1911 Encyclopædia Britannica/Vector Analysis
VECTOR ANALYSIS, in mathematics, the calculus of vectors. The position of a point B relative to another point A is specified by means of the straight line drawn from A to B. “It may equally well be specified by any equal and parallel line drawn in the same sense from (say) C to D, since the position of D relative to C is the same as that of B relative to A: A straight line conceived in thisway as having a definite length, direction and sense, but no definite location in space, is called a vector. It may be denoted by AB (or CD), or (when no confusion is likely to arise) simply by'AB. Thus a vector may be used to specify a displacement of translation (without rotation) of a rigid body. Again, a force acting on a particle, the velocity or momentum of a particle, the state of electric or magnetic polarization at a particular point of a medium, are examples of physical entities which are naturally represented by vectors. The quantities, on the other hand, with which we are familiar in ordinary arithrnetical algebra, and which have merely magnitude and sign, without any intrinsic reference to direction, are distinguished as scalars, since they are completely specified by their position on the proper scale of measurement. The mass of a body, the pressure of a gas, the charge of an electrified conductor, are instances of scalar magnitudes. It is convenient to emphasize. this distinction by a difference of notation; thus scalar quantities may be denoted by italic type, vectors (when they are represented by single symbols) by “ black ” or “ Clarendon ” type.,
There are certain combinations of vectors with one another, and with scalars, which have important geometrical or physical significance. Various systems of “vector analysis ” have been devised for the purpose of dealing methodically with these; we shall here confine ourselves to the one which is at present in most general use. Any such calculus must of course begin with definitions of the fundamental symbols and operations; these are in the first instance quite arbitrary conventions, but it is convenient so to frame them that the analogy with the processes of ordinary algebra may as far as possible be maintained. As already explained, two vectors which are represented by equal and parallel straight lines drawn in the same sense are regarde as identical. Again, the product of a scalar m into a vector A is naturally defined as the vector whose direction is the same as that of A, but whose length is to that of A in the ratio m, the sense (moreover) being the same as that of A or the reverse, according as m is positive or negative. We denote it by mA. The particular case where m=-I is denoted by-A, so that a change of sign simply f€V€l'S€S' th€ S€l'lS€ of 3 vector.
As regards combinations of two vectors, we have 1n the first place the one suggested by composition of displacements in kinematics, or of forces or couples in statics. Thus if a rigid body, receive in
- }
succession two translations represented by AB and BC, the final
- 'f
result is equivalent to the translation represented by AC. It is convenient, therefore, to regard AC as in a sense the “ geometric sum " of AB and BC, and to write
-b *-P ' P
AB+BC=AC.
This constitutes the definition of vector addition; and it is evident at once from fig. I that
—U '-'9 — —5 'if *4 *P
BC-I-AB =AD+DC =AC =AB-l-BC.
Hence, A and B being any two vectors, we have - A-l-B ==B+A, ..... (1)
i.e. addition of vectors, like ordinary arithmetical addition, is subject to the “ commutative law.” As regards subtraction, we define A — B as the equivalent of A+(-B); thus in fig. I, if lH3=A, l§ ?I=B, we have,
A-l-B=AC, A-B=DB.
When the sum (of difference) of two vectors is to be further dealt with as a single vector, this ma be indicated by the use of curved brackets, e.g. (A -|-B). It is easdy seen from a figure that (A-l-B)-{-C=A-I-(B+C), .... (2)
and so on; i.e. the “ associative law ” of addition also holds. Again, if rn be any scalar quantity, we have m(A -{-B) =mA+mB, .... (3)
or, in words, the multiplication of a vector sum by a scalar follows the “ distributive law.' The truth of (3) is obvious onireference to the similar triangles in fig. 2, where O P==A, PQ=B, o1>'=mA, P'Q'==mB.
Q
C .
D Q
- ll
B VP
A 0
Fic. x. F rc. 2.
It willlbe noaiced (halt tge prgofs of (1) and (3) involve the fundamental postu ate o the ucli ean eomet .
The definition of “ work ” in mechanics gives us another important mode of combination of vectors. The product of the absolute magnitudes A, B (say) of two vectors A, 'B into the cosine of the angle 9 between their directions is called the scalar product of the two vectors, and is denoted by A .B or simply AB. Thus AB=ABcos0=BA, .... (4)
so that the “ commutative law of multiplication " holds here as in grdinary algebra. The “ distributive law " is also valid, for we ave
A(B~}-C) =AB -I-AC, .... (5)
the proof of this statement being identical with that of the statical theorem that the sum of the works of two forces in any displacement of a particle is equal to the work of their resultant. For an illustration of the next mode of combination of vectors we may have recourse to the geometrical theory of the rotation of a rigid body about a fixed point O. As explained under MEEANICS, the state of motion at any instant is specified by a vector OI representing the angular velocity. The instantaneous velocity of any other point P oi) the body is completely determined by the two ~' }
vectors OI and OP, viz. it is a vector normal to the plane of OI and OP, whose absolute magnitude is OI .OP. sin 0, where 0 denotes the inclination of OP to OI, and its sense is that due to a right-handed rotation about OI. A vector derived according to this rule from any two given vectors A, B is called their vector product, and is denoted by A XB or by [AB]. This type of combination is frequent in electro-magnetism; thus if C be the current and B the magnetic induction, at any point of a conductor, the mechanical force on the latter is represented by the vector [CB]. It will be noticed in the above kinematical exam le that if the roles of the two vectors OI, OP were interchanged: the resulting vector would have the same absolute magnitude as before, but its sense would be reversed. Hence .
AB] = - [BA], ..... (6)
so that the commutative law does not hold with respect to vector products. On the other hand, the distributive law applies, for we have lA(B+C)] =[AB]-l-[ACL ~ —~~ (7),
as may be proved without difficulty by considering the kinematical interpretation.
Various types of triple roducts may also present themselves the most important being the scalar roduct of two vectors, one oi which is itself given as a vector prodiuct. Thus A[BC] is equal in absolute value to the volume of the parallelepiped constructed on three ed es OA, OB, OC drawn from a point O to represent the vectors Ii, B, C respectively, and it is positive or negative 'accordin as the lines OA, OB, OC follow one another in right- or left-handed cyclical order. It follows that
A[BC] =B[CA] = -B{AC] =&c .... S?)
In order to exhibit the correspondence between the short and methods of vector analysis and the more familiar formulae of Cartesian geometry, we take a right-handed system of three mutually perpendicular axes Ox, Oy, Oz, and adopt three fundamental unit vectors i, j, /1, having the positive directions of these axes respectively. As regards the scalar products of these unit-vectors, we have, by (4), {'2=j2=k2=I, .j/{=[;]= /i=()
Any other vector A is expressed in terms of its scalar projections Al, Ag, A; on the co-ordinate axes by the formula f A=fA1'l“fA2"l-/{As. ..... (IO)
For the scalar product of any two vectors we have AB = (fAi+jAz+/1Aa) (fB1-f'°fB2+lfBs) =A1B1+AgBg+A;B3, (I I) as appears on developing the product and making use of (9). In particular, forming the scalar square of A we have A2=A12'l-A22'l-A32, . . . . (12)
where A denotes the absolute value of A., Again, the rule for vector products, applied to the fundamental units, gives
I l1"l = [Pl = I/f”l =0»
U/fl = -l/U] =l, I/fl] = ~lI/fl =/» lu] = '° UI] =/Li (13) Hence
lABl =l(iA1'l'fA2~l'ffA:)(iB1 ~l-1132+/fB=)l =l(AgB3 'A, |B2) -A1B,) *AzB;)
= - [BA]. ........... (14)
The correspondence with the formulae which occur in the analytical theory of rotations, &c., will be manifest. If we forrr: the scalar product of a third vector C into [AB], we obtain C[AB] = Al, Bl, C1
Ag, BQ, C2 . ....
Aa, Ba, Cs
in agreement with the geometrical interpretation already given. In such subjects as hydrodynamics and electricity we are introduced to the notion of scalar and vector fields. With every point P of the region under consideration there are associated certain scalam (e.g. density, electric or magnetic potential) and vectors (e.g. fluid velocity, electric or magnetic force) which are regarded as functions of the position of P. If we treat the partial-differential opera tort 6/6x, 8/6y, 6/az, where x, y, z are the co-ordinates of P, as if they were scalar quantities, we are led to some remarkable and signin cant expressions. Thus if we write
3 6 0
V= IEC-l-fa;-l-/l%, . . . (16)
and operate on a scalar function qi, we obtain the vector 6 6 6
v4> =l5§ +j£+h£. .... (17)
This is called the gradient of ¢ and sometimes denoted by “grad qi "; its direction is that in which ¢> most rapidly increases, and its magni tude is equal to the corresponding rate of increase. Thus 2
3¢ 92 £92
ei + <..> + <..>- » t ~ <»>
A repetition of the operation V gives 2 2
v2¢=%+f§ -3*-2+-§ ', $. ~ (19) In the theory of attractions this expression is interpreted as measuring the degree of attenuation of the quantity φ at P; if we reverse the sign we get the concentration, −∇2φ.
Again, if we form the scalar product of the operator V into a vector A we have
∇A＝ -1-j'-gl;-l-ki) <iA.+fA2+1fA, >=~'?;§ %+5°, ,4'§ +%. . (20)
If A represent the velocity at any point (x, y, z) of a fluid, the latter expression measures the rate at which fluid is flowing away from the neighbourhood of P. By a generalization of this idea, it is called the divergence of A, and we write
| ∇A＝div A. | (21) |
The vector product [VA] has also an important significance. We find
vm = [ 0%-I-ji, -I-kg; <fA.+/A.+/fA, >
6A3 BA;, GA, BA; GA2 BA,
"I ély 02) +-l<6z dx) +k<6x By) ' (22)
If A represent as before the velocity of a fluid, the vector last written will represent the (doubled) angular velocity of a fluid element. Again if A represent the magnetic force at any point of an electro-magnetic field, the vector [V ] will represent the electric current. In the general case it is called the curl, or the rotation, of A, and we write
vA]=curl A, or rot A. . . . (23)
These definitions enable us to give a compact form to two important theorems of C. F. Gauss and Sir G. G. Stokes. The former of these may be written,
div A . dV=fAndS, . . . (24)
where the integration on the left hand includes all the volume elements dV of a given region, and that on the right includes all the surface-elements dS of the boundary, n denoting a unit vector drawn outwards normal to dS. Again, Stokes's theorem takes the form /'Ads=f curl A ndS, . . . (25)
where the integral on the right extends over any open surface, whilst on the left ds is an element of the boundin curve, treated as a vector. A certain convention is implied as to tliie relation between the positive directions of n and ds.
It is to be observed that the term “vector ” has been used to include two distinct classes of geometrical and physical entities. The first class is typified by a displacement, or a mechanical force. A polar vector, as it is called, is a magnitude associated with a certain linear direction. This may be specified by any one of a whole assemblage of parallel lines, but the two “senses” belonging to any one of the lines are distinguished. The members of the second class, that of axial vectors, are primarily not vectors at all. An axial vector is exemplified by a couple in statics; it is a magnitude associated with a closed contour lying in any one of a system of parallel planes, but the two senses in which the contour may be described are distinguished. It was therefore termed by H. Grassmann a Plangrosse or Ebenengrosse. ]ust as a polar vector may be indicated by a length, regard being paid to its sense, so an axial vector may be denoted by a certain area, regard being paid to direction round the contour. A theory of “ Plangrossen ” might be developed throughout on independent lines; but since the laws of combination prove to be analogous to those of suitable vectors drawn perpendicular to the respective areas, it is convenient for mathematical purposes to include them in the same calculus with polar vectors. In the case of couples this procedure has been familiar since the time of L. Poinsot (1804). In the Cartesian treatment of the subject no distinction between polar and axial vectors is necessary so long as we deal with congruent systems of co-ordinate axes. But when we pass from a right-handed to a left-handed system the formulae of transformation are different in the two cases. A polar vector (e.g. a displacement) is reversed by the process of reflection in a mirror normal to its direction, whilst the corresponding axial vector (e. a couple) is unaltered.
References.-The methods of vector analysis are chiefly used as a means of condensed expression of various important relations which are of frequent occurrence in mathematical physics, more especially in electricity. They are freely employed, for example, in many recent German treatises. The historical development of the subject can only be briefiy referred to. The notions of scalar and vector products originated independently with Sir W. R. Hamilton (1843) (see Quaternions) and H. Grassmann (1844), but were associated with various other conceptions of which no use is made in the simplified system above sketched. The present currency of this latter Zystem is due mainly to the advocacy of O. Heaviside and ]. W. ibbs, although for the systematic hysical interpretation of the various combinations of symbols which constantly recur in electricity and allied subjects we are indebted primarily to the classical treatise of J. C. Maxwell on Electricity and Magnetism (1873). For further details and applications of the calculus reference may be made to the following: O. Heaviside, Electro-Magnetic Theory (London, 1894); J. W. Gibbs, Vector Analysis (2nd ed., New York, 1907); M. Abraham, Die Maxwellsche Theorie d. Elektrizität (Leipzig, 1904); the articles by H. E. Timerding and M. Abraham in vol. iv. of the Encycl. d. Math. Wiss. (Leipzig, 1901–2); A. H. Bucherer, Elemente d. Vektor-Analysis (Leipzig, 1905). For an account of other systems of vector analysis see H. Hankel, Theorie d. complex en Zahlensysteme (Leipzig, 1867); and A. N. Whitehead, Universal Algebra, vol. i. (Cambridge, 1898). (H. Lb.)