicists, (Second Edition), Academic Press, NY, 1970 p.32: In section 1.6 V was defined as a vector operator. Now, paying careful attention to both its vector and its differential properties, we let it operate on a vector. First, as a vector we dot it into a second vector to obtain V V a - + + a Vz 9x 9y O z p.35: Another possible operation with the vector operator V is to cross it into a vector. The presentation makes the false products so firmly established as legitimate products. 2. Borisenko, A. I. and I. E. Tarapov, Vector and Tensor Analysis, Revised English Edition of the Russian text (Moscow, 1966), translated by Richard A Silverman, Dover Publications, NY, 1979 p.157: A coordinate-free symbolic representation of the operator is ()-) V n(...1)1 ( It should be observed that their representation of the operator requires (*v*). For the divergence their V(.) is exactly the same as Gibbs's V. which consists of two symbols used to denote the divergence. p.180: Find V(A * B) Solution. Clearly V(A * B) = V(Ac * B) + V(A * Bc) 1

where the subscript has the same meaning as on p.170 (Ac is constant in V(Ac-B)) According to formula (1.30) c(a. b) = (a c)b- a x (b x c) hence, setting a=Ac,b= B,c=V we have V(Ac B) = (Ac V)B + Ac x (V x B) In vector algebra a x (b x c) = -a x (c x b) but a x (b x V) #4 -a x (V x b). The author is playing a game. We must remember that the 'vector product' V x B does not exist. It is an assembly. V x B is merely a notation due to Gibbs'. 3. Chambers, L. G., A Course in Vector Analysis, Chapman and Hall, London, 1969 p.90: It may be regarded as the scalar product of the vector operator V and the vector A. The curl is treated in a similar manner. The author is not too sure about the concept of this scalar product because he used the words "may be regarded". 4. Coburn, Nathaniel, Vector and Tensor Analysis, Macmillan, NY, 1955 p.47: From the definition of V in (17.2) and the definition of the star product we may define V * a in cartesian orthogonal coordinates by.O 0 0 V*a = -*a +j- *a+k *a. If we assume that differentiation and the star product are interchangeable (a result which is easily verified when a is scalar or vector field) we may write the equation for V * a in the form. aa. Oa Oa V * a =t i * - =* + k *. ax ay az There are two errors in this statement; the first is the interchange of the differentiation and the star sign and secondly, such an interchange cannot be verified at all. What the author had in his mind is that the manufactured expressions for the divergence and the curl can be verified by other means such as by the flux model but one can never verify 9/O9x * f =.* af/Ox. 5. Cole, R. J., Vector Methods, Van Nostrand, NY, 1944 p.64: The divergence of a differential vector field is a scalar field defined by V.f =. Of + f _ af O + y z+ ax fy Oz 2

The first line and the third line can be used as the definition for the divergence as Gibbs originally suggested. But the second line is wrong. 6. Eisenman, Richard, Matrix Vector Analysis, McGraw-Hill, NY, 1963 p.99:...by definition, F, V-r = (8r,2 y ) F2 F3 = OF + OyF2 + aF3 V * F is similar to a dot product. It is not a dot product because V is not a vector and, e.g., V- F -F-V. The word 'similar' was not explained. In fact, it cannot be explained. 7. Franklin, Philip, Methods of Advanced Calculus, McGraw-Hill, NY, 1944 p.308: Since the volume of the box is dx dydz, the divergence or rate of flow outward per unit volume is O9Q OQ; OQ Qv. Ox + y '+ O The designation of V * q as div Q or the divergence anticipated this interpretation of V * Q as divergence per unit time per unit volume. In the last sentence the words 'as divergence per' presumably should be 'as charge per'. The main error is to designate V * Q as the sum of the partial derivatives instead of as a notation for this quantity. 8. Gans, Richard, Einfuhrung in die Vektor-Analysis, B.G. Teubner, Leipizig, 1905. English translation of the sixth edition by W.M.Deans, Blakie, London, 1932. The seventh German edition was published in 1950 p.47(English edition): Thus, the operator V denotes a differentiation, Seeing that VV = grad V has the components aV/Ox, 9V/y, V/Oz, that (V * A) = div A = (O/Ox)As + (aO/y)Ay + (aO/z)A,, and that [V, A] curl A has the components aA,/Oy - OAy/Oz, etc. We may formally regard the operator as a vector with components 0/Ox, O/8y, O/Oz. Gans used only linguistic notations (grad f, div f and curl f) in the previous editions of his book. This is the first time Gibbs's notations were used except that he used [V, A] instead of V x A to denote a vector product. Nevertheless, he also consider V as a constituent of the divergence and the curl. 3

A 9. Hay, George E., Vector and Tensor Analysis, Dover Publications, NY, 1953 p.108: If b is a vector then V.-_( O 9Zb br V * b = (Eir b = E * =.=E The curl is treated in a similar manner. 10. Haskell, Richard E., Introduction to Vector and Cartesian Tensors, Prentice Hall, Englewood, NY, 1972 p.229: If we write F=F u(j), then 0 aOF a_ a Fj V.F= (i)az, F(j) = U() () I= b-, j = a 11. Hassani, Sadri, Foundations of Mathematical Physics, Allyn and Bacon, Needham Heights, MA, 1991 p.48: The divergence can be written more compactly if we recall that the vector operator V has components (a/ax,.* * ) and note that the expression in parentheses (Ax/O9x + * * *) looks like a dot product of this operator with the vector A. Thus, OAx OAY OAZ V * A =- + -F + Ax ay +Oz p.56: In fact, using the mnemonic determinant form of vector product, we can write ex ey ez VxA= a a a Ax Ay Az 12. Hollingsworth, Charles A., Vectors, Matrices, and Group Theory for Scientists and Engineers, McGraw-Hill, NY, 1967 p.36: ( 0 0 9 9a\ Ou, Ouy Ou, v*U = o( ' ao (U uy -' u) = 9 y + - -y a 13. Kaplan, Wilfred, Advanced Calculus, Addison Wesley, Cambridge, MA, 1952 p.145: Formula (3.15) can be written in the symbolic form div = V-v; 4

for, treating V as a vector, one has VXt = (a i+***)*(v.i+ ) = V, V V) avz. + -5y + ax a y a = divt In the first place V should not be written as V = (a/9x)i + * because (a/ax)i is the conventional notation for a differential function or ai/9x which happens to be equal to zero in this case. Secondly, the false product error has been committed. On p.146, the curl is treated in a similar way. The presentation remains the same in the last edition published in 1991. 14. Kemmer, N., Vector Analysis, Cambridge University Press, Cambridge, 1977 p.84: Thus the Cartesian form of curl f is the cross product of the V symbol with f. Just as for the gradient, we see that here again V behaves like a vector. p.95:...We insert this result to (1), go to the limit and find that div f = V * f in terms of the Cartesian operator V, and to be quite explicit div f -V * f = Wf/ + Af + f3 dvfV.f= ay az V is truly a differential operator for the gradient. But in Gibbs's notation for the divergence, V f, V is not a differential operator, the same for V x f. 15. Korn, Granino and Therasa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill, NY, 1961 p.154: The linear operator V (del or nabla) is defined by =.a.a ax ay az Its application to a scalar point function *(r) or a vector function E(r) corresponds formally to a noncommutative multiplication operation with a vector having the rectangular Cartesian "components" 1a/x, a9/y, a/az; thus in terms of righthanded rectangular Cartesian coordinates x, y, z, a Vi - grad - +- + ax VaFZ V.F-=divF 0 +... I OXz -— (t-f) 5

4 16. Kovach, Ladis D., Advanced Engineering Mathematics, Addison Wesley, Reading, MA, 1982 p.312: Another way in which we can use the del operator is in applying it to a vector function. Since the operator has the form of a vector we must use either dot or cross multiplication. Given the vector function v, we define V V. - o'V + V -z Ax ay az This scalar is called the divergence of v and is also written div v. The author evidently treats V v as a dot multiplication. The curl is treated as a cross product on p.313. Then on p.320, V in cylindrical coordinate system is written as V = e~, + - e, + e, No distinction has been made between a differential operator and a differential function. 17. Kreyszig, Erwin, Advanced Engineering Mathematics (Fifth Edition), John Wiley, NY, 1983 p.397: The function divv = Z ai is called the divergence of v or the divergence of the vector field defined by v. Another common notation for the divergence of v is V v, av19 av 92 av3 div v =V.v= Ti +..(vi + * * *)= -- + a + OV3 with the understanding that the 'product' (a/ax)v, in the dot product means the partial derivative 9vl/Ox etc. This is a conventional notation but nothing more. Note that V * f means the scalar div v where Vf means the vector grad f defined in Sec.8.8. The more the author tries to explain the more a student gets confused. It is impossible to make an assembly meaningful. By writing V in the form of (a/ax)i + (a/9y)j + (a/9z)k is in conflict with the role of V in Vf. 18. Krishnamurtz, Karamcheti, Vector Analysis and Cartesian Tensor, Holden-Day, San Francisco, 1967 p.50: The quantity V * A is a scalar, whereas V x A is a vector. For reasons that will become known later, V * A is known as the divergence of A and denoted by div A, V x A is known as the curl of A and denoted by curl A. Thus we have divA =V.A 6

curl A = V x A It may be verified that In Cartesian system we obtain divA=-VA= +.. ax p.53: V2A = graddivA-curlcurl A = VVA-V x (Vx A) This identity may be verified by expansion in Cartesian system or by expand V x (V x A) according to the formula for a vector triple product. The author appears to be very sure V * A and V x A are two valid products. For the Laplacian of A he is using a mixed language and treats the identity as one in vector algebra. 19. Lagally, Max, Vorlesungen uber Vektor-rechnung, Akademische Verlagsgesellschaft, Leipzig, 1928 p.123: Divergence of b=v.= (-+...).(ib+...) or O9u Ov Ow div b=- + - - Oz ay Oz As the rotation of b one define the vector product of V with a field function b: rotb=Vxb= (i +...) x (iu+...)=i () + ay 09 +"' The above passage is an English translation of the original text in German. 20. Lass, Harry, Vector and Tensor Analysis, McGraw-Hill, NY, 1950 p.45: We postpone the physical meaning of the curl and define i j k curlf=Vxf= a U V W 21. Moon, P. and D. E. Spencer, Vectors, Van Notrand, Princeton, NJ, 1965 The two authors interpreted V * f in a curvilinear system, pp. 325-326, as = h1 0 (Ui f) 7

and then concluded that it does not yield the correct result. This observation actually shows the evidence that the 'scalar product' between V and f does not exist. If such a product does exist then it would be invariant to the choice of the coordinate system. On the other hand, V, V and V are three independent invariant differential operators[l]. 22. Pipes, Louis A., Applied Mathematics for Engineers and Physicists, McGraw-Hill, NY, 1946 p.343: The scalar product of the vector operator V and a vector A gives a scalar that is called the divergence of A; that is, 9Ax A Ay OAA V. A = -- + a7 + + Z = divergence of A ax ay 8z p.349: The curl is defined as the vector function of space obtained by taking the vector product of the operator V and A. The same presentation is found in the Third Edition published in 1970. 23. Pomey, J. B., Elements de Calcul Vectorial, Ganthier-Villars, Paris, 1934 p.33: One calls divergence the scalar quantity which is the scalar product of V and a vector (X, for example) divX = (VX) = - + +' dxi &x2 dx3 rot X: It is given by the vector product of V and X. This is the English translation of the French text. 24. Porter, Merle C., Mathematical Methods in the Physical Sciences, Prentice Hall, Englewood, NJ, 1978 p.221: The dot product of V operator with u is written in rectangular coordinates as V.u= ( +*) (u= +.)_a+a + a Z It is known as the divergence of the vector field u. The author treats the curl as the cross product between V and u and the Laplacian as the dot product between two dels. 25. Rektorys, Karel (Editor), Survey of Applicable Mathematics, English translation of a Czechoslovakian book by Rudolf Vyborny et al. edited by Staff of the Department of Mathematics, University of Survey, M.I.T. Press, Cambridge, MA, 1969 8

p.272: The divergence of a vector a is the scalar Clai div a = V * a = a p.275:... we note that the operator V is given in vector form by (20); for example, T72 v - ~7: aj: a\ V =V.V=(Za ).(Za-zj =E 2 When a Czechoslovakian book is translated into English by members of an English university and published in the U.S.A. nobody would question its qualification. 26. Schey, H. M., Div, Grad, Curl, and All That, W. W. Norton & Company, NY, 1973 p.43: If we take the dot product of V and some vector function F..., we get v.r=(.- +- ) (iF+...)= +. F (. +oy +Oz Now we interpret the 'product' of d/Ox anf F, as a partial derivative, that is, a aFx 0/Ox is a differential operator. When it operates on F, it yields 9Fr/9x. The problem is the false product manipulation not the subsequent interpretation. p.82 You can verify for your self that curl F = V x F wich is read "del cross F". 27. Sokolnikoff, I. S. and R. M. Redheffer, Mathematics of Physics and Modern Engineering (Second Edition), McGraw-Hill, NY, 1966 p.395: In terms of the differential operator 0.0 09 V =i +j- + kc a O ay az introduced in Sec.2, we can consider a symbolic scalar product v, = (S + * * * * ( +I) - a + + ay aZ On comparing this with (6-6) we see that divv = V.v. 9

We can define the Laplacian operator V2 by the formula vS = (i x+* *) iTa + *-)= 2 + a + 2 and observe that if v = Vu divVu = V. Vu = V2u. V x v Is defined by the rule for computing vector products. 28. Thomas, George B. Jr., Calculus and Analytical Geometry (Third Edition), AddisonWesley, Reading, MA, 1960 p.719: The 'curl' of a vector F = if + jg + kh is defined to be del cross F, that is, curlF - V x F and the 'divergence' of a vector V = iu + jv + kw is defined to be del dot V, that is, a u Ov Ow divV E-V V=-+-+ox ay Qz When a popular textbook makes such a positive assertion it becomes an unquestionable truth. 29. Wilson, E. B., Vector Analysis, Charles Scribner's Sons, NY, 1901 p.150: Although the operation VV has not been defined and cannot be at present, two formal combinations of the vector operator V and a vector function V may be treated. These are the (formal) scalar product and the (formal) vector product of V and V. They are v.v= i + j + k -V V X V w= T(i + j +k X V. V * V is read del dot V; and V x V, del cross V. The differentiators /Ox/a, O/y, O/Oz, being scalar operators, pass by the dot and the cross. That is OV.OV OV V * V = i * -' + j * - + k * z OV. OV OV VxV=ix - +j x +k x aThese may be expressed in terms of the components V1, V2, V3 of V... From some standpoints objections may be brought forward against treating V as a symbolic vector and introducing V V and V x V respectively as the symbolic scalar and vector products of V into V. These objections may be avoided by simply 10

laying down the definition that the symbol V. and Vx, which may be looked upon as entirely new operation operators quite distinct from V, shall be. V O V V v 'V-i'x + y + k dv O V 9 V V x V = i x -+ j x - + k x - But for practical purposes and for remembering formulae it seems by all means advisable to regard.0.0 0 V = is-+ j +k9x dy 9z as a symbolic vector differentiator. This symbol obeys the same laws as a vector just in so far as the differentiators 0/Ox, 9/dy, d/dz obey the same laws as ordinary scalar quantities. This is the most influential writing by a pioneer in vector analysis which has been adopted by many authors in this country. The operators 0/Ox, d/dy, d/z can operate on functions, scalar or vector, directly but they cannot pass by a dot or a cross. A simple proof of this misunderstanding is to consider the reversal operation. We start with the expression divV = - + a +Vy + V dx dy dZ which can be written as divV = (V.)+ (V. ) + (V.) x + V +. aV aV aV = ~ 5- 5+ +' +zx dy az which is obviously not equal to ix +y57Y.Z V In other words, the divergence is equal to the sum of the components of the directional derivatives of a vector function; it is not the scalar product between the del operator and the vector function. Like Gibbs, Wilson did consider V. and Vx as two new operators but his mistake is to create the concept of scalar and vector products between V and V. Even the use of the word 'formal' does not justify such a manipulation. Heaviside treated V * V and V x V as two legitimate products without using the word 'formal' in his presentation. We have to blame both Heaviside and Wilson for the false product error. 30. Wylie, C. R., Advanced Engineering Mathematics, (Third Edition), McGraw-Hill, NY, 1966 11

p.554: If F is a vector whose components are functions of x, y, and z, this leads to the combinations 9.= i 9 \. F, 9F2 3F V.F= (i +... ).li(+...)= F1 +F F3 which is known as the divergence of F, and VxF= ( i - ) +... x( Fi +-)=i( a+ F - xy az + "' which is known as the curl of F. 3 Books on Electromagnetic Theory and Physics 1. Argence, Emile and Thea Kahn, Theory of Waveguides and Cavity Resonentors, Hart Publishing Company, NY, 1967 (English translation of an original book in French) p.19: The scalar product V v* defines the divergence (div) of a vector v - / 0O Ovvy Ovz div -V.v= (eVa +...).(e +...)= v vy O The vector product V x v defines the curl,... Finally, we have the relations Af = div grad f V Vf = 2. Boast, Warren B., Principles of Electric and Magnetic Fields, Harper and Brothers, NY, 1956 p.391: Let the vector operator V operate as a scalar product upon some vector N. Letting the V operator replace the vector M in Eq.21.01 (M * N = MxNz + * * ) gives V N = aN y + ANz ax ' y a o 3. Chen, Hollis C., Theory of Electromagnetic Waves, McGraw-Hill, NY, 1983 p.49: The divergence of a vector function V*A = aigi aO i V X it + -jik = Eijk 9jILk 0x3 12

4. Durney, C. H. and Curtis C. Johnson, Introduction to Modern Electromagnetics, McGraw-Hill, NY, 1969 p.45: The dot product is formed in the obvious way. V operator cannot be defined in other coordinate system. p.57: V * V is treated as a product. V is an invariant operator. It is well defined in any curvilinear coordinate system, even in a non-orthogonal system. 5. Elliott, Robert S., Electromagnetics, McGraw-Hill, NY, 1966 p.606: Indeed, only in cartesian coordinates, and only because in that system hl =h = h3 =1, do the gradient and divergence operators turn out to be identical. p.610: Comparison with (V.14) suggests the notation curl A = V x A in which V is the del operator defined by (V.67). 6. Fano, Robert M., Lan Jen Chu, and Richard B. Adler, Electromagnetic Fields, Energy, and Forces, John Wiley, NY, 1960 After the authors derived the expression of the divergence by the flux model, they commented that in the simple case of cartesian coordinates div A = 9A+ A+ AZ =V.A Oz +y - 9z where it is recognized that the result expression can be interpreted as the scalar product of the del operator and the vector A. A similar interpretation was given to curl A with the remark that by inspection, curlA= V x A. 7. Harrington, Roger F., Introduction to Electromagnetic Engineering, McGraw-Hill, NY, 1958 p.34: div A = V A = + + (2.40) This is the differential form of divergence in rectangular coordinates and components. It is a scalar, as called to mind by the dot product symbolism of V * A. We can view V (del) as the differential operator V = u - + uy-T, + U,-z (2.41) Eq.(2.40) results from an application of our formal rules for scalar multiplication between the operator V and a vector A. However, V is not a vector. It has meaning only when it operates on a function according to (2.40) or according to other equations we shall consider later. 13

The author derived the expression for div A by the flux model first and then he added the above remarks which represents the false product error. 8. Hans, Hermann A. and James R. Melcher, Electromagnetic Fields and Energy, Prentice Hall, Englewood, NJ, 1989 p.48: The definition (2) (the flux model) is independent of the choice of coordinate system. On the other hand, the del notation suggests the mechanics of the operation in cartesian coordinates. We will have it both ways by using the del notation in writing equations in cartesian coordinates, but using the name divergence in the text. If one understands the meaning of Gibbs's notation for the divergence it is perfectly all right to use it to denote the divergence in any coordinate system. It is merely a notation. There is no need to use both (V.) and div. 9. Hayt, William H. Jr., Engineering Electromagnatics, McGraw-Hill, NY, 1958 p.73: Considering V * D, signifying V-D = ( a+ -a + az (D.a + Do y + Dzaz) += + (Do)+ z) where the parentheses are now removed by operating or differentiating: V.D= D + OD + ODz ax ay az p.74: Vu = a. + a ay + a-az)u au au au = aars + a ay + a az The operator V does not have a specific form in other coordinate systems. It is very unfortunate that the author writes V in the form (a/ax)a~ +... instead of a.(a/ax) +.... The former is a differential function (=0) and the later is a differential operator. The differential operator V is well defined in any coordinate system for the gradient. It is not a constituent for the divergence nor the curl. 10. Heaviside, Oliver, Electromagnetic Theory, Vol.1, first published in 1893, reproduced by Dover Publications, NY, 1950 14

p.127: When the operand of V is a vector, say, D, we have both the scalar product and the vector product to consider. Taking the formula alone first, we have diviv = ViD1 + V2D2 + V3D3 This function of D is called the divergence and is a very important function in physical mathematics. Many of Heaviside's manipulations are found in [2]. 11. Javid, Mansour and Philip Marshall Brown, Field Analysis and Electromagnetics, McGraw-Hill, NY, 1963 The authors show on pp.477-479 Appendix II that the volume-integral definition of the operator given by V{ } = li 1 fi{ } ds leads to the expression V{ } 1 [ (i>7l<})+ a2 (;~*<})+ -a3 [ 3 )] where fl = hlh2h3, the product of the metric coefficients. Then by letting { } = /, *F, xF they found the expressions for the gradient, the divergence, and the curl in curvilinear systems and concluded that the operator may be treated as a vector quantity. Actually, the authors have already identified V as the del operator. The operation V. and Vx are Gibbs's notations for the divergence and the divergence and the curl. V alone is not a constituent of the divergence nor the curl. The authors fail to recognize that their definition of V{ } is merely a general notation. When { } = {I}, we indeed have the gradient. When { } = {-F}, V{-F} = V * F is merely a notation for the divergence defined by the volume-integral. Unlike the symbolic expressions Vf, V * f and V x f their V{ } for the divergence and the curl are not the same as the operators V and V. This is one of the most delicate features of the method of symbolic vector in contrast to Javid/Brown's definition of V( *... ) originally formulated by Gans in the cartesian coordinates. 12. Johnk, Carl T. A., Engineering Electromagnetics, John Wiley, NY, 1975 p.71: Upon taking the dot product of V with F in the rectangular coordinates one finds F This is the basis for the equivalent symbolisms divF = V- F Similar interpretation for the curl is found on p.82. 13. Jordan, E. C., Electromagnetic Waves and Radiating Systems, Prentice Hall, Englewood, NJ, 1950 15

p.9: 2. If A is a vector function, we can apply Eqs.(10), (12), and (18) (the formulas for A * B and V) and get V. A = OAX + -Ay + AZ AOx ay 9z The operation is called the divergence and is abbreviated V * A = div A. 3. If A is a vector function, we can apply (15), (17), and (18) (the formulas for A x B and V) to show that Vx A= (OAza OA)i+... This operation is called the curl and can be written as V x A = curl A There is no change in the second edition coauthored with Keith G. Balmain, published in 1968. 14. Kraus, John D., Electromagnetics (Fourth Edition), McGraw-Hill, NY, 1992 After deriving correctly the expression for the divergence by the flux model it was stated on p.169 that the divergence of D can also be written as the scalar, or dot, product of the operator V and D. On p.249, it was stated that curl H is conveniently expressed in vector notation as the cross-product of the operator del (V) and H. 15. Marion, Jerry B., Classical Electromagnetic Radiation, Academic Press, NY, 1965 p.451: The divergence of a vector A is defined by div A = V * A = E OAi/9xi. The curl is defined by curl A = V x A = Eijk ei i,j,k 3 16. Mason, Max and Warren Weaver, The Electromagnetic Field, The University of Chicago Press, Chicago, 1929 p.336: The differential operator V can be considered formally as a vector of components 8/9x, 9/9y, 9/9z, so that its scalar and vector products with another vector may be taken. Afterwards, the authors considered the flux model as an alternative definition of div V. 17. Neff, Herbert P. Jr., Introductory Electromagnetics, John Wiley, NY, 1991 p.16: The del operator must operate on some quantity, and it is easy to show V. = ( +...) - (a,, +...) 16

OD OD D = lim f fas DIs ' ds VD = + + ny + ff- =limV (1.29) Ox 9y Oz AV-+o AV The left and right sides of (1.29) gives us a completely general mathematical definition of del as an integral operator: Vo lim f fs dso " AV-+O AV If the small circle (o) becomes a dot, we obtain the divergence of a vector. If the small circle becomes a cross, we obtain the curl of a vector. This presentation is similar to that of Gans and the one of Javid and Brown. The author did not realize that the left side of his (1.29) is a notation and the right side is truly a defining expression for the divergence. 18. Nussbaum, Alien, Electromagnetic Theory for Engineers and Scientists, Prentice Hall, Englewood, NJ, 1965 p.35: This operator (V) can be used in conjunction with the definition of the scalar product to generate other fundamental operations. These are the divergence of a vector such as the field ~, which is defined and denoted by divE-.= V - +x 9y + z p.184: In rectangular coordinates, this (the curl) is defined as curl A = V x A so that by (4.26) i j k curlA = a a a ax ay F9z Ax Ay Az 19. Page, Leigh and Norman Ilsley Adams, Electromagnetics, Van Nostrand, NY, 1940 p.27: If we form the scalar product av: av~ ov, v x a y Oz we obtain a proper function called the divergence of V. p.29: If V is a proper vector function of coordinates we may form the vector product of V and V so as to obtain another proper vector function known as the curl or rotation of V. This is i j k V=i oy oz + V a v, vy VZ 17

p.31: The divergence of the gradient of a proper scalar function is 7 02 29 a 2 02, V V = V * i * * = -- + Ox y + z2 \9 * ) 9 Oy2 z2 As the same result is obtained by allowing the scalar product 02 02 02 VV 2= + y2 + z The interpretation of the Laplacian as the scalar product of two del's or nabla's is found in many books. It is, of course, a false product error. 20. Panofsky, Wolfgang K. H. and Melba Phillips, Classical Electricity and Megnetism (Second Edition), Addison Wesley, Reading, MA, 1962 In p.470 the authors stated that the relations involving the vector operator V may be derived formally from the vector identities (1) through (5) if one remembers that V is a differential operator as well as a vector and thus does not commute with functions of the coordinates. In the example given by the authors, they wrote Vx(AxB) = (V.B)A-(V.A)B = (V. B)A + (V. B)Ac - (V. Ac)B - (V. A)Bc where subscript c indicates that the function is held constant and may be permuted with the vector operator, with due regard to sign changes if such changes are indicated by ordinary vector relations. It is seen that their (V * B)A is not equal to (div B)A or (VB)A. Rather, it is equal to (V * Bc)A + (V * B)AC. If Bc is held constant div Bc = 0. The use of vector algebraic relations to derive vector identities has no mathematical foundation. The contradiction is similar to Shilov's work. The proper way to do this exercise is to prove first that V x (A x B) = V * (BA - AB) (Gibbs's notation) or V(A x B) = V(BA - AB) (new notation) where AB is a dyadic and BA its transpose, then by means of dyadic analysis one finds V (AB) = (V A)B - A VB V (BA) = (V B)A - B VA hence V x (A x B) = (V B)A - B * VA - (V * A)B + A. VB 18

A relatively simple method of deriving this identity is to apply the method of symbolic vector. The treatment is given in [1] and reproduced in [2]. We want to point out that a relation like V x (A x B) = (V * B)A - (V. A)B as given by Panofsky and Phlllips is in contradiction to the accepted rules of vector analysis. To change (V - B)A to V * (BA) is similar to the 'pass by' manipulation of Wilson. We shall mention that the formula V x (A x B)= V (BA- AB) is listed in Appendix B of the monograph by Paul Penfield Jr., and Hermann A. Hans on Electrodynamics of Moving Media, M.I.T. Press, 1966, p.249 without derivation nor comment. 21. Paul, Clayton R. and Syed A. Nasar, Introduction to Electromagnetic Fields, McGraw-Hill, NY, 1982 p.40: Again we may write, grad f as Vf, but the operator is not simply defined in spherical coordinates. p.48: In terms of the del operator defined for a rectangular coordinate system... we may write (90) as div F = V * F as a simple expression will show. The gradient operator in spherical coordinate is well defined and there is no expansion for V. F. 22. Pender, Harold and S. Reid Warren, Jr., Electric Circuits and Fields, McGrawHill, NY, 1943 p.246: The concept of a dot product may also be expended to the vector operator V..., then in the notation just defined V * A signifies the scalar quantity V.A= + --- 0x Similarly, the dot product of V by itself usually written V2, is the scalar operator 02 02 02 V2 =V.V= + z ' 9x2 +9y2 '9z2' 23. Plonsey, Robert and Robert E. Collin, Principles and applications of Electromagnetic Fields, McGraw-Hill, NY, 1961 The authors derived the expression for the divergence by the flux model first then they stated on 19

p.14: Since the operator (V) defined earlier has the formal properties of a vector, we may form its product with any vector according to the usual rules. Thus, if we write the scalar product OF. 8Fy OF, V *= or + ~ + Oz ax 9y 9Z we discover a convenient representation of div F which leads to correct expression in cartesian coordinates. Accordingly, we adopt the notation V F = div F This is a typical example of presenting the divergence by many authors for the 'convenience' of memorizing a formula. 24. Plonus, Martin A., Applied Electromagnetics, McGraw-Hill, NY, 1978 p.37: The above [div D = D/9x + * * * (1.91d)] can be written in simple form by employing the del operator which was first used in the gradient operation. Equation (1.91d) can be seen to be the dot product of the operator V with D; that is, div D=V * D=((9/9x)x +***)* (D +**). The author wrote V = (9/9x)x + * * *, the same as done by Kaplan. This is very unfortunate because V is a differential operator and (a9/9x)x + * * * is a differential function which happens to be equal to zero because & is a constant. In other words, the position of the unit vectors and the differentiators cannot be interchanged in V. 25. Ramo, Simon, John R. Whinnery, and Theodore von Duzer, Fields and Waves in Communication Electronics, John Wiley, NY, 1965 The expression for the divergence was derived by the flux model first, like many other authors, then they stated on p.83: Consider the expression for the dot or scalar product, Eq.2.10(1), and the definition of V above. Then (5) indicates that div D can correctly be written as V * D. It should be remembered that V is not a true vector but rather a vector operator. p.116: It is noted that the above (curl) can logically be written as 'del cross F'. As a vector operator V is only applicable to the gradient. 26. Roger, Walter E., Introduction to Electrical Fields, McGraw Hill, NY, 1954 The author first derived the expression for the divergence by the flux model, thus far denoted by V * D as V.D=O + 9 y + Az in cartesian coordinates. Then he remarked: 20

This explains the notation used for the shorthand. If V is thought of as something like a vector O.0 0 Ox ay az when it operates on another vector by "scalar multiplication", V.D=(i. +...).(iD.+... The result is Eq.(2). Unfortunately, the expression for the divergence in other coordinate systems is not so simple as this. It is very difficult to use (2) as defining equation for divergence and convert it suitably to other coordinate systems. For this reason, the "flux gain per unit volume" formulation is greatly to be preferred. The author is very cautious in expressing his view. He seems to be not aware of the invariance property of the divergence operator V which is defined by Gibbs as ZE x * (0/Oxi) in cartesian coordinates and denoted by V. but not as E Xi(9/9xi)-. By means of the method of gradient[l] one can readily transfer E x, * (0/9xi) to Z(ui/hi) * (/O9vi) to prove the invariance property of the divergence operator. The method of symbolic vector reveals the invariance property from the very definition of T(V). 27. Seeley, Sammuel, Electromagnetic Fields, McGraw-Hill, NY, 1958 p.71: Observe that in rectangular coordinates the divergence is obtained by the vector operation div E = V * E = + (iE +.***)*(i.+*). 28. Shen, Liang Chi and Kong Jin Au, Applied Electromagnetism (Second Edition), PNS Engineering, Boston, MA, 1987 p.21: The symbol (V) represents a vector partial-differentiation operator. The operation V x A is called the curl of A, and the operation V * A is called the divergence of A. p.22: Thus, V * A is the scalar product of the vector V and A. Later the authors used the flux model to interpret the divergence and the curl. 29. Skitek, G. G. and S. V. Marshall, Electromagnetic Concepts and Applications, Prentice Hall, Englewood, NJ, 1982 p.82: Example E-8. In rectangular coordinates, obtain the expanded form for V.A. Solution: V A = +...* * * * + *)= + + \ 9x O y Gz 21

S The authors originally derived the expression for the divergence correctly based on the flux model but then injected the false product error as an 'exercise'. 30. Smythe, William R., Static and Dynamic Electricity (Second Edition), McGraw-Hill, NY, 1950 p.48: Let the components of A in rectangular coordinates as A,, Ay, A, so that V A = Ax +Ay 8Az as + Qy 9z 31. Stratton, J. A., Electromagnetic Theory, McGraw-Hill, NY, 1941 p.49: Now in the analysis of the field we encounter frequently the operation V x V x F = VV F - V - VF No meaning has been atributed as yet to V * VF. In rectangular cartesian system of coordinates x1, x2, 23, it is clear that this operation is equavalent to 3.F VF 61(2F, 82KF 12 F\ VV F(xl)2 (+ (X2)2 + 8(3)2) i.e., the Laplacian acting on the rectangular components of F. In generalized coordinates V x V x F is represented by the determinant... The vector V * VF may now be obtained by substraction of (85) [the expression for V x V x F] from the expression of VV * F, and the result differs from that which follows a direct application of the Laplacian operator to the curvilinear components of F. Actually, the Laplacian of a vector function, div grad F, is a well defined function in any curvilinear coordinate system. The relation V x V x F = VV * F - V * VF is an identity. An analytical proof of this identity is given in [1]. The misunderstanding of the meaning of div grad F is partly due to Gibbs's notation for the Laplacian in the form of V * V or V2. There are two independent operators involved in this double operation. In terms of our new notation it is VVF. Stratton's presentation was followed by P. Moon and D. E. Spencer ["The meaning of the vector Laplacian", J. Franklin Inst., 256, 1953, p.551]. They even introduced a special notation for the Laplacian of a vector function in the form of * F. The Laplacian of a vector function does not require a special symbol. Of course, one must remember that VF is a dyadic function, and the divergence of a dyadic function is a well defined vector function which is the Laplacian of F in this case. 32. Ware, Lawrence A., Elements of Electromagnetic Waves, Pitnam, NY, 1949 p.26: If V is written before a vector using a dot to indicate a scalar product, a scalar results, as will be shown, and it is defined as "divergence". Given a vector V, the divergence is written This is immediV=Vately seen as a scalar.+ (iV+ ) + This is immediately seen as a scalar. 22

33. Weber, Ernest, Electromagnetic Fields, Vol.1, John Wiley, NY, 1950 p.541: As a vector V can be applied to a field vector either in scalar or in vector product form in accordance with (3) and (5) (V. W, V x W), respectively. The results in these two cases are: V-V=-+-+-=divV 8x ay dz VX V= y i z 9V) +... =curlV. 34. Whitmer, Robert M., Electromagnetics (Second Edition), Prentice Hall, Englewood, NJ, 1962 p.41: The integrand of the right side of Eq.(2.52) (Gauss Theorem) is the divergence of the vector r. It is obvious that this integrand is just the scalar product of the vector operator V with r. p.51:... we find that the curl is the vector product of the operator V with the vector r. 35. Zahn, Markus, Electromagnetic Field Theory, John Wiley, Somerset, NJ, 1979 p.24: It (, OAi/Oxi) can be recognized as the dot product between the vector del operator of Sec.1-3-1 and the vector A. p.30: The partial derivatives in (4) and (5) (OAk/Oxj - OAj/Oxk) are just components of the cross product between the vector del operator of Sec.1-3-1 and the vector A. 4 Conclusion Our conclusion is very simple. The symbol V (del, nabla, Hamilton operator) can be used only for the gradient of a scalar or vector function. It is not a constituent of the divergence and the curl. In vector analysis there are three distinct differential operators. The scalar product and the vector product between V and a vector function do not exist, whether it is formal or informal. We do not need a 'convenient' but false manipulation to teach students to remember the formulas of the divergence and the curl in cartesian system. 5 References [1] C. T. Tai, Generalized Vector and Dyadic Analysis, IEEE Press, Piscataway, NJ, 1992 [2] C. T. Tai, "Problems in vector analysis", to be published 23