IB IIA 1 1 r, θ, φ 1 (r, θ, φ)., r, θ, φ 0 r <, 0 θ π, 0 φ < 2π., θ, φ, φ, r, r, θ r, θ, φ,., r, θ, φ,,,. r, θ, φ r, θ, φ, 3 (r, θ, φ)., r, θ, φ e r, e θ, e φ., e r, e θ, e φ e r, e θ, e φ.,., (r, θ, φ) e r (r, θ, φ), e θ (r, θ, φ), e φ (r, θ, φ), (r, θ, φ) e r, e θ, e φ., 2 (r, θ, φ) dr (r, θ, φ) (r dr, θ dθ, φ dφ), dr h r dr e r h θ dθ e θ h φ dφ e φ (1). h r, h θ, h φ h r 1, h θ r, h φ r sin θ (2). r, θ, φ, h r (r, θ, φ), h θ (r, θ, φ), h φ (r, θ, φ), h r, h θ, h φ. 1
1. z dr er r sinθ dϕ eϕ r dθ eθ dr θ dr dθ r x 0 ϕ r sinθ dϕ r sinθ dϕ y dr dr er r dθ eθ r sinθ dϕ eϕ 2. (r, θ, φ) 2 dr 1 h r dr 1 e r h θ dθ 1 e θ h φ dφ 1 e φ dr 2 h r dr 2 e r h θ dθ 2 e θ h φ dφ 2 e φ (3) 2
z r sinθ dϕ eϕ r dθ eθ ds dθ θ r x 0 ϕ r sinθ dϕ r sinθ dϕ y 2 ds r dθ eθ x r sinθ dϕ eϕ r sinθ dθ dϕ er 3 ( ). z dr er r sinθ dϕ eϕ θ dθ r r dθ eθ x 0 ϕ dϕ y 2 dv r sinθ dr dθ dϕ 4. ds dr 1 dr 2 h θ h φ dθ 1 dφ 1 dθ 2 dφ 2 e dφ r h φ h 1 dr 1 r dφ 2 dr 2 e dr θ h r h 1 dθ 1 θ dr 2 dθ 2 e φ (4).. 3
,., r. dθ, dφ 3 2 dr 1 h θ dθ e θ r dθ e θ dr 2 h φ dφ e φ r sin θ dφ e φ (5). 2 ds h θ dθ e θ h φ dφ e φ h θ h φ dθ dφ e r r 2 sin θ dθ dφ e r (6)., ds n ds n e r, ds h θ h φ dθ dφ r 2 sin θ dθ dφ.,, θ, ds h φ dφ e φ h r dr e r h φ h r dφ dr e θ r sin θ dφ dr e θ (7),,, φ, ds h r dr e r h θ dθ e θ h r h θ dr dθ e φ r dr dθ e φ (8)., dr, dθ, dφ. 4, dr, dθ, dφ (r, θ, φ) 3, dr 1 h r dr e r dr e r dr 2 h θ dθ e θ r dθ e θ dr 3 h φ dφ e φ r sin θ dφ e φ (9) dv dr 1 (dr 2 dr 3 ) h r dr e r (h θ dθ e θ h φ dφ e φ ) h r h θ h φ dr dθ dφ r 2 sin θ dr dθ dφ (10). 4
2., r. V V V r (r, θ, φ)e r V θ (r, θ, φ)e θ V φ (r, θ, φ)e φ (11)., Γ {(r, θ, φ) a r b, θ θ 0, φ φ 0 } (12)., Γ (a, θ 0, φ 0 ) (b, θ 0, φ 0 ). Γ θ, φ,, dθ 0, dφ 0, Γ dr h r dr e r (13)., Γ (r, θ 0, φ 0 ) V dr V dr V r (r, θ 0, φ 0 )h r (r, θ 0, φ 0 ) dr (14), Γ V Γ V dr b a b a V r (r, θ 0, φ 0 )h r (r, θ 0, φ 0 ) dr V r (r, θ 0, φ 0 ) dr (15)., V r cos θ cos φ e r r 2 sin θ sin φ e θ r 3 tan θ tan φ e φ (16) Γ V dr b a r cos θ 0 cos φ 0 dr 1 2 (b2 a 2 ) cos θ 0 cos φ 0 (17). r, θ Γ {(r, θ, φ) r r 0, a θ b, φ φ 0 } (18) 5
V dr Γ b a b a V θ (r 0, θ, φ 0 )h θ (r 0, θ, φ 0 ) dθ V θ (r 0, θ, φ 0 )r 0 dθ (19)., φ Γ {(r, θ, φ) r r 0, θ θ 0, a φ b} (20) V dr Γ b a b a V φ (r 0, θ 0, φ)h φ (r 0, θ 0, φ) dφ V φ (r 0, θ 0, φ)r 0 sin θ 0 dφ (21).., Γ., p r r(p), θ θ(p), φ φ(p) (22) Γ (r(p), θ(p), φ(p)). V V V r (p)e r V θ (p)e θ V φ (p)e φ (23)., V r (r(p), θ(p), φ(p)) V r (p)., dr dr dp dp, dθ dθ dp dp, dφ dφ dp (24) dp, Γ { dr h r (p) dr dp e r h θ (p) dθ dp e θ h φ (p) dφ } dp e φ dp (25) 6
., h r (r(p), θ(p), φ(p)) h r (p)., Γ (r(p), θ(p), φ(p)) V dr { V dr V r (p)h r (p) dr dp V θ(p)h θ (p) dθ dp V φ(p)h φ (p) dφ } dp (26) dp, Γ V { V dr V r (p)h r (p) dr Γ Γ dp V θ(p)h θ (p) dθ dp V φ(p)h φ (p) dφ } dp dp { V r (p) dr dp V θ(p)r(p) dθ dp V φ(p)r(p) sin θ(p) dφ } dp dp Γ (27)... V V V r (r, θ, φ)e r V θ (r, θ, φ)e θ V φ (r, θ, φ)e φ (28)., Σ {(r, θ, φ) r r 0, a θ b, c φ d} (29)., Σ r. 1 (6), ds h θ h φ dθ dφ e r (30)., Σ (r 0, θ, φ) V ds V ds V r (r 0, θ, φ)h θ (r 0, θ, φ)h φ (r 0, θ, φ) dθ dφ (31), Σ V Σ V ds d b c a d b c a V r (r 0, θ, φ)h θ (r 0, θ, φ)h φ (r 0, θ, φ) dθ dφ V r (r 0, θ, φ)r 2 0 sin θ dθ dφ (32) 7
., V r cos θ cos φ e r r 2 sin θ sin φ e θ r 3 tan θ tan φ e φ (33) Σ V ds d b c a r 3 0 cos θ sin θ cos φ dθ dφ 1 2 r3 0(cos 2 a cos 2 b)(sin d sin c) (34)., Σ {(r, θ, φ) c r d, θ θ 0, a φ b} (35) V ds Σ d b c a d b c a V θ (r, θ 0, φ)h φ (r, θ 0, φ)h r (r, θ 0, φ) dφ dr V θ (r, θ 0, φ)r sin θ 0 dφ dr (36)., Σ {(r, θ, φ) a r b, c θ d, φ φ 0 } (37) V ds Σ d b c a d b c a V φ (r, θ, φ 0 )h r (r, θ, φ 0 )h θ (r, θ, φ 0 ) dr dθ V φ (r, θ, φ 0 )r dr dθ (38).., Σ., p, q r r(p, q), θ θ(p, q), φ φ(p, q) (39) 8
Σ (r(p, q), θ(p, q), φ(p, q)). V V V r (p, q)e r V θ (p, q)e θ V φ (p, q)e φ (40)., V r (r(p, q), θ(p, q), φ(p, q)) V r (p, q)., 1 (4) θ dr 1 dp, dθ 1 dp, dφ 1 p q p q p θ dr 2 dq, dθ 2 dq, dφ 2 q q q p p q p dp dq (41), Σ { (θ, φ) ds h θ (p, q)h φ (p, q) (p, q) e (φ, r) r h φ (p, q)h r (p, q) (p, q) e θ (r, θ) h r (p, q)h θ (p, q) (p, q) e φ } dp dq (42)., h r (r(p, q), θ(p, q), φ(p, q)) h r (p, q)., α (α, β) (p, q) p β p q q α q β q., Σ (r(p, q), θ(p, q), φ(p, q)) V ds { (θ, φ) V ds V r (p, q)h θ (p, q)h φ (p, q) (p, q) (φ, r) V θ (p, q)h φ (p, q)h r (p, q) (p, q) } (r, θ) V φ (p, q)h r (p, q)h θ (p, q) dp dq (p, q) p p (43) (44) 9
, V Σ { (θ, φ) V ds V r (p, q)h θ (p, q)h φ (p, q) Σ Σ (p, q) (φ, r) V θ (p, q)h φ (p, q)h r (p, q) (p, q) } (r, θ) V φ (p, q)h r (p, q)h θ (p, q) dp dq (p, q) { V r (p, q)r 2 (θ, φ) (p, q) sin θ(p, q) Σ (p, q) (φ, r) V θ (p, q)r(p, q) sin θ(p, q) (p, q) } (r, θ) V φ (p, q)r(p, q) dp dq (p, q). (45) Ω f. 1, (10)., f dv Ω. Ω Ω f(r, θ, φ) h r (r, θ, φ)h θ (r, θ, φ)h φ (r, θ, φ) dr dθ dφ (46) f(r, θ, φ) r 2 sin θ dr dθ dφ 3 f f f dr df (47)., dr. dr t ds f t df ds 10 (48)
. (48) f t. (47), (r, θ, φ) f f 1 f h r f e r 1 f e θ 1 f e φ (49) h θ θ h φ f e r 1 f e θ 1 f e φ (50) r θ r sin θ. (49). f r, θ, φ r e r h r, θ e θ h θ, φ e φ h φ (51)., f f f r θ θ f φ (52)., f (49). (r, θ, φ) f dr h r dr f e r h θ dθ f e θ h φ dφ f e φ (53). f f df dr f f dθ dφ (54) θ,, r f e r 1 h r ( f ) (55)., θ, φ, f (49). V V ( V ) S 11 σ V dr (56)
., σ, S. S n S ( V ) n 1 S σ V dr (57). (57) V n. (56), (r, θ, φ) V { V 1 ( Vφ ) h φ h θ h φ θ { 1 ( Vr ) h r h φ h r { 1 ( Vθ ) h θ h r h θ } Vθ h θ } Vφ h φ } Vr h r θ e r e θ e φ (58) { ( Vφ ) } 1 r sin θ Vθ r V r 2 sin θ θ { 1 ( Vr ) } Vφ r sin θ e θ r sin θ { 1 ( Vθ ) } r Vr e φ r θ e r (59). (58). V e r h θ h φ e θ h φ h r θ e φ h r h θ V r h r V θ h θ V φ h φ (60). f (49) ( f) 0 (61) 12
.,. V, f f(r, θ, φ) r r 0 V r (r, θ 0, φ 0 )h r (r, θ 0, φ 0 ) dr θ φ V θ (r, θ, φ 0 )h θ (r, θ, φ 0 ) dθ V φ (r, θ, φ )h φ (r, θ, φ ) dφ θ 0 φ 0 (62), V V f., V (58)., (56) σ 5 γ 1 : A B, γ 2 : B C, γ 3 : C D, γ 4 : D A (63) 4., σ σ γ 1 γ 4., γ 1, γ 3., γ 1 t e θ, γ 3 t e θ V dr V dr γ 1 γ 3 θ0 θ θ 0 {v θ (r 0, θ, φ 0 ) v θ (r 0, θ, φ 0 φ)} dθ., v θ V θ h θ., v θ (r 0, θ, φ 0 φ) φ 2 vθ v θ (r 0, θ, φ 0 φ) v θ (r 0, θ, φ 0 ) φ (65) 0 r 0 θ (64) θ0 θ vθ V dr V dr φ dθ γ 1 γ 3 θ 0 0 r 0 θ vθ Vθ h θ θ φ θ φ 0 r 0 θ 0 0 r 0 θ 0 (66)., 2 θ θ θ 0., γ 2, γ 4., γ 2 t e φ, γ 4 t e φ 13
V dr V dr γ 2 γ 4 φ0 φ φ 0 {v φ (r 0, θ 0 θ, φ) v φ (r 0, θ 0, φ)} dφ (67)., v φ V φ h φ., v φ (r 0, θ 0 θ, φ) θ 2 vφ v φ (r 0, θ 0 θ, φ) v φ (r 0, θ 0, φ) θ (68) θ 0 0 φ0 φ vφ V dr V dr θ dφ γ 2 γ 4 φ 0 θ 0 0 vφ Vφ h φ θ φ θ φ θ 0 φ 0 r 0 θ 0 φ 0 r 0 (69)., 2 φ φ φ 0., σ σ σ { ( Vφ ) h φ V dr θ 0 φ 0 r 0 Vθ h θ 0 r 0 θ 0 } θ φ (70)., σ n e r, S h θ h φ θ φ, (57), ( V ) e r 1 h θ h φ { ( Vφ ) h φ θ 0 φ 0 r 0 Vθ h θ., ( V ) e θ, ( V ) e φ, r 0, θ 0, φ 0 r, θ, φ, V (58). 0 r 0 θ 0 } (71) V V V V 14 ω V ds (72)
z 2 S r sinθ θ ϕ er A D B x S C θ 0 ϕ0 A( r0, θ0, ϕ0 ) B( r0, θ0 θ, ϕ0 ) C( r0, θ0 θ, ϕ0 ϕ) D( r0, θ0, ϕ0 ϕ) θ0 ϕ r0 y 5 V r. z θ θ0 r0 r σ2 x σ1 0 ϕ0 σ4 σ3 ϕ σ6 σ5 y 6 V. 15
., ω V, ω ω. (72), (r, θ, φ) V { ( Vr ) } 1 h θ h φ Vθ h φ h r Vφ h r h θ V (73) h r h θ h φ θ V 1 r 2 sin θ { ( Vr r 2 ) sin θ Vθ r sin θ θ } Vφ r (74). (73). V (58) ( V ) 0 (75).,., V (73)., (72) ω 6 σ 1 : (r 0 r, θ, φ ) σ 2 : (r 0, θ, φ ) σ 3 : ( r, θ 0 θ, φ ) σ 4 : ( r, θ 0, φ ) σ 5 : ( r, θ, φ 0 φ) σ 6 : ( r, θ, φ 0 ) r 0 r r 0 r, θ 0 θ θ 0 θ, φ 0 φ φ 0 φ (76) 6., ω ω σ 1 σ 6., σ 1 σ 2., σ 1 n e r, σ 2 n e r V ds V ds σ 1 σ 2 φ0 φ θ0 θ φ 0 θ 0 {v r (r 0 r, θ, φ) v r (r 0, θ, φ)} dθ dφ., v r V r h θ h φ., v r (r 0 r, θ, φ) r 2 vr v r (r 0 r, θ, φ) v r (r 0, θ, φ) r (78) 0 (77) 16
φ0 φ θ0 θ vr V ds V ds r dθ dφ σ 1 σ 2 φ 0 θ 0 0 vr r θ φ 0 θ 0 φ ( 0 ) Vr h θ h φ r θ φ 0 θ 0 φ 0 (79)., 2 θ, φ θ θ 0, φ φ 0. σ 3 σ 4, σ 5 σ 6 V ds ω { ( Vr ) } (80) h θ h φ Vθ h φ h r Vφ h r h θ r θ φ 0 θ 0 0 θ 0 φ 0 φ 0 r 0., V h r h θ h φ r θ φ, r 0, θ 0, φ 0 r, θ, φ, V (73). r 0 θ 0 17
{ 2 1 h θ h φ f f h r h θ h φ h r 1 f r 2 r2 ( h φ h r f θ h θ θ 1 r 2 sin θ ) ( h r h θ h φ ) ( f sin θ θ θ ) } f 1 r 2 sin 2 θ ( 2 ) f 2 (81). 18