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VECTOR ANALYSIS
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VECTOR See also:ANALYSIS , in See also:mathematics, the calculus of vectors. The position of a point B relative to another point A is specified by means of the straight See also:line See also: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 See also:translation (without rotation) of a rigid See also:body. Again, a force acting on a particle, the velocity or momentum of a particle, the See also:state of electric or magnetic polarization at a particular point of a See also:medium, are examples of See also:physical entities which are naturally represented by vectors. The quantities, do the other See also:hand, with which we are See also:familiar in See also:ordinary arithmetical See also:algebra, and which have merely magnitude and sign, without any See also:intrinsic reference to direction, are distinguished as scalars, since they are completely specified by their position on the proper See also:scale of measurement. The See also:mass of a body, the pressure of a See also:gas, the See also: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 See also:italic type, vectors (when they are represented by single symbols) by " See also:black " or See also: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 See also:present in most See also:general use. Any such calculus must of course begin with See also:definitions of the fundamental symbols and operations; these are in the first instance quite arbitrary conventions, but it is convenient so to See also:frame them that the See also: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 (more-over) being the same as that of A or the See also:reverse, according as m is See also:positive or negative. We denote it by mA. The particular See also:case where m=—I is denoted by—A, so that a See also:change of sign simply reverses the sense of a vector. As regards combinations of two vectors, we have in the first See also:place the one suggested by See also:composition of displacements in See also:kinematics, or of forces or couples in See also:statics. Thus if a rigid body receive in See also:succession two See also:translations represented by AB and BC, the final result is See also: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 See also: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 See also: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 See also:geometry. The definition of " See also:work " in See also:mechanics gives us another important mode of See also:combination of vectors. The product of the See also:absolute magnitudes A, B (say) of two vectors A, B into the cosine of the See also: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 See also:proof of this statement being identical with that of the statical theorem that the sum of the See also:works of two forces in any displacement of a particle is equal to the work of their resultant. For an See also: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 See also: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 See also:plane of OI and OF, whose absolute magnitude is 01 .0P. See also:sin B, where 0 denotes the inclination of OP to 01, and its sense is that due to a right-handed rotation about 01. A vector derived according to this See also: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 electro-See also:magnetism; thus if C be the current and B the magnetic See also:induction, at any point of a conductor, the See also: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 See also:pro-ducts. 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 See also: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 See also: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 See also:left-handed cyclical See also:order. It follows that A[BC] = B[CA] _ — B[AC] = &c. . (8) In order to exhibit the See also:correspondence between the shorthand methods of vector analysis and the more familiar formulae of Cartesian geometry, we take a right-handed See also:system of three mutually perpendicular axes Ox, Oy, Oz, and adopt three fundamental unit-vectors ',Lk, having the positive directions of these axes respectively. As regards the scalar products of these unit-vectors, 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, See also:A2, As on the co-See also:ordinate axes by the See also: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 See also: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 See also:analytical theory of rotations, &c., will be See also:manifest. If we See also: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 See also:hydrodynamics and See also:electricity we are introduced to the notion of scalar and vector See also:fields. With every point P of the region under See also:consideration there are associated certain scalars (e.g. See also: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-See also:differential operator*, a/ax, a/ay, a/az, where x, y, z are the co-ordinates 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 See also:function 0, we obtain the vector =iax~-See also: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 See also: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 (IAi-FIA2+kA3) = —ax + --ay + az . . (20)
If A represent the velocity at any point (x, y, z) of a fluid, the latter expression See also:measures the rate at which fluid is flowing away from the neighbourhood of P. By a generalization of this See also: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 See also:element. Again if A represent the magnetic force at any point of an electro-magnetic See also: See also:Stokes. The former of these may be written f 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 See also:surface-elements 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 See also:curve, treated as a vector. A certain See also:convention is implied as to the relation between the positive directions of n and ds. It is to be observed that the See also: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 See also: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 See also:area, regard being paid to direction See also:round the contour. A theory of " Plangrossen " might be See also:developed through-out on See also:independent lines; but since the See also: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 See also:procedure has been familiar since the See also:time of L. See also:Poinsot (18o4). In the Cartesian treatment of the subject no distinction between polar and axial vectors is necessary so See also:long as we See also: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 See also:process of reflection in a See also:mirror normal to its direction, whilst the corresponding axial vector (e.Rp a couple) is unaltered. See also:Abraham in vol. iv. of the Encycl. d. Math. Wiss. (See also:Leipzig, 19o1-2); A. H. Bucherer, Elemente d. Vektor-Analysis (Leipzig, 1905). For an See also:account of other systems of vector analysis see H. Hankel, Theorie d. complexen Zahlensysteme (Leipzig, 1867) ; and A. N. See also:Whitehead, Universal Algebra, vol. i. (See also:Cambridge, 1898). (H. Additional information and CommentsThere are no comments yet for this article.
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