General Relativity and Einstein's Theory. 403 



Now let 



(^P ^ 2k dx 

 ' a a a 



be a vector element (four dimensional) in S. This is a general- 

 ized type of vector, and does not necessarily represent a directed 

 element of length. For instance, in polar coordinates in a plane 



dp"' = k^. dr + k^ d0 . 



Let dp be the same vector as measured by an observer in 

 S'. Then it follows from (33) and (34) that 



dp'"=Dydp'^, (35) 



^/p"= D'" • dp'^. (36) 



p. 



Consequently it is seen that Do and D' are transformation 



dyadics which enable us to determine the measured value of a 

 vector in one system from its value in another system. 

 Consider the vector differential operator 



Evidently 



d,=d; ■ D„. (37) 



D/s = d^d;. (38) 



It is seen that the transformation used here is the reverse of 

 that employed in the case of dp^ . The transformation of (35) 

 and (36) is known as contravariant, and dp^ is called a contra- 

 variant vector. On the other hand, the transformation defined 

 by (37) and (38) is covariant, and the vector Do is a covariant 



