96 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



4.408 The gradient of a true scalar is a polar vector; the gradient of a pseudo- 

 scalar is an axial vector. 



4.409 The divergence of a polar vector is a true scalar; of an axial vector a 

 pseudo-scalar. 



4.6 Linear Vector Functions. 



4.610 A vector Q is a linear vector function of a vector R if its components, 

 Qi> Qz, Qs, along any three non-coplanar axes are linear functions of the com- 

 ponents Ri, R 2 , Rs of R along the same axes. 



4.611 Linear Vector Operator. If co is the linear vector operator, 



Q = coR. 

 This is equivalent to the three scalar equations, 



Ql = COn^l + C0i2#2 + C0i3#3, 

 + C022#2 + C0 23 #3, 



4.612 If a, b, c are the three non-coplanar unit axes, 



ton = S.acoa, co 2 i = S.bcoa, co 3 i = S.ccoa, 

 C0i2 = S.acob, co 2 2 = S.bcob, 0)32 = S.ccob, 

 cois = S.acoc, co 2 3 = .S.bcoc co 3 3 = S.ccoc. , 



4.613 The conjugate linear vector operator ' is obtained from co by replacing 

 co^by u kh ] h, k = i, 2, 3. 



4.614 In the symmetrical, or self-conjugate linear vector operator, denoted 

 by co, 



co = J(& -|- co r ). 

 Hence by 4.612 



S.acob = S.bcoa, etc. 



4.615 The general linear vector function coR may always be resolved into the 

 sum of a self -con jugate linear vector function of R and the vector product of 

 R by a vector c: 



coR = coR + F.cR, 

 where 



co = J(co + co'), 

 and 



c = J(w 3 2 - co 23 )i + |(coi3 - co 3 i)j + J(co 2 i - coi 2 )k, 

 if i, j, k are three mutually perpendicular unit vectors. 



4.616 The general linear vector operator co may be determined by three non- 

 coplanar vectors, A, B, C, where, 



