48 VECTOK ANALYSIS. 



99. To assist the memory of the student, some of the principal 

 results of Nos. 93-98 may be expressed as follows : 



Let w 1 be any solenoidal vector function of position in space, a> 2 any 

 irrotational vector function, and u any scalar function, satisfying the 

 conditions that their potentials have in general definite values. 



With respect to the solenoidal function to v j Lap and Vx are 

 inverse operators ; i.e., 



7 Lap Vxee>! = Vxj Lap ^ = a> r 



! 



Applied to the irrotational function o> 2 , either of these operators gives 

 zero; i.e., 



Lap 2 = 0, Vx o> 2 = 0. 



With respect to the irrotational function &> 2 , or the scalar function u, 

 j New and V . are inverse operators ; i.e., 



47T 



-; New V.ft) 9 = ft) 9 , V. -: New u = u. 



4-7T 4-7T 



Applied to the solenoidal function co v the operator V. gives zero ; i.e. 



V. ! = (). 



Since the most general form of a vector function having in general 

 a definite potential may be written a^-ho^, the effect of these operators 

 on such a function needs no especial mention. 



With respect to the solenoidal function o^, -r Pot and VxVx are 

 inverse operators ; i.e., 



-:- Pot Vx VXft>i = Vx-^ Pot Vxa)i = VxVx^- Pot ,=<,. 

 4?r - - 



With respect to the irrotational function o) 2 , j Pot and VV. are 



inverse operators; i.e., 

 JL 



* -*. **s v T T wn T M j. v^ v T V-K/O 



- Pot VV.w^ - V Pot V.coo= - VV. Potw 2 = ft) 2 . 



With respect to any scalar or vector function having in general a 

 definite potential j Pot and V. V are inverse operators ; i.e., 



-^- Pot V.Vu= -V.-i- Pot Vu= -V.V-^- 



4-7T 4?T 4?T 



With respect to the solenoidal function & lt V.V and VxVx are 

 equivalent; with respect to the irrotational function w 2 V.V and VV. 

 are equivalent ; i.e., 



