Electromagnetic Theory, Vol. 1 (1893) — part 12 of 31

To do this, we may remark that the angle between two planes is the same as the angle between the normals to the planes, if the normals coincide when the planes do, so that the three quantities in ( )'s in (41), which are the scalar pro- jections of c on the axes, are also the areas of the projections of the area V0ab on the planes perpendicular to them. We have therefore only to verify that the projection of any plane on the plane normal to one of the axes is given correctly in sign by (41). Take a rectangle, parallel to the j, k plane, sides parallel to j and k, that is, «2j and 63k ; so that

Here «263 is the projection of the rectangle on a parallel plane, and is correctly positive.

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 161

Now go back to (41). We have

Vab = i(a263-a362)-i- (45)

Similarly Vac = i (a2c3 - a3c2) + (46)

Add these together. We get

Vab + Vac = i[a2(63 + c3)-a3(62 + c2)] + (47)

But the right member, by definition of a vector product, or by (41 ), means Va (b + c). That is,

Va(b + c) = Vab + Vac. .... (48)

Similarly Vd (b + c) = Vdb + Vdc (49)

So, by adding these again,

V (a + d) (b + c) = Vab + Vac + Vdb + Vdc, . (50)

a very remarkable and important formula. It shows that the vector product of two vectors, A and B, equals the sum of all the vector products which can be made up out of the com- ponent vectors [(a + d) and (b + c) in (50)] into which we may divide A and B, provided we keep the components of A always before those of B. The necessity of this proviso of course follows from the reversal of Vab with the order of a and b. Although (50) only shows this for two components to each primary vector, yet the process by which it was obtained evi- dently applies to any number of components.

Subject to the limitation named, the formula (50) is pre- cisely similar to (17) §108, relating to the scalar products. Now the truth of (17) could be seen without much trouble. A similar proof of (50), on the other hand, would not be at all easy to follow, owing to the many changes of direction in- volved in the vector products. This is why I have done it through i, j, k, which are auxiliaries of the greatest value. When in doubt and difficulty, fly to i, j, k.

On the other hand, although the establishment of (50) by geometry without algebra is difficult, and there is preliminary trouble in fixing the direction of a vector product, yet we see from (50) that vector products are nearly as easily to be manipulated algebraically as scalar products.

162 ELECTROMAGNETIC THEORY. CH. III.

Examples relating to Vector Products.

§ 114. By means of the formula (41) we can at once obtain the cartesian expansion of the parallelepipedal cVab. Use equation (23), applied to (41), remembering that c is now any vector, and we get

cVab = ^(0363 - a362) + c^a^ - a^) + c^a^ - aj)^ . (51)

By rearrangement of terms on the right side of this, putting the a's outside the brackets, we obtain aVbc, and putting the b s outside, we obtain bVca, and thus verify (40) again.

The vector VcVab is, perhaps, most simply expanded through i, j, k. First write d for Vab, then by (41) we have

Vcd = icd

Next put for dv d2, d3 their values in terms of a's and b's, given in (41), and we have

Next, add and subtract a^c^ and we get

VcVab = ifc^bc - ^ac] + . . . . Lastly, reconvert fully to vectors, and we have

VcVab = a.bc-b.ca ..... (52)

2 ^ "i. ^ % I 2

This important formula should be remembered. That VcVab is in the plane of a and b is evident beforehand, because Vab is perpendicular to this plane, and multiplying by Vc sends it back into the plane. It is, therefore, expressible in terms of a and b, as in (52), which shows the magnitude of the com- ponents.

In some more complex formulae it is sufficient to remember the principle upon which they are founded, as by its aid they can be recovered at any time. Thus in the case of

r=/a+#b + Ac, ..... (53)

already treated, equation (31), if we use now Vbc for 1, which was, before, any multiple of it, and similarly for m and n, we

have r

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 163

There is no occasion to put a formula like this in the memory, because it is so simply obtained at any time from (53) by multiplying by the auxiliary vectors so as to isolate /, <7, h in turn and give their values.

Notice that the three denominators in (54) are equal. Also that by exchanging a and Vbc, b and Vca, c and Vab, we obtain

r- " Vbc+ * Vca+-£-Vab, . (55) aVbc bVca cVab

and this is also true, as we may at once prove by multiplying it in turns by a, b, and c, each of which operations nullifies two terms on the right. (55) is the expression of r in terms of three vectors, which are normal to the three planes of any three independent vectors, a, b, c, taken in pairs. Here, again, such a formula can be immediately recovered if wanted by attending to the principle concerned.

To obtain the cartesian expansion of any formula containing vectors is usually a quite mechanical operation. The cartesian, or semi-cartesian, representatives of a few fundamental functions being remembered, their substitution in the vector formula is all that is required. If the formula be scalar (although involving as many vectors as we please) the result is a single scalar for- mula in cartesians. But if it be a vector formula, it reduces to a semi-cartesian vector formula involving i, j, k, giving three scalar equations, one for each component.

The converse process, to put a scalar cartesian investigation into vectorial form, is less easy, though dependent upon the same principles. Here the three component equations have to be reduced to one vector equation. It is very good practice to take a symmetrically written-out cartesian investigation and go through it, boiling it down to a vector investigation, using the unit reference vectors i, j, k, whenever found to be con- venient.

The Differentiation of Scalars and Vectors.

§ 115. In the preceding account of vector addition, and of the

scalar product and the vector product, the reader has nearly all

that is needed for general purposes in geometry and in the usual

physical mathematics involving vectors, so far as the algebra

M 2

164 ELECTROMAGNETIC THEORY. CH. III.

itself is concerned. For the addition of differentiations does not usually introduce anything new into the algebra. Thus, in the analysis of varying vectors, we have the same vector algebra with new vectors introduced, these being derived from others by the process of differentiation.

The ideas concerned in the differentiation of a vector with respect to a scalar are essentially the same as in the differentia- tion of a scalar. Thus, u being a scalar function of x, i.e., a quantity whose value depends on that of x, we know that if AM is the increment in u corresponding to the increment A# in x, then the ratio Aw/A# usually tends to a definite limiting value when A# is infinitely reduced, which limit, denoted by du/dxy is called the differential coefficient of u with respect to x. This I should be strongly tempted to call the " different iant " of u to x, were I not informed on the highest authority that the expression " differentiant " for "differential coefficient" is objectionable. But the differentiating operator djdx which acts on the operand u may be termed the " differentiator," as has perhaps been done by Sylvester and others. This way of regarding a differential coefficient, splitting du/dx into (d/dx) and u, sometimes leads to great saving of labour, though we are not concerned with it immediately.

The differential coefficient is thus strictly the rate of increase of the function with the variable, or the increase of the func- tion per unit increase of the variable, .on the tacit assumption that the rate of increase of the function keeps the same value throughout the whole unit increment in the variable as it has for the particular value of the variable concerned. This plan, referring to unit increment, is often very useful. The reservation involved, though it must be understood, need not be mentioned.

There is also the method of differentials, which is of some importance in vector analysis, as it can be employed when differential coefficients do not exist. Thus, u being a function of x, whose differential is dx, the corresponding differential of

du=f(x + dx)-f(x); . .'.. . (56) which, by expanding/ (x + dx), reduces to

dudx, (57)

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 165

provided we neglect the squares and higher powers of dx, that is, regard dx as infinitesimal, and therefore du also. And if there be two variables, as in u=f(x, y), then we have

f(x,y), . . (58)

leading to du = —dx + —dy, ..... (59)

dx dy

expressing the differential of u in terms of those of x and y.

Now let the operand be a vector function of xt say, E. For instance, x may mean distance measured along a straight line or axis in an electrostatic field, where E, the electric force, will usually change as we pass along the line. It may change con- tinuously in direction as well as in magnitude ; in any case, the change itself in E between two points on the line is a vector, being the vector AE which must be added to the E at the first point to produce that at the second point. Dividing by Ax, the increment in x, we get the vector AE/A# ; and this, when the increments are taken smaller and smaller, approximates towards a definite limiting vector denoted by dE/dx, which is, by the manner of its construction, the rate of increase of E with x, or the increase in E per unit increase in x.

We have also, in differentials,

-f(x), .... (60)

leading to dE=^dx, ...... (61)

dx

understanding that the differentials dx and dH are infinitesimal. Similarly, the rate of increase of the vector d'E/dx with x is the second differential coefficient d2E/dx2 ; and so on.

Semi-Cartesian Differentiation. Examples of Differentiating Functions of Vectors.

§ 116. In semi-Cartesian form we have

E = iE1+jE2 + kE3 ..... (62)

Here the scalars EI} E2, E3 are functions of x, whilst the reference vectors i, j, k are constants ; so we have

^? = i^l+j^2 + k^3j . (63)

dx dx dx dx

showing the components of the vector dEjdx.

166 ELECTROMAGNETIC THEORY. CH. III.

Snarly, i+J . . . (64)

dx* ax2 ax2 ax*

shows the components of the second differential coefficient, obtained by differentiating (63).

Now, this semi-Cartesian process is quite general. Any vector function may be expressed in the form (62), and when the right member is differentiated, the differentiations are performed upon scalar functions. The same remark applies to (63) and (64), &c. ; so we see that the rules for differentiating vectors, and functions of vectors, with respect to scalar variables, are the same as those for differentiating similar functions of scalars, so far as the independent action of a differentiator on the separate members of a product goes.

Thus, in differentiating a scalar product, AB, with respect to a variable scalar, t, we have, by (23),

= (A! B! + A2B2 + A3 B3) + (Al Bx + A2 B2 + A3 B3) ; or, re-transforming to vectors,

(65) just as if A and B were scalars. Similarly, we have

.... (66)

as we may see immediately by differentiating the semi-Cartesian expansion of VAB, equation (41). We have also

^AVBC = AVBC + AVBC + AVBC, . . . (67)

and A VAVBC = VAVBC + VAVBC + VAVBC. . . (68)

But, although we proceed formally, as in the ordinary differentiation of scalars, as regards the properties peculiar to differentiation, yet we must always remember those which are peculiar to the vector algebra. For example, we must not only on the left sides, but also on the right sides of (67) and

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 167

(68), remember the reversal of the sign of a vector product with the order of the letters.

Independently of i, j, k, we may proceed thus, using differen- tials : —

d (AB) = (A + cf A) (B + dB) - AB = AdB + Bo?A, . (69)

omitting the infinitesimal dAdE of the second order. Dividing by dt, and proceeding to the limit, we obtain (65). Similarly we have

d (VAB) = V(A + dA.) (B + dB) - VAB = V(</A)B + VAdB, . (70)

omitting the second differential WAc?B. So, dividing by dt, we obtain (66).

In getting (69) we have, of course, used the distributive law of scalar products, equation (17) ; and, in getting (70), the similar law of vector products, equation (50).

Motion along a Curve in Space. Tangency and Curvature ; Velocity and Acceleration.

§ 117. Some spacial and motional examples maybe here use- fully inserted to illustrate the differentiation of vectors. Let

FIG. 7.

there be any curve in space, and let s be length measured along it from some point on the curve. Also let r be the vector from any fixed origin 0 to a point P on the curve. We have then r =/(s). With the form of the function we have no concern.

Now, consider the meaning of the first differential coefficient dijds. First, if the increment As is finite, say the arc PQ, the

168 ELECTROMAGNETIC THEORY. CH. III.

corresponding increment in Ar is the vector chord PQ, the new vector of the curve being OQ, that is r + Ar, instead of OP orr. We see that Ar/As is a vector parallel to the chord, whose tensor is the ratio of the chord to the arc. Now reduce the incre- ments to nothing. In the limit the direction of dr/ds is that of the tangent to the curve at P, and its tensor is unity, since the chord and the arc tend to equality. That is,

J-I-irff + j^ + k*, . . . .(71) as as as as

where T is the unit vector tangent, and the semi-Cartesian expression (got by expanding r) is also given. In the figure the size of T is quite arbitrary, since the unit of length is arbitrary.

Next, differentiate again with respect to s to obtain the second differential coefficient d2T/ds2 or dT/ds. It is the change in the unit tangent per unit step along the curve. Now, as the unit tangent cannot change its length, it can only change its direction ; and, moreover, this change is a vector perpendicular to the tangent. It and the two tangents, initial and final, are in some plane, usually termed the osculating plane. It is the plane of the curve for the time being, unless, it be a plane curve, when it is the plane of the whole curve. Thus dT/ds, being at right angles to T, points from the curve towards the centre of curvature for the time being. Moreover, its tensor measures the curvature. For the usual measure is d6/dst where dO is the angle between the tangents at the ends of ds, and it may be readily seen that this is the tensor of dT/ds.

Now the reciprocal of the curvature is the radius of curva- ture. If then R be the vector from the curve to the centre of curvature C (Fig. 7), we have

& -5 3

and the vector from the origin to the centre of curvature is r + R.

Now, referring to the same figure, let a point move along the curved path. Consider its velocity and acceleration, taking t the time for independent variable. We have

• = **-t,T = v,. (73)

dt ds dt l ;

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 169

by using (71), and denoting the velocity by v. Its tensor is v or ds/dt. The interpretation of (73) is sufficiently obvious, since it says that the motion is (momentarily) along the tangent, at speed v.

Differentiate again to t. We get

. (74)

dt2 dt dt

That is, the vector rate of acceleration is exhibited as the sum of two vectors, the first being the tangential component, whose tensor equals the rate of acceleration of speed, whilst the second we may expand thus, by introducing the intermediate vari- able s : —

&-,£_ ..... (75) dt ds dt R

where the third form is got by using (72). This shows the rate of acceleration perpendicular to the motion. It is towards the centre of curvature, of amount -y2/R, the well known result.

Tortuosity of a Curve, and Various Forms of Expansion.

§ 118. Again referring to Fig. 7, if the curve be not a plane curve, or be a tortuous curve, the osculating plane undergoes change as we pass along the curve. It turns round the tan- gent, and the measure of the tortuosity is the amount of turn- ing per unit step along the curve. It is d<f>/ds, if d<J> is the angle between the two osculating planes at the extremities of ds. It is equivalently denoted by the tensor of dN/ds, if N be the unit normal to the osculating plane. This is similar to the equivalence of dd/ds and the tensor of dT/ds in measuring the curvature before noticed, the equivalent plane in that case being the plane normal to T.

The three vectors, T, N, and Rlf form a unit rectangular system, for we have

N = VTR1, ...... (76)

since T and Rx are perpendicular unit vectors. As we move along the curve this system of axes moves as a rigid body, since there is no relative change. They may, in fact, be imagined to be three perpendicular axes fixed in a rigid body moving

170 ELECTROMAGNETIC THEORY. CH. III.

along the curve in such a way that these axes keep always coincident with the T, Rx, and N of the curve. As it moves, the body rotates. It rotates about the N axis only when the curve is plane, but about the T axis as well when tortuous. The vector change (infinitesimal) in T is perpendicular to it, in the osculating plane — that is, the plane whose normal is N, and is directed towards the centre of curvature. And the vector change (infinitesimal) in N is perpendicular to it, also in the osculating plane, and directed towards the centre of curva- ture, the vector R from the curve to the centre of curvature being on the intersection of the two planes mentioned.

Various expressions for the tortuosity may be obtained. Let it be Y the tensor of Y, given by

• •'• • -<77)

Then, by (76) and the context, (77) gives one expression. A second is got by performing the differentiation, giving

i . . . (78) ds ds

But the first term on the right is zero, by (72), the two vectors after V being parallel. So

Y = VT^i ..... (79) gives another expression. Or

Y = V0T^, ..... (80)

considering the tensor only. But, since Y is parallel to Ru we have YRj = Y ; therefore, by (79),

-1 . . (81)

as

is a third and entirely different expression. That Y is parallel to R! we may prove by (77) and (79). Multiply (77) by N.

Then YN = N^ = j4N2=0, . (82)

ds ds

since N2 = l ....... (83)

This shows Y is perpendicular to N. It is also, by (79), per- pendicular to T. So its direction is that of Rr

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 171

By the parallelepipedal property (40) we may also write (81) thus,

(84)

as Here again, we may write

R, dE, I dR, , ^ d 1 /Q~X

L = -- = -- + K -— — .... (oo)

ds ds-R R ds dsR

Substituting this compound vector in (84), we see that the first of the resulting vector products vanishes because RJ and R"1 are parallel. There is then left

-^-L ..... (8G) ds R

f™, . . . (37) s as*

by (72), or, in terms of r and derivatives,

which is a known symmetrical form, but whose meaning is less easy to understand than several preceding expressions.

Since, by the definition of a vector product we have V^b = 6 when B,^ and b are perpendicular, ax being a unit and b any vector, we may, from the first equation (77) conclude that

, ..... (89) as

which is equivalent to a form given by Thomson and Tait.

Similarly, Y = V0T^, ..... (90)

as

Y being perpendicular to T as well as to N.

The reader may, perhaps, find the above relating to tortuosity hard to follow, though the previous matter relating to tangency, curvature, velocity, and acceleration may be sufficiently plain. The hardness lies in the intrinsic nature of tortuosity, as to which see works on analytical geometry. But the reader who wishes to get a sound working knowledge of vectors should go through the ordinary Cartesian investigations, and turn into

172 ELECTROMAGNETIC THEORY. CH. III.

vectors. Also, convert the above to Cartesian form through i, j, k, and verify all the results, or correct them, as the case may be. Space is too short here for much detail.

Hamilton's Finite Differentials Inconvenient and Unnecessary.

§119. It is now desirable to say a few words regarding the method of treating vector differentiation employed by Hamilton, and followed by Tait. The latter speaks in his treatise (chapter L, §33, third edition) of the novel difficulties that arise in quaternion differentiation ; and remarks that it is a striking circumstance, when we consider the way in which Newton's original methods in the differential calculus have been decried, to find that Hamilton was obliged to employ them, and not the more modern forms, in order to overcome the characteristic difficulties of quaternion differentiation. (For the word "quaternion," we may read vector or vectorial here, because a vector is considered by Hamilton and Tait to be a quaternion, or is often counted as one. This practice is sometimes confusing. Thus the important physical operator y is called a quaternion operator. It is really a vector. It is as unfair to call a vector a quaternion as to call a man a quadruped ; although, four including two, the quadruped might be held (in the matter of legs) to include the biped, or, indeed, the triped, which would be more analogous to a vector. It is also often inconvenient that the name of the science, viz., Quater- nions, should be a mere repetition of the name of the operator. There is some gain in clearness by preserving the name " quaternion " for the real quaternion — the quadruped, that is to say.) The matter is illustrated by the motion of a point along a curve. But it does not appear, from this illustration, where the obligation to depart froni common usage comes in.

The subject is, however, returned to more generally in Chapter IV., on Differentiation, where we are informed that we " require to employ a definition of a differential somewhat different from the ordinary one, but coinciding with it when applied to functions of mere scalar variables." Here again, however, a most searching examination fails to show me the necessity of the requirement ; or, at least, that the necessity is demonstrated.

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 173

To examine this matter, consider for simplicity a function u of a single scalar variable a and a single vector variable r, say

«=/(r,«).

Following the common method of infinitesimal differentials, we have

du=f(r + dr,a + da)-f(r,a). ... (91)

That is, on giving the infinitesimal increments dx and da to r and a we produce the infinitesimal increment du in the function. Here du, di and da are the differentials, and it is important to remark, in connection with the following, that they are infini- tesimal, and that du is really the increment corresponding to dr and da.

Now, Hamiltonian differentials have a different meaning. Though they would be denoted by du, di and da, yet, being different, we shall here dignify them with brackets, and denote them by (du), (di), and (da), to distinguish them from the differentials in (91). The Hamiltonian differential (du) is then defined by

(du) - »[/(r + (£>,« + to) -/(r, «)], . (92)

on the understanding that n is infinity, and that (on which stress is laid) the differentials (di) and (da) are finite and perfectly arbitrary.

That this process is circuitous is obvious, but is it necessary ? Let us examine into its meaning, and compare (92) with (91). First, divide (92) by n. We then have

• (>-/(,, a). . . (93)

Now, we see at a glance, that (93) and (91) become iden- tical if

*>-&. -() ,-<> (94)

n n n

The division of the finite differentials (da) and (di) by n pro- duces infinitesimal results, and the finite differential (du) is similarly reduced to an infinitesimal. Why then not employ the infinitesimal differentials at once, and avoid the circuitous ness?

174 ELECTROMAGNETIC THEORY. OH. HI.

When we proceed to form a differential coefficient, as du/dt, if r and a are functions of t, or du/da, if r is a function of a, we see that their values are equivalently expressed by (du)/(dt) and (du)l(da), using the finite differentials; because this merely says that the value of the fraction du/dt is unaffected when the numerator and denominator are each multiplied by the same quantity.

In the above, the reasoning is the same, whether u is a scalar or a vector function of r and a.

But whilst the differential coefficient comes out the same, yet the differential (du) is not the increment in u belonging to the finite increments (di) and (da\ nor is it anything like it, excep- tions excepted.

The above is as fair and as clear a statement as I can make of the difference between common and finite differentials as applied to vectors ; and we see that the Hamiltonian plan is only a roundabout and rather confusing way of expressing what is done instantly with infinitesimal differentials. But then the Hamiltonian plan is said to be obligatory. There's the rub ! I can only give my own personal experience. After muddling my way somehow through the lamentable quater- nionic Chapter II., the third chapter was tolerably easy ; but at the fourth I stuck again, on this very matter of the obliga- tory nature of the finite differentials. I am no wiser now about it than I was then. Perhaps there is some very profound and occult mystery involved which cannot be revealed to the vulgar in a treatise ; Hamilton is a name to conjure by. Or perhaps I " began the study of quaternions too soon !" Or perhaps it is only a fad, after all. Anyway, I am willing to learn, and if I live long enough, may reach a suitable age. At present let us return to common infinitesimal differentials.

Determination of Possibility of Existence of Differential Coefficients.

§ 120. Although we may employ the method of differentials in any equation involving vectors, using the symbol d, and although we may turn it into an equation of differential coeffi- cients by changing d to the scalar differentiator d/dt (for example), as in § 116, yet we cannot usually turn differentials into differential coefficients when the variable is a vector.

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 175

This is not because we cannot form the expressions du\dt or dH/di, but because they are not differential coefficients when the denominators are vector differentials. This will be easily seen from typical examples. (The failure has nothing to do with the previous question of finiteness or otherwise of differentials.) Let

w = ab ........ (95)

and let the vector b vary/ Then,

...... (96)

and, therefore, — = (dT>)-l(B,db) = (db)l[a,(db)1l . . (97) db

where (db)l means the unit db. Here the imitation differential coefficient du/db is a fraud, because its expression on the right side of (97) is not clear of the differential db, whose direction it involves. Nor is equation (97) of any particular use, com- pared with (96).

Again, in the vector equation

R = Vab ...... (98)

let b be variable. Here we have

..... (99)

and from this equation of differentials we may form dH/db and VdR/db ; but neither of them, nor any combination of them, is a differential coefficient clear of differentials. Thus,

. . (100) contains the differential on the right side. And

(101)

also contains the differential. As a special result the value of (100) is zero (a parallelepiped in a plane) : and by equation (52) we may develope (101) to

(rfb^a^b^-a ..... (102)

But it is (99) that is the really useful fundamental equation, the developments (100) to (102) being useless for ditferentiating purposes, although true and interpretable geometrically,

176 ELECTROMAGNETIC THEORY. CH. III.

Generally, if

.... (103) is a function of

(104)

then Rj, R2, R3 are functions, usually different, of x, y and 2, and the ratio of dR to di (even in the most general quater- nionic sense), cannot possibly be freed of the differentials dx, dy, dz.

Even when we reduce the generality of the problem by making R and r coplanar, say in the plane of i, j, with z = 0 and R3 = 0, dR/dr is not usually a definite differential co- efficient. It only becomes one by special relations between the differential coefficients of Rj and R2 with respect to x and y, viz: —

=__^52. . . (105)

^ __

dx dy ' dy dx

This is the vectorial basis of the large and important branch of modern mathematics called the Theory of Functions, which appears in electric and magnetic bidimensional problems.

Variation of the Size and Ort of a Vector.

§ 121. Sometimes it is convenient to have the variation of a vector exhibited in terms of the variations of its tensor and unit vector, or its size and ort. Thus, from

we produce, by differentiating,

dR = EdEl+HldU ..... (106) If there be no variation of direction, then

^R = R^R; ..... (107)

and if the tensor does not vary, whilst the direction does, then dH = ~RdRl ..... . (108)

Dividing (107) by the scalar c?R, we get a proper differential coefficient

=E» ...... <109>

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 177

provided dR1 = 0. From (108),

provided dE = 0, which makes c?R perpendicular to E. But it is as well to avoid forms of this kind (with a vector variation in the denominator), even when by some limitation, as above, we can get rid of the differential. Of course, in (106), without any limitation, we can change d to d/dt (or other scalar diffe- rentiator) ; thus,

R = RR1 + R1R ...... (Ill)

It is this (scalar) kind of differentiation that nearly always occurs in physical applications, and there is no abstrusity about it. See §116. Nor is there about differentials, provided we do not attempt to make differential coefficients with respect to vectors, or make the differentials themselves finite.

We have also R2 = R2,

which, on differentiation, gives

RrfR = RdR; ..... (112) or, dividing by R,

..... (113)

which says that the increment in the tensor is the component along the vector of its vector change dR, whose full expression is given by (106). Using (106) in (113) we get

= 0 ....... (114)

Using (113) in (106) we get

JR = RdK1 + R1(R1^),. . . . (115) or, dividing by R,

^ = (/R1 + ^5, ..... (116)

it xx.

all of which results may be geometrically interpreted.

In differentiating the reciprocal of a vector, proceed thus : —

— ,<m)

where, for d may be written djdt, <kc.

178 ELECTROMAGNETIC THEORY. OH. III.

Another way is

It may easily be shown, by (106), that these results are equivalent.

Enough has now been said about the meaning and effect of differentiating vectors. The important vector differentiator V deserves and demands a separate treatment.

Preliminary on V- Axial Differentiation. Differentiation referred to Moving Matter.

§ 122. The Hamiltonian vector V occurs in all physical mathematics involving three dimensions, when treated vec- torially. It is denned by

V = i-^+j/+k^, .... (119) dx dy dz

the second form being merely an equivalent one, often more convenient, and easier to set up. We see that the Hamiltonian is a fictitious vector, inasmuch as the tensors of its components are not magnitudes, but are differentiators. As, however, these differentiators are scalar — not scalar magnitudes, but scalar operators, having nothing vectorial about them — the Hamiltonian, in virtue of i, j, k, behaves just like any other vec- tor, provided its differentiating functions are simultaneously attended to. Of course, an operand is always implied, which may be either scalar or vector.

The Hamiltonian has been called Nabla, from its alleged resemblance to an Assyrian harp — presumably, only the frame thereof is meant, without any means of evoking melody. On the other hand, that better known musical instrument, the Triangle, perhaps equally well resembles V> and- it does not want strings to play upon.

First notice that

iV = Vi, JV=-V2, kV = V8, - . (120) and that these are all special cases of the scalar product

3, • • . (121)

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 179

where N is any unit vector. Here NV» or its equivalent Cartesian expansion, means differentiation with respect to length, say s, measured along the axis of N, usually denoted by

.....

ds dx dy dz

or by similar forms. This operation was termed by Maxwell differentiation with respect to an axis, which we may more conveniently call axial differentiation. We see, therefore, that N Vj or the N component (scalar) of V> is an axial differentiator, the axis being that of N. Thus,

= , . . (123) ds ds

P being scalar and A vector. Also

NV.AB=A.NV.B+B.NV.A, . . (124) NV.VAB=V(NV.A)B+VA.NV.B, . (125)

and so on, simply because N V is a differentiator. Expanded in Cartesians, (124) contains the eighteen differential coefficients of the components of A and B with respect to x, y^ and z. The last two equations will also serve to illustrate notation. The dots are merely put in to act as separators, and keep the proper symbols connected. Mere blank spacing would do, but would be troublesome to work. We may, however, use brackets equivalently.

Thus, A.NV.B may be written A(NV)B, and this form may be used with advantage if found easier to read. Of course, N V being scalar, on its insertion between the two members of AB, the result remains a scalar product. Similarly, VA.NV.B, arising from VAB, is still a vector product.

A somewhat more general operator is vVj where v is any vector. It is plainly v times the corresponding axial differen- tiator, or /y.v1V- Thus, expanded,