Electromagnetic Theory, Vol. 1 (1893) — part 13 of 31

vV-D = KVi + vaV2 + %V3) (iDi + JD2 + kD8), . (126)

which we may separate into i, j, k terms if we like.

When v is the velocity of moving matter, vV comes frequently into use. Let, for example, w denote the measure of some

N2

180 ELECTROMAGNETIC THEORY. CH. III.

property of the moving matter (here a scalar, but it may equally well be a vector), and, therefore, a function of position and of time. Its rate of change with the time at a fixed point in space is dw/dt, and the matter to which this refers is changing. But if we wish to know the rate of time-change of w for the same portion of matter, we must go thus. Let Swfit denote the result, then

Sw __ dw dw dx dw dy dw dz Stdi ~dxdi ~dydi ~

by elementary calculus. Or

where v is the velocity of the matter.

This is the equation extensively employed in hydrokinetics. In elastic solid theory the term vV-w is generally omitted, being often small compared with the first term on the right. But, in special applications, it may happen that any one of the three terms in (127) vanishes. Thus the term on the left side obviously vanishes when w keeps the same in the same matter. If, however, for w we substitute a vector function, say A, even though it may not suffer change at first sight in the same matter, yet the rotational part of the motion will alter it by turning it round, so that SA/St will not vanish.

The operator 8/8t is distributive, like dfdt and vV> as exemplified in (124), (125).

Motion of a Rigid Body. Resolution of a Spin into other

Spins.

§ 123. If the moving matter is so connected that it moves as a rigid body, we have a further development. If we set a rigid body spinning about a fixed axis whose direction is defined by the unit vector a1} with angular speed a, the speed of any point P in the body equals the product of its perpendicular distance from the axis into the angular speed, and the direction of P's motion is perpendicular to the plane through the axis containing P. That is, a given particle describes a circle round the axis. This is obviously necessitated by the rigid con-

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 181

nection of the parts of the body (or detached bodies). The velocity of P is therefore expressed by

v = Var, ..... (128)

if a = aa1, and r is the vector to P from any point 0 on the axis. This is by the definition of a vector product.

Here we may observe a striking advantage possessed by the vectorial method. For (128), obtained by elementary consider- ations, proves without any more ado that angular velocities about different axes compound like displacements, translational velocities, or forces, or, iu short, like vectors. Thus, in (128) we see that the velocity of P is the vector product of a and r. The latter fixes the position of P. The former depends on the spin, its tensor being the angular speed, and its direction that of the axis of spin. Now every vector, as before seen, may be expressed as the sum of others, according to the rule of vector addition. But every vector has its species. Thus, r being a vector distance, its components are vector distances. Similarly, a being a. vector axis of spin with tensor equal to the angular speed, the components of a, for instance, the three terms on the right of

are vectors of the same nature. Therefore any spin may be replaced by other spins about other axes, according to the vector law. Of course the body cannot spin about more than one axis at once, but its motion is the same as if it could and did. This property is notoriously difficult to understand by Cartesian mathematics.

The most general kind of motion the rigid body can possess is obtained by imposing upon the rotational any translational velocity common to all parts. Let q be this translational velocity, then, instead of (128),

v = q + Var ...... (129)

Here observe that q is the velocity of the point 0, or of any other point on the axis of rotation passing through the point 0. We may infer from this that if we shift the origin 0 to Q, a point on a parallel axis, equation (129) will still be true, pro- vided r means the vector from the new origin Q to P, and q means the translational velocity of Q itself. Of course, the

182 ELECTROMAGNETIC THEORY. CH. III.

axis of rotation is now through Q. We may formally prove it thus. Take

r = h + R, (130)

where h is the vector from 0 to Q and R the vector from Q to P. Put (130) in (129), then

v = q + Vah + VaR (131)

But here a + Vah is, by (129), the velocity of Q, say u, so we have

v = u + VaR, (132)

showing the velocity of P to be the sum of the translational velocity u at any point Q, plus the velocity at P due to rota- tion about the axis through Q.

All motion is relative. But observe the absolute character of the spin a. It is absolute just because it involves and depends entirely upon relative motions.

The translational velocity u of the point Q consists of a motion along the axis of rotation, combined with a transverse motion. The latter may be got rid of by shifting the axis. For if w is the velocity of Q transverse to the axis, if we go to . the distance w/a from the axis through Q, keeping in the plane through the axis perpendicular to w, we shall reach a place where the velocity due to the rotation about the Q axis is either + w or — w, according to which side of the Q axis we go. Choosing the latter, where the transverse motion is can- celled, and transferring the axis to it. we see that the general motion of a rigid body consists of a spin about a certain axis (which is termed the central axis), combined with a translation along this axis. That is, it is a screw motion. It may, however, be more convenient not to employ the screw motion with a shifting axis, but to use (132), and let u and a be the translational and rotational velocities of a point Q fixed in the body.

We may also have simultaneous spins about axes which do not meet. Thus, let R be the vector from any origin 0 to a point Q about which there is a spin a, and r the vector from 0 to any other point P. Then the velocity of P due to the spin is, by the preceding,

v = Va(r-R) (133)

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 183

Next let there be any number of points of the type Q, each with its spin vector a. The velocity at P is then the sum of terms of the type (133), in which r is the same for all. The result is therefore

v = VAr-2VaR, .... (134)

where A is the sum of the a's, or the spin of the body, whilst the minus summation expresses the velocity at 0. For example, two equal spins, a, in opposite senses about parallel axes at dis- tance h combine to make a mere translational velocity perpen- dicular to the plane of the axes; speed ah. I give these examples, in passing, to illustrate the working of vectors. These transformations are effected with a facility and a sim- plicity of ideas which put to shame the Cartesian processes. It may, indeed, be regarded as indicative of a mental deficiency to be unable to readily work the Cartesian processes — which is in fact my own case. But that does not alter the fact that if a man is not skilful in Cartesians, he may get along very well in vectors, and that the skilful mathematician who can play with Cartesians of great complexity, could as easily do far more difficult work in vectors, if he would only get over the elements, and accustom himself to the vectors as he had to do to the other method.

Motion of Systems of Displacement, &c.

§ 124. Going back to (127), we should, when the property whose time-variation is followed belongs to matter moving as a rigid body, employ in it the special reckoning of v of equation (132), giving the velocity of any point P in terms of u and a at an invariable point Q in the body. Thus, we have

S=i!r(n+VaIl)v-w- • • • (135)

But equation (127) applies not merely to the case of matter moving through space, carrying some property with it, but also when the matter itself is fixed, whilst some measurable pro- perty or quality moves through the matter, or is transferred from one part of the matter to another. And here we may leave out the consideration of matter altogether, and think only of some stationary medium which can support and through which the phenomenon can be transferred.

184 ELECTROMAGNETIC THEORY. OH. III.

Thus, we may have a system of electric displacement D in a dielectric, which may be moving through it independently of any motion it may possess, and apply (127) to calculate D. It is, however, only in relatively simple cases that this can be followed up. For it is implied that we know the motion of the displacement and how it changes in itself, and these may be things to be found out. »

But should the displacement system move as a rigid body, the matter is greatly simplified. This will occur when the sources of the displacement (electrification, for instance) move as a rigid body, and the motion is of a steady type, so long continued that the displacement itself (in the moving system) is steady. Writing D for w in (135), we shall have

~=VaD-(u + VaR)V.D .... (136) at

Here VaD is the rate of time-variation of the displacement in a moving element of the displacement system, caused by the rotation. It is zero when the displacement is parallel to the axis of spin, and a maximum when it points straight away from the axis, like the spoke of a wheel.

The displacement system is, however, not necessarily or usually the same when in steady motion as when at rest. In the latter case only can it be regarded as known initially. Set it steadily moving and it will (by reason of the self- iuduction) be changed to another displacement system. Never- theless (136), along with the electromagnetic equations, enable us to find the new displacement system.

Should, however, the velocities of the connected sources be only a very small fraction of the speed of propagation of dis- turbances through the medium, the re-adjustment of the dis- placement as the sources move takes place practically instantaneously (as if the speed of propagation were infinite), so that the displacement system remains unchanged ; preserving its stationary type, whilst it moves through the medium as a rigid body, in rigid connection with the sources. (This is obviously quite incorrect at a great distance from the sources, but there the effects themselves are insensible.) In this case, too, it is clearly unnecessary that the motion of the sources should be steady. Thus, in (136), D becomes the known

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 185

displacement system (of equilibrium), moving as defined by u and a, which may vary anyhow.

Similar considerations apply to systems of induction, moving with their sources. A general caution, however, is necessary when there are conductors or other bodies in the field, not containing sources. To keep them from having currents induced in them, or in other w°ys upsetting the regularity of the moving system, they should also partake in the motion, by rigid connection.

Motion of a Strain-Figure.

§ 125. Equation (136) also applies to the motion through a medium of a " strain-figure," treated of by Dr. C. V. Burton in the current number of the Phil. Mag. (February, 1892). It is similar to that of slow motion of an electrical displacement system. Imagine a stationary infinite elastic medium with inertia, somehow set into a strained state, and let the speed of propagation of disturbances of strain be practically infinite. If then the strain-figure can move about through the medium, it will do so as a rigid body, provided the sources of the strain do so. In the above, D may signify the displacement (ordinary), whose variation in space constitutes the strain. Then D is the velocity of displacement, and is expressed in terms of the velocity of the strain-figure. From the expres- sion for the kinetic energy of the complete strain-figure, the mechanical forces concerned can be deduced. As this is to be done on dynamical principles involving Newton's laws, we may expect beforehand that a strain-figure sym- metrical with respect to a centre will behave as a New- tonian point-mass ; as does a similar electrical displacement figure. But, in general, without symmetry, the calcu- lation of the forces concerned would be very troublesome indeed.

But the real difficulty appears to me to be rather of a physical than a mathematical nature. We have first to get some idea of how the strain-figure is kept up. Let it be stationary first. Then the strain-figure is referred to a forced or unnatural state of the medium in certain places. At any rate, we require intrinsic "sources" somewhere, and perhaps it might for this reason be convenient to consider

186 ELECTROMAGNETIC THEORY. CH. Ill

these portions of the medium (with the sources) to constitute the atom or molecule, rather than the whole strain-figure.

Now, having got a stationary strain-figure, how is it to be set moving? Three ways suggest themselves. First, a bodily motion of the medium carrying the strain-figure with it. This is plainly inadmissible. Next a motion of the atomic portions only of the medium (with the sources) through the rest of the medium, either disturbing it to some extent near by, or not disturbing it at all — slipping through, so to speak. This would carry on the strain-figure. But it is inadmissible, if it be our object not to move the medium at all in any part. Thirdly, we may keep the whole medium at rest, and cause the sources themselves to move through it, so that the atomic portions of the medium change. But there is no means for doing this presented to our consideration. It lies beyond the dynamical question of the forces on an atom brought into play if the strain-figure can move in the manner sup- posed. The second course above, on the other hand, is more intelligible, although it implies something akin to liquidity.

These objections are, however, only made suggestively. The matter is sufficiently important to be deserving of a thorough threshing out.*

Space- Variation or Slope VP of a Scalar Function.

§ 126. The simplest and most easily understood effect of V upon a function is when it acts upon a scalar, say P. This implies that P is a scalar function of position, or of x, y> z. In the most important applications P is single-valued, as when it represents density, or pressure, or temperature, or electric potential, or the corresponding magnetic potential of magnets. But P may also be multiplex, as when it is the magnetic potential outside a linear electric current. At present let P be simplex.

By (119) we have

VP = LV1P+j.V2P + k.V3P. . . . (137)

Of course, since v is vector and P scalar, the result is a vector. Its meaning is easily found. From the fact indicated in (137) that the rectangular scalar components of VP are the rates of increase of P along the axes of i, j, k, we may conclude that

[* In case of eolotropy, add to the right member of (26) the term Sa, where S is .the torque and a the spin.]

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 187

the component of vP in any direction is the rate of increase of P in that direction. Thus, N being any unit vector, we have, by (137),

3P. . (138)

Comparing with (121), we see that

. . . (139)

by (123), s being length measured along N.

The identity of NvP and Nv-P is tolerably obvious algebrai- cally. It is not, however, true when for P is substituted a vector, as we shall see later.

If we take P = constant, we obtain the equation to a surface. If, then, T is any tangent to the surface at a chosen point, that is, any line in the tangent plane perpendicular to the normal, we shall have TvP = 0. The direction of vP itself is therefore that of the normal to the surface; or the lines of vP cut the equipotential surfaces perpendicularly, and pass from one to the next (infinitely close) by the shortest paths. The distance between any two consecutive equipotential sur- faces of a series having a common difference of potential varies inversely as the magnitude of vP, and vP itself is the vector showing at once the direction and the rate of the fastest increase of P. No perfectly satisfactory name has been found for this slope, or space- variation of a scalar function. Com- paring P with height above the level on a hillside, vP shows the greatest slope upwards. But the illustration is inadequate, since on the hillside we are confined to a surface.

The tensor of vP is given by

(VP)2 = (V1P)2 + (V2P)2 + (V3P)2, . . (140)

as with any other vector.

We may also here notice the vector product VNv in its effect on a scalar. We have, by the semi-Cartesian formula for a vector product,

VNV = i(N2V3 - N3V,) + j(N8V1 - NiVg) + k(NxV2 - N^). (141) Thus, when the operand is scalar, as P, we shall have

(VNV)P = VNVP ...... (142)

188 ELECTROMAGNETIC THEORY. CH. III.

The result is a vector perpendicular to the plane of N and vP, *s in any other vector product, vanishing when they are parallel, and a maximum when they are perpendicular.

Scalar Product VD. The Theorem of Divergence.

§ 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 former alone first, we have

divD = VD = V1D1 + V2D2 + V3D3. . . (143)

This function of D is called its divergence, and is a very impor- tant function in physical mathematics. Its general signification will be best appreciated by a consideration of the Theorem of Divergence.

Let liquid be in motion. The continuity of existence of the matter imposes certain restrictions on the motion. The current is mq per unit area, whore m is the density and q the velocity. But, to further simplify, let m = 1, making the liquid incompressible. Then the current is simply q, which measures the amount of liquid crossing unit. area of any surface per- pendicular to q, per second. But if the surface be not perpen- dicular to q, the effective flux is only Nq, the normal component of q, if N denotes the unit normal to the surface. Therefore, 2 Nq, or the summation of Nq over any surface, expresses the total flux of liquid through the surface.

Now suppose the surface (fixed in space) is closed. Then the summation represents the amount of liquid leaving the enclosed region per second through its boundary, if N be the outward normal. This amount is evidently zero, because of the assumed incompressibility. If then we observe, or state, that 2 Nq is not zero, either the fluid is compressible, or else there must be sources of liquid within the region. Adopting the latter idea, because it simplifies the reasoning, we see that the summation 2 Nq is the appropriate measure of the total strength of the sources in the region, since it is the rate at which liquid is being generated therein, or the rate of supply. The position of the internal sources is quite immaterial, and so is. the shape of the region, and its size, or the manner in which a source sends out its liquid (i.e., equably or not in all directions). So we have

'ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 189

if p represents the strength of a source, and the 2 includes them all.

When the sources are distributed continuously, so that p is a continuous function of position, its appropriate reckoning is per unit volume. That this is the divergence of q is clear enough, according to the explanations relating to divergence already given, (§ 51). That it is the same as vq, as in (143), we may prove at once by the Cartesian form there exhibited.

We have to reckon the flux outward through the sides of a unit cube. Take its edges parallel to i, j, k respectively. Then there are two sides whose outward normals are - i and i, and their distance apart is unity. The outward fluxes due to them are therefore - iq, or - qv and ql + v^, whose sum is v^. Similarly the two sides whose normals are ±j contribute V2g2, and the remaining sides contribute V3^3. Comparing with (143), we verify the Cartesian form of the divergence of a vector.

But we may have any number of other special forms, accord- ing to the co-ordinates we may choose to employ for calculating purposes, such as spherical, columnar, &c., and the most ready way to find the corresponding form is by the immediate appli- cation of the idea of divergence to the volume-element concerned. For purposes of reasoning, however, it is best to entirely eliminate the idea of co-ordinates. Divergence is independent of co-ordinates.

The sources need not be so distributed as to give rise to a finite volume-density. We may have, within the region concerned, surface-, line-, or point-sources, and the principle concerned is the same throughout. Thus the density of a surface-source is measured by the sum of the normal fluxes on its two sides per unit area, that is, by 2 Nq applied to the two sides of the unit area, if the flux wholly proceeds outwards. This is, however, not fully general, as there may be a flux in the surface itself, so that the full measure of the surface-density has to include the divergence of the surface-flux, to be found by calculating the flux leaving the unit area across its bounding line. Similar considerations apply to linear sources. In the case of a point-source the measure of the strength is the flux outward through a closed surface enclosing the point infinitely near it — the surface of the point, so to speak. Any other-

190 ELECTROMAGNETIC THEORY. CH. III.

surface enclosing the point will do, provided there are no other sources brought in by the change. There may also be multiplex sources. Thus, a pair of equal point-sources, one a source, the other a sink, would be equivalent to no source at all if brought infinitely near one another ; but if the reduction in distance be accompanied by a corresponding increase in strength of the sources, the final result is not zero. This is the case of a magnetised particle on the theory of magnetic matter ; but it is not necessary or desirable to complicate matters by entering upon special peculiarities of discontinuity in considering the Divergence Theorem. Its general form is

2ND = 2divD, (145)

when the vector D, to which it is applied, admits of finite differentiation ; and the special meanings to be attached to the divergence of D, in order to satisfy the principle concerned, may be understood.

Although a material analogy, as above, is very useful, it is not necessary. Any distributed vector magnitude will have the same peculiarities of divergence as the flux of a liquid. If it be the motion of a real expansible fluid that is in question, then the divergence of its velocity represents the rate of ex- pansion. It would, however, be very inconvenient to have to carry out this analogy in electric or magnetic applications ; an incompressible liquid, with sources and sinks to take the place of expansions and contractions, is far more manageable.

Extension of the Theorem of Divergence.

§ 128. The following way of viewing the Divergence Theorem, apart from material analogies, is important. Consider the summation 2 ND of the normal component of a vector D over any closed surface. Divide the region enclosed into two regions, Their bounding surfaces have a portion in common. If, then, we sum up the quantity ND for both regions (over their boundaries, of course), the result will be the original 2 ND for the complete region. The normal is always to be reckoned positive outwards from a region, so that on the surface common to the two smaller regions N is + for one and - for the other region, and 2 ND for one is the negative of that for the other, so far as the common surface goes.

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 191

Since this process of division may be carried on indefinitely, we see that the summation 2ND for the boundary of any region equals the sum of the similar summations applied to the surfaces of all the elementary regions into which we may divide the original. That is,

(146)

where, on the right side, we have a volume-summation whose elementary part <£(D) is the same quantity 2ND as before, belonging now, however, to the elementary volume in question. We have already identified <£(D) with the divergence of D.

But if this demonstration be examined, it will be seen that the validity of the process whereby we pass from a surface- to a volume-summation, depends solely upon the quantity summed up, viz., ND, changing its sign with N. We may therefore at once give to the Divergence Theorem a wide extension, making it, instead of (146), take this form : —

2F(N) = 2/(N) ..... (147)

Here, on the left side, we have a surface-, and on the right side a volume- summation. The function F(N), where N is the out ward normal, is any function which changes sign with N. The other function /(N), the element of the volume-summation, is the value of 2F(N) for the surface of the element of volume. Thus, by considering a cubical element,

/•(N)=V1F(i) + V2F(j) + V3F(k) . . (148)

is the Cartesian form, should F(N) be a scalar function, or the semi-Cartesian form should it be a vector function. A few examples of the general theorem (147) will be given later. In the meantime the other effect of v should be considered.

Vector Product WE, or the Curl of a Vector. The Theorem of Version, and its Extension.

§ 129. The vector product of v and a real vector, say E, is given in semi-Cartesian form by

WE = i(V2E3 - V3E2) + j^ -

= curlE.

192 ELECTROMAGNETIC THEORY. CH. III.

As before, with respect to the divergence of a vector, we can best appreciate the significance of this formula by the general property involved, expressed by the Theorem of Version.

On any surface draw a closed curve or circuit. Let, for distinctness, E be electric force. Calculate the voltage in the circuit due to E. The effective force per unit length of circuit is the tangential component of E, or TE, if T is the unit tangent. The voltage in the circuit is, therefore, 2 TE, the summation being circuital, or a line-integration extended once round (along) the closed curve.

Now draw on the surface a line joining any two points of the circuit. Two circuits are thus made, having a portion in common. Reckon up the voltage in each of the smaller circuits and add them together. The result is the voltage in the first circuit, if we rotate the same way in both the smaller circuits as in the original, because the common portion contributes voltage equally and oppositely to the two smaller circuits.

This process may be carried on to any extent by drawing fresh lines on the surface. We therefore have the result that the voltage in the circuit bounding a surface equals the sum of the voltages in the elementary circuits bounding the elements of surface. Or

2TE = 20(E), (150)

where on the left side we have a circuital summation, and on the right side an equivalent surface-summation, in which 0(E), the quantity summed, is the value of 2 TE, that is, the voltage, in the circuit bounding the particular element of surface con- cerned.

To find the form of 0 in terms of i, j, k, &c., we need only calculate the voltages in unit square elements of surface taken successively with edges parallel to j, k, to k, i, and to i, j. In the first case, when the normal to the square circuit, or the axis of the circuit, is i, the voltage is

V2E3-V3E2,

that is, by (149), the i component of VvE, and therefore the normal component, since here N = i. Similarly when N=j, and the axis of the circuit is j, the voltage in it is expressed bv the

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 193

coefficient ofj in (149). And when N = k, the voltage is the coefficient of k in (149). Thus, in any case, the voltage in an elementary circuit of unit area is the normal component of VvE, that is, NVvE. So (150) becomes

, . . (151)

expressing the Theorem of Version, sometimes termed Stoke's Theorem.

When E is not electric force, 2 TE is not circuital voltage, but circuital something else ; but this does not affect the general application of the theorem.

This theorem is particularly important in electromagnetism because it is involved in the two fundamental laws thereof, what were termed the First and Second Circuital Laws, or Laws of Circuitation, connecting together the electric and magnetic forces and their time-variations. (See §§ 33 to 36, Chap. II., and later.)

If the vector whose curl is taken be velocity q in a moving fluid, then curl q represents twice the spin or vector angular velocity of the fluid immediately surrounding the point in question ; its direction being that of the axis of rotation, and magnitude twice the rate of rotation. But I have not made use of the fluid analogy in describing and proving the Version Theorem, because it is not of material assistance.

Since the validity of the process whereby we pass from a circuital summation of the tangential component of a vector to an equivalent surface-summation depends upon the fact that TE changes sign with T, we may generalise the theorem thus : —

2F(T) = 2/(T), ..... (152)

where on the left we have a circuital and on the right a surface- summation, and F(T) is such a function of T as to change sign with T ; whilst /(T), the element of the surface-summa- tion, is the value of the former 2 F(T) for the particular element of surface in question. Of this general theorem a few examples will be given later.

A few words regarding v, div and curl, terminologically con- sidered, may be useful. Since divergence and curl are expres- sible in terms of vex (a provisional name for v, which has been suggested to me), why not use the vex operator only, like

o

194 ELECTROMAGNETIC THEORY. CH. HI.

the quaternionists ? The reasons, which are weighty, should be obvious.

In the first place, we require a convenient language for de- scribing or referring to processes and results, expressing approxi- mately their essential meaning without being too mathematical. Now V alone is not convenient for this purpose. The scalar product of v and D conveys no such distinct idea as does divergence ; nor does the vector product of v and E speak so plainly as the curl or rotation of E.

Besides, the three results of V, exemplified in vP, and vD, and VvE, are so remarkably different in their algebraical development and in their meaning, that it is desirable, even in the algebra, to very distinctly separate them in representation. Therefore, in the preceding part of the present work (as in all former papers), the symbol V is only prefixed to a scalar, as in V P, the space variation of P, whilst for the scalar and vector product are employed div and curl, in the formulae as well as in descriptive matter.

There are, however, cases when it may be desirable to use v and Vv applied to vectors in formulae, namely, when the combinations of symbols are not so simple that their meaning and effect can be readily seen, and when it is required to perform transformations also not readily recognisable. The utility of V in its vectorial significance then becomes apparent, for one may use it alone, temporarily if desired, and work it as a vector, remembering, however, its other functions. Of this, too, some examples should be given.

Five Examples of the Operation of V in Transforming from Surface to Volume Summations.

§ 130. Returning to the theorem (147), or

2/, ...".. (153)

let us take a few of the simplest cases that present themselves. Given any odd function F of N, the normal outwards from a closed surface over which the summation on the left side ex- tends, we convert it to an equivalent summation throughout the enclosed region by making/, the quantity summed, be the

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 195

value of 2 F(N) for the surface of the element of volume. This last is most conveniently a unit cube, so that

/=V1F(i) + V2F(j) + V3F(k),. . . (154) as in (148).

(a). The simplest case of all is F(N) = N itself. Then, by (154), or by considering that the vector normals to the six faces of a cube balance one another in pairs, we have

2N = 0, ...... (155)

expressing that a closed surface has no resultant orientation ; or, that a normal pull applied to every part of a closed surface, of uniform amount per unit area, has no resultant.

(b). If we multiply by p, any scalar function of position, we have the same case again if p be constant. When negative, it makes a well-known elementary hydrostatic result. But when p is not constant, then, by (154),

Vjp, . . . (156) so that we have

2Np = 2V# ...... (157)

These are, of course, vector summations. The sum of the surface tractions equals the sum of the bodily forces vp arising from the space- variation of the internal tension. Take p nega- tive to indicate pressure.

(c). Take F(N) = ND, where D is a vector function of position. This gives the most valuable theorem of divergence,

= 2divD, . . . (158) already discussed.

(d). Take F(N) = NDP, where P is a scalar. Here, by (154), or by (158), we shall find

2NDP = 2V(DP) ...... (159)

Here V has to differentiate both D and P, thus,

, . . . . (160) 02

196 ELECTROMAGNETIC THEORY. CH. III.

so that the previous equation may be written

. . . (161)

which is a form of Green's Theorem relating to electrostatic energy. D may be the displacement in one system of electrifi- cation and P the potential in another. The quantity - 2DvP is their mutual energy ; and this is, by (161), equivalently ex- pressed by the sum of products of every charge in one system into the potential due to the other.

(e). Take F(N) = VNH. Then, by (154),

/= ViViH + V2VjH + V3VkH.

But here we may shift the v's to the other side of the V'sa because they are scalars ; this produces

/=VVH, so that we have