of Four-Dimensional Vector Analysis. 



401 



§ 11. The last of the three theorems occupies a place inter- 

 mediate between the other two and connects an integral in 

 two dimensions with one in three dimensions. 



Fiir. 2. 



so that 



Proceeding as in the other two cases, if p defines the centre 

 of the element bounded by the three vectors dp x , dp 2 , and dp d i 



tdv.F{p) = 



V 2 dp 2 dp 3 . V ^ 1 D 1 . F(p) - Vidptdpi.YodfrDx . F( P ) 



-Y 2 dp 1 dp 2 .V dp z D 1 .F(p). 



It is easy to show that 



Y 2 (y s a. 1 a 2 a 3 ) . u= + V 2 a 2 a 3 . \ u l u -\- Y 2 a^^ . V a 2 « 



+ Y 2 c/ l a 2 . V a 3 «, (H"l) 



Zdv . F(p) = -{Y 2 (Y,d Pl dp 2 dp d ) . D t }F(p) 



= +Y 2 (drJ) 1 ).F(p). 



Thus by extension as in the previous cases to regions of 

 finite extent : 



^■d,¥(p)=^V 2 ldTD i ).F(p). , . (11-2) 



If the summation ^dv'F(p) is required, since 



dv'=-Idv, 

 we have : 



jjrfv'F(p) = -jjjIV 2 (rfrI) 1 )F(p). . (11-3) 



Sommerfeld defines the component of the vector divergence 

 normal to dr of a six- vector P 2 by the operation : 



Lim -jj\V (dv'V 2 ) 

 TSt-XT T8t ~ 



where ToY is the magnitude of the three-dimensional volume 

 St. This expression is called the component normal to St. 



