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 this way 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, do the other hand, with which we are familiar in ordinary arithmetical 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 regarded 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 reverses the sense of a vector.
As regards combinations of two vectors, we have in 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 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
AB+BC=AC.
This constitutes the definition of vector addition; and it is evident at once from fig. i that
BC+AB=AD+DC=AC=AB+BC. Hence, A and B being any two vectors, we have
A+B=B+A, (I)
i.e. addition of vectors, like ordinary arithmetical addition, is subject to the " commutative law." As regards subtraction, we define A 8
as the equivalent of A+(—B); thus in fig. I, if AB=A, BC=B, we have
A+B=AC, A—B=DB.
When the sum (or difference) of two vectors is to be further dealt with as a single vector, this may be indicated by the use of curved brackets, e.g. (A+B). It is easily seen from a figure that
(A+B)+C=A+(B+C), . . . . (2) and so on; i.e. the " associative law " of addition also holds. Again, if m 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 on reference to the similar triangles in fig. 2, where
OP=A, P'Q=B, OP'=mA, P'Q'=mB.
It will be noticed that the proofs of (I) and (3) involve the fundamental postulate of the Euclidean geometry.
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 a between their directions is called the scalar product of the two vectors, and is denoted by A .B or simply AB. Thus
AB=AB cosh=BA, . . . . (4)
so that the " commutative law of multiplication " holds here as in ordinary algebra. The " distributive law " is also valid, for we have
A(B+C) =AB+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 arigid body about a fixed point O. As explained under MEC:IANICS,
the state of motion at any instant is specified by a vector 01 representing the angular velocity. The instantaneous velocity of any other point P of the body is completely determined by the two
vectors OI and OF, viz. it is a vector normal to the plane of OI and OF, whose absolute magnitude is 01 .0P. sin B, where 0 denotes the inclination of OP to 01, and its sense is that due to a righthanded rotation about 01. A vector derived according to this rule from any two given vectors A, B is called their vector product, and is denoted by A X B or by [AB]. This type of combination is frequent in electromagnetism; 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 example 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
[A(B+C)] = [AB]+[AC], . . . (7)
as may be proved without difficulty by considering the kinematical interpretation.
Various types of triple products may also present themselves the most important being the scalar product of two vectors, one of _ which is itself given as a vector product. Thus A[BC] is equal in absolute value to the volume of the parallelepiped constructed on three edges OA, OB, OC drawn from a point 0 to represent the vectors A, B, C respectively, and it is positive or negative according as the lines OA, OB, OC follow one another in right or lefthanded cyclical order. It follows that
A[BC] = B[CA] _ — B[AC] = &c. . (8)
In order to exhibit the correspondence between the shorthand methods of vector analysis and the more familiar formulae of Cartesian geometry, we take a righthanded system of three mutually perpendicular axes Ox, Oy, Oz, and adopt three fundamental unitvectors ',Lk, having the positive directions of these axes respectively. As regards the scalar products of these unitvectors, we have, by (4),
I'=f =k2=1, Jk=kJ=1i=o. . . . (9)
Any other vector A is expressed in terms of its scalar projections As, A2, As on the coordinate axes by the formula
A=iAi+JA2+kAs (lo)
For the scalar product of any two vectors we have
AB = (iA,+JAs+kAs) (FBi I JBZ+kBs) =A1BI+A:Bs+AsBs,(I I)
as appears on developing the product and making use of (9). In particular, forming the scalar square of A we have
A2 =Ai2+A22+AP, (12)
where A denotes the absolute value of A.
Again, the rule for vector products, applied to the fundamental units, gives
A repetition of the operation p gives
v2O = a +.57 o. (19)
ax ay az=
[Pl— LP] =WI —oi
[jk] = [ki1= — [ik] =J, WI= — [Ji] =k. c (13)
Hence
[AB] _ [(iAi+JAs+kAs) (iB, +JB2+kB,)]
=i(A,B3—AsB2) +J(A,Bi —A,Bs)+k(AIB%—A,Bi)
=[BA] (14)
The correspondence with the formulae which occur in the analytical theory of rotations, &c., will be manifest. If we form the scalar product of a third vector C into [AB], we obtain
C[AB] = Bs, Cl
, Bs, C2 . . . (15)
As, B,, 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 scalars (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 partialdifferential operator*, a/ax, a/ay, a/az, where x, y, z are the coordinates of P, as if they were scalar quantities, we are led to some remarkable and signifi cant expressions. Thus if we write
v= (riff ey+kaz) , (16)
and operate on a scalar function 0, we obtain the vector
=iax~Jay+kaa. . (17)
This is called the gradient of 4, and sometimes denoted by " grad 4, "; its direction is that in which 4, most rapidly increases, and its magni tude is equal to the corresponding rate of incre1ase. Thus
(1$)
In the theory of attractions this expression is interpreted as measuring the degree of attenuation of the quantity 4 at P; if we reverse the sign we get the concentration,—v24.
Again, if we form the scalar product of the operator v into a vector A we have
(a a a 1 aA, aA2 0As
vA = t — +jay+kaz J (IAiFIA2+kA3) = —ax + ay + az . . (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
vA=div A. . (21)
The vector product [VA] has also an important significance. We find
a a a
[vA] = [ (ix+Jay+kaz) (iAI% Ai+kAs)
—_~ (aA3 aA2) () k aA2 ) ay az +~ z ax + ax a0AI
y (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 electromagnetic field, the vector [VA] 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
f div A . dV = fAndS, . . . (24)
where the integration on the left hand includes all the volumeelements dV of a given region, and that on the right includes all the surfaceelements dS of the boundary, n denoting a unit vector drawn outwards normal to dS. Again, Stokes's theorem takes the form
f 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 bounding curve, treated as a vector. A certain convention is implied as to the 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. Just 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 (18o4). 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 coordinate axes. But when we pass from a righthanded to a lefthanded 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.Rp a couple) is unaltered.
Abraham in vol. iv. of the Encycl. d. Math. Wiss. (Leipzig, 19o12); A. H. Bucherer, Elemente d. VektorAnalysis (Leipzig, 1905). For an account of other systems of vector analysis see H. Hankel, Theorie d. complexen Zahlensysteme (Leipzig, 1867) ; and A. N. Whitehead, Universal Algebra, vol. i. (Cambridge, 1898). (H. LB.)
End of Article: VECTOR ANALYSIS 

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