104 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



5.23 Conical Coordinates. 



The three orthogonal surfaces are: the spheres, 



i. 



the two cones: 



2. 



+ f + z 2 = u 2 , 



- b 2 v 2 - c 

 V 8 s 2 



= o. 



o. 



C 2 >v 2 >b 2 >w 2 . 



a? = 



u 2 (v 2 -b 2 ) 



, _ u 2 (v* - c 2 ) (w 2 - c 2 ) 



C 2 (C 2 - b 2 ) 

 g fr? _ ff) (^2 _ ^2) ? g ffl _ W 2) (^2 _ ^2) 



A A' A ( 2A \ , 



6. div A = -5 (tr-4 J H 



2 



~ b 2 ) (C 2 - &) 



V ^ - w 2 ^ 

 o?; 



(c* - 



u(v 2 - w 2 ) 



- b 2 } (c 2 - 



u 2 (v* - w 2 ) 



curl A 



curl v A = 



- V(b 2 - w 2 ) (c 2 - w 2 ) f Vv 2 - w 2 A*] 1 , 

 V(6 2 - w 2 } (c 2 - w 2 } dA u i a 



1 A T d I 4 \ V^ 2 ~b 2 } (C 2 ~ V 2 ) dA u 



curlu, A = - - [uA v ] - = -\ 



( U dU\ U^- W 2 dV 



5.30 Elliptic Cylinder Coordinates. 



The three orthogonal surfaces are: 

 i. The elliptic cylinders: 



, 



C 2 U 2 ' C 2 (U 2 - 



I. 



