12 BELL SYSTEM TECHNICAL JOURNAL 



It transforms upon a rotation of axes as does an ordinary vector: 



V' = a\J (4.13) 



grad M = V Mc, a matrix (4.14) 



div u —\7cU,a, scalar (4.15) 



curl u =V X w, a matrix (4.16) 



grad radius vector ='\/p = I, the idemfactor (4.17) 



{abc ...)"'= ... c'^b'^a'^ (4.18) 



{ahc . . .) cC = ... Ccbcac (4.19) 



an ...\~' n/an . 



022 ... \ =/ 1/^22 .... 



033 1/033 .. ' ^ ^ ^ 



,„ , . . s-1 (matrix) ,. ^.. 



(Scalar times matrix) = (.4.21; 



scalar 



SECTION 5 



The Geometry of Rotations 



As a first application of the matrix algebra let us compute the a matrix 

 for a few general rotations. Although we can consider a general rotation 

 as one of angle </> about the unit vector* .4 , it is easier to consider a general 

 rotation as three successive rotations about coordinate axes. 



A study of Fig. 34 shows that for a counterclockwise rotation about .Ti, 

 the new components of a vector V are: 



V[=Vi 



V2 — Vi cos + F3 sin (j) 



V's = —V2 sin 4> -\- V3 cos (j) 



whence V = aV where 



(5.1) 



* A general rotation of amount <j> about the unit axis A is given by 

 a = AAc + (I — AAc) Cos ^ + Sin <^ ^ 

 See Vector Analysis (Gibbs Wilson, Yale Press) pp. 338. 



