Curl, Divergence, Gradient, and Laplacian in Cylindrical and Spherical Coordinate Systems

Transcription

1 A P P E N D I X B Curl, Divrgnc, Gradint, and Laplacian in Cylindrical and Sphrical Coordinat Systms 788 In Chaptr 3, w introducd th curl, divrgnc, gradint, and Laplacian and drivd th xprssions or thm in th Cartsian coordinat systm. In this appndix, w driv th corrsponding xprssions in th cylindrical and sphrical coordinat systms. Considring irst th cylindrical coordinat systm, w rcall rom Sction.3 that th ininitsimal box dind by th thr orthogonal suracs intrscting at point Pr,, z2 and th thr orthogonal suracs intrscting at point Qr + dr, + d, z + dz2 is as shown in Fig. B.. From th basic dinition o th curl o a vctor introducd in Sction 3.3 and givn by A c AC d a (B.) S: S n max w ind th componnts o Aas ollows, with th aid o Fig. B.: A2 r = d: dz: [A z ] r, + d2 - [A z ] r,2 d: r d Aabcda lim d: ara abcd dz: A z [A ] r,z2 r d + [A z ] r, + d2 dz - [A ] r,z + dz2 r d - [A z ] r,2 dz - A z r d dz dz: [A ] r,z2 - [A ] r,z + dz2 dz (B.2a)

2 Appndix B 789 c d h Q(r dr, d, z dz) dz b a P(r,, z) dr r d g (r dr) d FIGURE B. Ininitsimal box ormd by incrmnting th coordinats in th cylindrical coordinat systm. Aada A2 dz: ara ad dz: [A z] r,2 dz + [A r ], z + dz2 dr -[A z ] r + dr,2 dz - [A r ],z2 dr dr dz (B.2b) dz: [A r ],z + dz2 - [A r ],z2 dz [A z ] r,2 - [A z ] r + dr,2 dr = A r z - A z r Aagba A2 z ara agb d: [A r],z2 dr + [A ] r + dr,z2 r + dr2 d - [A r ] + d,z2 dr - [A ] r,z2 r d r dr d d: (B.2c) [ra ] r + dr,z2 - [ra ] r,z2 r dr r ra 2 - A r r [A r ],z2 - [A r ] + d,z2 d: r d

3 79 Appndix B Curl, Divrgnc, Gradint, and Laplacian Combining (B.2a), (B.2b), and (B.2c), w obtain th xprssion or th curl o a vctor in cylindrical coordinats as A = c r A z - A z da r + c A r z + r c r ra 2 - A r da z a r a z a r r = 5 5 r z - A z r da (B.3) A r ra A z To ind th xprssion or th divrgnc, w us th basic dinition o th divrgnc o a vctor, introducd in Sction 3.3 and givn by # A = AS A # ds lim v: v (B.4) Evaluating th right sid o (B.4) or th box o Fig. B., w obtain # [A r] r + dr r + dr2 d dz - [A r ] r r d dz + [A ] + d dr dz - [A ] dr dz + [A z ] z + dz r dr d - [A z ] z r dr d A r dr d dz d: dz: [ra r ] r + dr - [ra r ] r r dr [A z ] z + dz - [A z ] z dz: dz A r ra r2 + r + A z z [A ] + d - [A ] d: r d (B.5) To obtain th xprssion or th gradint o a scalar, w rcall rom Sction.3 that in cylindrical coordinats, Thror, dl = dr a r + r d a + dz a z (B.6) d = r dr + d + z dz = a r a r + r a + z a zb # dr ar + r d a + dz a z 2 = # dl (B.7)

4 Appndix B 79 Thus, = r a r + r a + z a z (B.8) To driv th xprssion or th Laplacian o a scalar, w rcall rom Sction 5. that 2 = # (B.9) Thn using (B.5) and (B.8), w obtain 2 r ar r b + r a r b + z a z b r ar r b + 2 r z 2 (B.) Turning now to th sphrical coordinat systm, w rcall rom Sction.3 that th ininitsimal box dind by th thr orthogonal suracs intrscting at Pr, u, 2 and th thr orthogonal suracs intrscting at Qr + dr, u + du, + d2 is as shown in Fig. B.2. From th basic dinition o th curl o a vctor givn by (B.), w thn ind th componnts o Aas ollows, with th aid o Fig. B.2: d (r dr) sin u d d a P(r, u, ) dr c (r dr) du h Q(r dr, u du, d) r du r sin (u du) d FIGURE B.2 b g Ininitsimal box ormd by incrmnting th coordinats in th sphrical coordinat systm.

5 792 Appndix B Curl, Divrgnc, Gradint, and Laplacian Aabcda A2 r du: ara abcd d: du: d: A # Aada dl A2 u d: ara ad d: [A u] r,2 r du + [A ] r,u + du2 r sinu + du2 d - [Au] r, + d2r du - [A ] r,u2 d [A r ] u, + d2 - [A r ] u, 2 d: d r 2 sin u du d [A sin u] r,u + du2 - [A sin u] r,u2 du: du [A u ] r,2 - [A u ] r, + d2 d: d = u A sin u2 - A u [A ] r,u2 d + [A r ] u, + d2 dr -[A ] r + dr,u2 r + dr2 sin u d - [A r ] u,2 dr dr d (B.a) (B.b) = [ra ] r,u2 - [ra ] r + dr, u2 A r - r r dr r ra 2 Aagba A2 ara agb du: du: [A r] u,2 dr + [A u ] r + dr,2 r + dr2 du -[A r ] u + du,2 dr - [A u ] r,2 r du r dr du [ra u ] r + dr,2 - [ra u ] r,2 r dr [A r ] u,2 dr - [A r ] u + du,2 dr du: r du = r r ra u2 - A r r u (B.c)

6 Combining (B.a), (B.b), and (B.c), w obtain th xprssion or th curl o a vctor in sphrical coordinats as Appndix B 793 A = c u A sin u2 - A u da r + r c A r sin u - r ra 2da u + r c r ra u2 - A r u da (B.2) a r r 2 sin u = 5 r a u u a r 5 A r ra u A To ind th xprssion or th divrgnc, w us th basic dinition o th divrgnc o a vctor givn by (B.4), and by valuating its right sid or th box o Fig. B.2, w obtain # A du: d: [A r ] r + dr r + dr2 2 sin u du d - [A r ] r r 2 sin u du d c + [A u ] u + du r sinu + du2 dr d - [Au]u dr d s + [A ] + d r dr du - [A ] r dr du [r 2 A r ] r + dr - [r 2 A r ] r r 2 dr [A ] + d - [A ] d: d 2 r r2 A r 2 + r 2 sin u dr du d du: [A u sin u] u + du - [A u sin u] u du u A u sin u2 + A (B.3) To obtain th xprssion or th gradint o a scalar, w rcall rom Sction.3 that in sphrical coordinats, dl = dr a r + r du a u + d a (B.4)

7 794 Appndix B Curl, Divrgnc, Gradint, and Laplacian Thror, d = r = # dl dr + u du + d = a r a r + r u a u + a b # dr ar + r du a u + d a 2 (B.5) Thus, = r a r + r u a u + a (B.6) To driv th xprssion or th Laplacian o a scalar, w us (B.9), in conjunction with (B.3) and (B.6). Thus, w obtain 2 = r 2 ar2 r r b + u a r + a = r 2 ar2 r r b r 2 sin 2 u 2 b r 2 sin u u sin ub u asin u u b (B.7)