WILSON AND LEWIS. — RELATIVITY. 453 



Wo may theroforo write the following wjuations. The result of 

 applying <C> to a scalar function /' is a 1-vector 0^\ which might be 

 called the gradient of F. 



OF = ki „ + k. - + ka - — k4 - • 

 d.Vi d.v-2 0X3 d.V4 



The application of <0> to a 1-vector function f by inner multiplication 

 is a scalar, w-hich might be called the divergence of f. 



^ f dfi dfo dfs dfi 

 O.Vi 0X2 OXz 0X4 



The application of <0>, by outer multiplication, to the 1-vector f is a 

 2-vector function, which might be called the curl of f. 



Oxf = (f + f)k„ + (f + f )k. + (f + f )fc, 

 \dxi dxij \dxo axj \dX3 6x4/ 



+ (^'^' - ^■^'] ko3 + f ^-^ - ^^ ka, + f^ - ^^' ] k,o. 

 Vd.ro dxzJ "^ \dx3 dxj ^^ \dxi 6x2/ 



The expression <^ • F is a 1-vector. 



O-F = (^■^'' - ^'^'' + ^^''] ki + f^-^ - ^^ -f ^-^^^V" 

 \dx2 6x3 dXiJ ^ \dx3 dxi dx.J 



^ fdh _ dh ^ dU\ ^^ _ /a/u ^ a/24 _^ a/34V ^_ 



\dxi dx2 8x4/ ^ \dxi dx2 8x3/ 

 The product <^xP is a 3-vector. 



^ \a.r2 a.r3 a.r4/ V^-Vs dxi 8x4) 



\dxi 6x2 0X4 J \dxi dX2 0X3 J 



We might likewise expand <C> • if and O^ J^- 



The rules (30) and (24) for operation with the complement enable 

 us to write 



(O-a)* = - Oxo.*, (Oxa)* = O-a*, 



when a is a vector function of any dimensionality in four dimensional 

 space. 



It is important to note in all these equations that while (|uad 

 operates as a 1-vector, it is not a 1-vector in any geometrical sense. 



