By Parry Moon, Domina Eberle Spencer (auth.)

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**Additional info for Field Theory Handbook: Including Coordinate Systems, Differential Equations and Their Solutions**

**Example text**

1 (hyperboloids of one sheet, 0= const), tan1p = yj x (half planes, 1p = const) . Fig. 0 7. Obla t e spheroidal coordinates (Y}. 0, tp). Coordina te surfaces are oblate spheroids (1] = const) , hyperboloids of revolution (B = const ). and half-planes ('" = const ) The Stackel matrix may be written -1 o 1 1jcosh2'Y} - 1jsin2 0 . 1 Metric coefficients g~ = a3 (cosh 2'Y) 11 = cosh'Y} , - sin 2 0) cosh'Y} sin O. 13= a. l 81] e + a'" a cosh11] sin (j 8 rp -8;;;' Section 1. L 011 'I I 8 2 rp a0 2 E'I'cosh1]sinO + cot e orp fl o() 6 2 rp + -a coshT0s1nil-ea;P2 ' 2 SEPARATION OF LAPLACE'S EQUATION, 172 rp = 0, where IXl = 0 and Ul = H(y)), U2 = A (B), U3 = lJIClp) , General case 2H d-+tanhl)-----+ dH [ -P(P+1)+- q2---- 1H-o, _ d 112 d II cosh 2 )1 {222} d 2 (9 -({e il d (9 [ + cot Bilt!

Const) (~~ const), and byperboloids (8 = const, Metric coefficients (y/2 _ 02) (1]2 _ },2) (1]2 _ b 2 ) (1]2 - c2 ) , _ ((j2 _ A2) (1]2 - ( 2) g22 - (02-=7)2) (c2 _ (j2) , _ gll - _ (1]2-A2)((j2_A2) g33- (b 2-A2) (C 2-A2) , g! 2) ((j2 _ A2) .. 2)]t = [(1]2- b2) (1]2- C2)J~ , 12 = [(0 2 - b2) (C 2 - (2)J~, 13 = [(b 2 - 1. 2 ) (C 2 - A2)J~ . 11 Section I. /2f grad I}? 2)] 2 -(b2-A2)(C~A2f (dit) . /-2) div E 2 ]~ otp jiB [ (b 2 -A2) (c 2 -A2) ]~ otp + a. · (1)2-b2)&(1)2-C2)~ _~ [('f}2- (J2)~('f}2- it2)!

TRANSFORMATIONS in the complex plane, z =§(w) , where z=x+iy, w=u+iv; z=x-iy, w=u-iv. Equation No. Designation Fig. No. 02 Power Functions P1 P2 P3 P4 P5 Z= 1/w z=t w2 Z = i-w- 2 z = VZw! ln (_1_) k sn J 5 z=~lncnw J6 2Ka Z (w+z'K') +ta . w z=tw 2 I P1 I , 1- · I I coords'l Rect-I ang. 01 i Fig. , No. 06 i E 1 'I ' I • Ll~. 91 (e+ u)l - -! o~-I'--;~I - ~~-------1 - -- wh, ------- Transformation P2 I No. u)i e" sin v 1 (I) - I t -~- + + V 2 )3 a 2 e211 e2 1/ (u 2 + V2)-~ (u 2 + V2)-~ (u2 u 2 + v2 (u2 V 2 )2 gIl [el-(e"cosv+1)J2 - -1~ _________________4~1 a ---------_.