VECTOR ANALYSIS. 77 



164. The following equations between surface-integrals for a closed 

 surface and volume-integrals for the space enclosed seem worthy of 

 mention. One or two have already been given, and are here repeated 

 for the sake of comparison. 



a) 



(2) 

 (3) 

 (4) 



=fffdvVx, (5) 



(6) 



It may aid the memory of the student to observe that the transfor- 

 mation may be effected in each case by substituting fffdv V for Jfd<r. 



165. The following equations between line-integrals for a closed 

 line and surface-integrals for any surface bounded by the line, may 

 also be mentioned. (One of these has already been given. See 

 No. 60.) 



fdp u =Jfd<r x Vu t ( 1 ) 



fdp.co =ffd<r.Vxa>, (3) 



(4) 



(5) 



These transformations may be effected by substituting ff[da- X V] for 

 fdp. The brackets are here introduced to indicate that the multi- 

 plication of dor with the i y j, k implied in V is to be performed before 

 any other multiplication which ma'y be required by a subsequent sign. 

 (This notation is not recommended for ordinary use, but only sug- 

 gested as a mnemonic artifice.) 



166. To the equations in No. 65 may be added many others, as, p*'"- 



a) 



(2) 

 (3) 

 (4) 



(5) 

 (6) 

 (7) 

 etc. 



The principle in all these cases is that if we have one of the operators 

 V, V., Vx prefixed to a product of any kind, and we make any 



