VECTOR ANALYSIS. 



75 



CHAPTER IV. 



(SUPPLEMENTARY TO CHAPTKB II.) 



CONCERNING THE DIFFERENTIAL AND INTEGRAL CALCULUS OF VECTORS. 



159. If CD is a vector having continuously varying values in space, 

 and p the vector determining the position of a point, we may set 



p = xi+yj+zk, 



and regard <o as a function of /Q, or of x, y, and z. Then, 



rj i dw 

 dx 



that is, 

 If we set 



Here V stands for 



. d . d 



d() 



d_ 

 dz 



exactly as in No. 52, except that it is here applied to a vector and 

 produces a dyadic, while in the former case it was applied to a 

 scalar and produced a vector. The dyadic Vo> represents tfte nine 

 differential coefficients of the three components of co with respect to 

 x, y, and z, just as the vector Vu (where u is a scalar function of p) 

 represents the three differential 'coefficients of the scalar u with 

 respect to x, y, and z. 



It is evident that the expressions V.o> and Vxco already defined 

 (No. 54) are equivalent to {Veo} 8 and { V} x . 



160. An important case is that in which the vector operated on is 

 of the form Vu. We have then 



where 



dxdz 



ik 



dydz 



jk 



kk. 



dzdx'* dzdy'^ dz 2 

 This dyadic, which is evidently self-conjugate, represents the six 



