76 VECTOR ANALYSIS. 



differential coefficients of the second order of u with respect to x, y, 

 and z* 



161. The operators Vx and V. may be applied to dyadics in a 

 manner entirely analogous to their use with scalars. Thus we may 

 define Vx3? and V.3? by the equations 



.. 



dy dz 



Then, if $ = ai + /3j + yk, 



Or, if 3> = ia +j/3 + ky, 



& _ . rdy d/3~\ . rda dy 



~ 



. 



dx dy dz 



162. We may now regard V.V in expressions like V.Vo> as repre 

 senting two successive operations, the result of which will be 



in accordance with the definition of No. 70. We may also write 

 V.V$ for 



dx 2 dy z dz 2 



although in this case we cannot regard V.V as representing two 

 successive operations until we have defined V$.t 



That V.V<3? = V V.3? VxVx< will be evident if we suppose $ to 

 be expressed in the form ai+/3j+yk. (See No. 71.) 



163. We have already seen that 



u" u' =fdp .Vu, 



where u' and u" denote the values of u at the beginning and the em 

 of the line to which the integral relates. The same relation will hole 

 for a vector ; i.e., 



w'" CD' =fdp . 



* We might proceed to higher steps in differentiation by means of the triadics Ww, 

 VVVu, the tetradics VVVw, VVVVw, etc. See note on page 74. In like manner a dyadic 

 function of position in space ($) might be differentiated by means of the triadic V<, the 

 tetradic VV$, etc. 



t See footnote to No. 160. 



