Transcription

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4 ii A 71 A A.2 R A

5 x, y, z f f(x, y, z) ( ) f(x + x, y, z) f(x, y, z) lim x x 0 x yz (1.1) f x 1. y, z., () yz y, z., (x, y, z), x x, xf (1.2)., 2,. 1, f(x + x, y + y, z + z) ( ) f(x, y, z) + x + x yz ( ) y + y zx ( ) z + z xy (1.3), x, y, z x, y, z 1. (1.3) 2. x, y, z dx, dy, dz f ( ) ( ) ( ) df dx + dy + dz (1.4) x y z yz zx xy

6 2 1 f., g, h f (df) gh (1.5)., g f h f (df) gh 0 (1.6). ( ) ( ) ( ) (df) yz (dx) x yz + (dy) yz y yz + (dz) zx z yz xy ( ) (1.7) (dx) x yz yz, (dx) yz (df) yz (dx) yz ( ) x yz (1.8)., x, y, z 1 u, v, w x x(u, v, w), y y(u, v, w), z z(u, v, w) (1.9), v, w ( ) ( ) (df) vw (dx) x vw + y yz, (du) vw ( ) ( ) (df) vw (dx) vw + (du) vw x (du) vw y yz zx (dy) vw + zx (dy) vw (du) vw + ( ) (dz) z vw (1.10) xy ( ) (dz) vw, z xy (du) vw (1.11), ( ) u vw ( ) x yz ( ) x + u vw ( ) y zx ( ) y + u vw ( ) z xy ( z u ) vw (1.12)

7 v, w, ( ) ( ) ( ) ( ) ( ) ( ) x y + + v x v y v z wu ( ) w uv yz ( ) x yz wu ( ) x + w uv zx wu ( ) ( ) y + y zx w uv xy ( ) z v wu (1.13) ( ) ( ) z z xy w uv (1.14). (1.12), (1.13), (1.14),. 1.2 a b a b a b cos θ (1.15)., a a,, θ a b. (1) a b b a (2) a a a 2 (3) a b 0 a b (4) (λa) b a (λb) λ (a b) (5) a (b + c) a b + a c (1.16)., a a a 1 e 1 + a 2 e 2 + a 3 e 3 (1.17),, e i e j δ ij (1.18)

8 4 1 e 1, e 2, e 3., (1.17) e i a i a i. a a a a 2 (1.19) 3,,. a b a 1 b 1 + a 2 b 2 + a 3 b 3 (1.20) a b (a 0, b 0) c 3 (1) a c b c (2) c a b sin θ (3) det[ a b c ] 0 (1.21)., (2) θ a b., (3) det[ a b c ], e x, e y, e z e 1, e 2, e 3., a, b 0 c 0. (1.21) (1), (2), (3) c a b c a b. (2) a b a b,, (3) a b., a b a 2 a 3 b 2 b 3 e 1 + a 3 a 1 b 3 b 1 e 2 + a 1 a 2 b 1 b 2 e 3 (1.22). a b e 1 e 2 e 3 a 1 a 2 a 3 b 1 b 2 b 3 (1.23)

9 (1) a b b a (2) a a 0 (3) a b 0 a b (4) (λa) b a (λb) λ (a b) (5) a (b + c) a b + a c (1.24)., (1.21) (1), (2), (3) (1.22)., (1) a 1 c 1 + a 2 c 2 + a 3 c 3 0 b 1 c 1 + b 2 c 2 + b 3 c 3 0 (1.25). 2 (a 1 b 2 a 2 b 1 ) c 1 (a 2 b 3 a 3 b 2 ) c 3 0 (a 1 b 2 a 2 b 1 ) c 2 + (a 3 b 1 a 1 b 3 ) c 3 0 (1.26)., c 1 : c 2 : c 3 a 2 a 3 b 2 b 3 : a 3 a 1 b 3 b 1 : a 1 a 2 b 1 b 2 (1.27), t ( a 2 a 3 c t b 2 b 3 e a 3 a b 3 b 1 e 2 + a 1 a 2 b 1 b 2 ) e 3 (1.28)., (2) c 2 a 2 b 2 sin 2 θ a 2 b 2 a 2 b 2 cos 2 θ (a a a 2 3)(b b b 2 3) (a 1 b 1 + a 2 b 2 + a 3 b 3 ) 2 (a 2 b 3 a 3 b 2 ) 2 + (a 3 b 1 a 1 b 3 ) 2 + (a 1 b 2 a 2 b 1 ) a 2 a 3 a 3 a 1 a 1 a b 2 b 3 b 3 b 1 b 1 b 2 (1.29)

10 6 1, (1.28) t +1 1., (3) a 1 a 2 a 3 a 2 a 3 det[ a b c ] b 1 b 2 b 3 b c 1 c 2 c 3 2 b 3 c a 3 a b 3 b 1 c a 1 a b 1 b 2 c t a 2 a 3 a 3 a 1 a 1 a b 2 b 3 b 3 b 1 b 1 b 2 (1.30), t +1., (1.21) (1), (2), (3) (1.22) a, b, c a (b c) (1.31) 3. 3 a 1 a 2 a 3 a (b c) b 1 b 2 b 3 c 1 c 2 c 3. (1.32) a (b c) b (c a) c (a b) (1.33)., 3 a, b, c., 3. a (b c), a, b, c, a (b c), a, b, c., a (b c) (1.34) 3. 3, a (b c) (a c)b (a b)c (1.35)

11 ,,, a (b c) (a b) c.

12

13 r f(r), f., r V (r), V., r., r r r + dr dr. dr, ds., dr t dr t ds (2.1).,,. r 2 dr 1, dr 2 ds dr 1 dr 2 (2.2)

14 10 2 z r dr r + dr x 0 y 2.1:. z dr2 ds r dr1 x 0 ds dr1 x dr2 y 2.2:.

15 z dr3 dr2 dv r dr1 x 0 y dv dr1 (dr2 x dr3) 2.3:. dr 1, dr 2. ds. ds, ds., ds n ds n ds (2.3).. r 3 dr 1, dr 2, dr 3 dv dr 1 (dr 2 dr 3 ) (2.4) dr 1, dr 2, dr 3. dv V Γ, Γ

16 12 2 V Γ V dr (2.5)., Γ, (2.5) Γ V. V Σ, Σ V V ds (2.6) Σ., Σ, (2.6) Σ V.. Σ Σ, Σ Σ., Σ Σ. f Ω, f Ω f dv (2.7) Ω., Ω Ω., Ω Ω. 2.4 f f f dr df (2.8)., dr. dr t ds f t df ds (2.9)

17 (2.9) f t., f f,., f df/ds 0, f f., f. f r a r b f dr df f(r b ) f(r a ) (2.10) r a r b r a r b., f f,., f (2.10) 0., V f. V V ( V ) S σ V dr (2.11)., σ, S. S n S ( V ) n 1 V dr (2.12) S. (2.12) V n.,., V V,., V 0. (2.11), V f σ 0,, V 0.,, σ ( f) 0 (2.13)

18 14 2. V V V V ω V ds (2.14)., ω V, ω ω. V 1 V ds (2.15) V,.,., V 0., V V.,. 2 f 2. ω ( V ) 0 (2.16) 2 f ( f) (2.17)

19 x, y, z 1 (x, y, z)., x, y, z < x <, < y <, < z <., y, z, z, x, x, y x, y, z,., x, y, z yz, zx, xy,. x, y, z x, y, z, 3 (x, y, z)., x, y, z e x, e y, e z., e x, e y, e z e x, e y, e z.,., (x, y, z) e x (x, y, z), e y (x, y, z), e z (x, y, z), (x, y, z) e x, e y, e z., 3.2 (x, y, z) dr (x, y, z) (x + dx, y + dy, z + dz),. dr dx e x + dy e y + dz e z (3.1)

20 16 3 z z z ez ex ey y x x x 0 y y 3.1:. z z dz ez dr dy ey dx ex x x 0 y y dr dx ex + dy ey + dz ez 3.2:.

21 z z dz ez 0 ds dy ey x x 0 y y ds dy ey x dz ez dy dz ex 3.3: (yz ). dz ez z z dx ex dy ey x x 0 y y dv dx dy dz 3.4:.

22 18 3 (x, y, z) 2 dr 1 dx 1 e x + dy 1 e y + dz 1 e z dr 2 dx 2 e x + dy 2 e y + dz 2 e z (3.2) ds dr 1 dr 2 dy 1 dz 1 dy 2 dz 2 e x + dz 1 dx 1 dz 2 dx 2 e y + dx 1 dy 1 dx 2 dy 2 e z (3.3).. yz,., yz x. dy, dz dr 1 dy e y, dr 2 dz e z (3.4). 2 yz ds dy e y dz e z dy dz e x (3.5)., ds n ds n e x, ds dy dz., zx, y, ds dz e z dx e x dz dx e y (3.6),, xy, z, ds dx e x dy e y dx dy e z (3.7)., dx, dy, dz. 3.4, dx, dy, dz (x, y, z) 3 dr 1 dx e x, dr 2 dy e y, dr 3 dz e z (3.8)

23 , dv dr 1 (dr 2 dr 3 ) dx e x (dy e y dz e z ) dx dy dz (3.9). 3.2., x. V V V x (x, y, z)e x + V y (x, y, z)e y + V z (x, y, z)e z (3.10)., Γ {(x, y, z) a x b, y y 0, z z 0 } (3.11)., Γ (a, y 0, z 0 ) (b, y 0, z 0 ). Γ y, z,, dy 0, dz 0, Γ dr dx e x (3.12)., Γ (x, y 0, z 0 ) V dr, Γ V Γ V dr V x (x, y 0, z 0 ) dx (3.13) V dr b a V x (x, y 0, z 0 ) dx (3.14)., V xyz e x + x 2 y 2 z 2 e y + x 3 y 3 z 3 e z (3.15)

24 20 3 V dr b Γ a xy 0 z 0 dx 1 2 (b2 a 2 )y 0 z 0 (3.16). x, y Γ {(x, y, z) x x 0, a y b, z z 0 } (3.17) V dr Γ b., z a V y (x 0, y, z 0 ) dy (3.18) Γ {(x, y, z) x x 0, y y 0, a z b} (3.19) V dr. Γ b a V z (x 0, y 0, z) dz (3.20)., Γ., p x x(p), y y(p), z z(p) (3.21) Γ (x(p), y(p), z(p)). V V V x (p)e x + V y (p)e y + V z (p)e z (3.22)., V x (x(p), y(p), z(p)) V x (p)., dx dx dp dp, dy dy dp dp, dz dz dp (3.23) dp

25 , Γ ( dx dr dp e x + dy dp e y + dz ) dp e z dp (3.24)., Γ (x(p), y(p), z(p)) V dr { V dr V x (p) dx dp + V y(p) dy dp + V z(p) dz } dp (3.25) dp, Γ V V dr Γ. Γ { V x (p) dx dp + V y(p) dy dp + V z(p) dz dp } dp (3.26). yz. V V V x (x, y, z)e x + V y (x, y, z)e y + V z (x, y, z)e z (3.27)., Σ {(x, y, z) x x 0, a y b, c z d} (3.28)., Σ x. 3.1 (3.5), yz ds dy dz e x (3.29)., Σ (x 0, y, z) V ds V ds V x (x 0, y, z) dy dz (3.30), Σ V d b V ds V x (x 0, y, z) dy dz (3.31) Σ c a

26 22 3., V ds Σ V xyz e x + x 2 y 2 z 2 e y + x 3 y 3 z 3 e z (3.32) d b c a x 0 yz dy dz 1 4 x 0(b 2 a 2 )(d 2 c 2 ) (3.33). yz, zx Σ {(x, y, z) c x d, y y 0, a z b} (3.34) V ds Σ d b., xy c a V y (x, y 0, z) dz dx (3.35) Σ {(x, y, z) a x b, c y d, z z 0 } (3.36) V ds. Σ d b c a V z (x, y, z 0 ) dx dy (3.37)., Σ., p, q x x(p, q), y y(p, q), z z(p, q) (3.38) Σ (x(p, q), y(p, q), z(p, q)). V V V x (p, q)e x + V y (p, q)e y + V z (p, q)e z (3.39)

27 , V x (x(p, q), y(p, q), z(p, q)) V x (p, q)., 3.1 (3.3) ( ) ( ) ( ) x y z dx 1 dp, dy 1 dp, dz 1 dp dx 2 p ) ( x q q dq, dy 2 p p ) ( y q q dq, dz 2 p p ) ( z q q dq p (3.40), Σ { } (y, z) ds (p, q) e (z, x) x + (p, q) e (x, y) y + (p, q) e z dp dq (3.41)., ( ) α (α, β) (p, q) p ( ) β p q q ( ) α q ( ) β q p p (3.42)., Σ (x(p, q), y(p, q), z(p, q)) V ds { } (y, z) V ds V x (p, q) (p, q) + V (z, x) y(p, q) (p, q) + V (x, y) z(p, q) dp dq (p, q) (3.43), V Σ V ds Σ { } (y, z) V x (p, q) Σ (p, q) + V (z, x) y(p, q) (p, q) + V (z, y) z(p, q) dp dq (p, q) (3.44). Ω f. 3.1

28 24 3, (3.9)., f dv f(x, y, z) dx dy dz (3.45). Ω Ω 3.3 (x, y, z) f ) ) f ( x yz e x + ( y e y + zx ( ) e z (3.46) z xy. (3.46). f x, y, z x e x, y e y, z e z (3.47)., ( ) ( ) f x + x y yz y + zx ( ) z (3.48) z xy., f (3.46). (x, y, z) f dr dx f e x + dy f e y + dz f e z (3.49). f ( ) ( df dx + x y yz ) dy + zx ( ) dz (3.50) z xy,, x f e x ( ) x yz (3.51)

29 z D A 0 y x S x0 C B z0 0 z S y z ex A( x0, y0, z0 ) B( x0, y0+ y, z0 ) C( x0, y0+ y, z0+ z) D( x0, y0, z0+ z) 3.5: V x.., y, z, f (3.46). y0 y (x, y, z) V { ( Vz ) ( ) Vy V y zx z { ( Vx ) ( ) Vz + z xy x { ( Vy ) ( ) Vx + x y. (3.52). e x e y e z V x y z V x V y V z yz xy yz zx } } } e x e y e z (3.52) (3.53)., f (3.46) ( f) 0,,

30 26 3. V, f f(x, y, z) + x x 0 V x (x, y 0, z 0 ) dx y z V y (x, y, z 0 ) dy + V z (x, y, z ) dz y 0 z 0 (3.54), V V f., V (3.52)., (2.11) σ 3.5 γ 1 : A B, γ 2 : B C, γ 3 : C D, γ 4 : D A (3.55) 4., σ σ γ γ 4., γ 1, γ 3., γ 1 t e y, γ 3 t e y V dr + V dr γ 1 γ 3 y0 + y y 0 {V y (x 0, y, z 0 ) V y (x 0, y, z 0 + z)} dy (3.56)., V y (x 0, y, z 0 + z) z 2 ( ) Vy V y (x 0, y, z 0 + z) V y (x 0, y, z 0 ) + z (3.57) z 0 x 0 y y0 + y ( ) Vy V dr + V dr γ 1 γ 3 y 0 z 0 ( ) Vy y z z 0 x 0 y 0 x 0 y z dy (3.58)., 2 y y y 0., γ 2, γ 4

31 , γ 2 t e z, γ 4 t e z V dr + V dr γ 2 γ 4 z0 + z z 0 {V z (x 0, y 0 + y, z) V z (x 0, y 0, z)} dz (3.59)., V z (x 0, y 0 + y, z) y 2 ( ) Vz V z (x 0, y 0 + y, z) V z (x 0, y 0, z) + y (3.60) y 0 zx 0 z0 + z ( ) Vz V dr + V dr y dz γ 2 γ 4 z 0 y 0 zx ( ) 0 Vz y z y 0 z 0 x 0 (3.61)., 2 z z z 0., σ σ { ( Vz ) ( ) } Vy V dr y z (3.62) y 0 z 0 σ z 0 x 0., σ n e x, S y z, (2.12), ( ) ( ) Vz Vy ( V ) e x (3.63) y 0 z 0 x 0 z 0 x 0 y 0., ( V ) e y, ( V ) e z, x 0, y 0, z 0 x, y, z, V (3.52). x 0 y 0 (x, y, z) V ( ) ( ) Vx Vy V + x y yz zx + ( ) Vz z xy (3.64)

32 28 3 z z0 y z x x x0 0 y0 y σ2 σ5 σ1 σ4 σ3 σ6 3.6: V.. (3.64). V (3.52) ( V ) 0 (3.65).,., V (3.64)., (2.14) ω 3.6 σ 1 : (x 0 + x, y, z ) σ 2 : (x 0, y, z ) σ 3 : ( x, y 0 + y, z ) σ 4 : ( x, y 0, z ) σ 5 : ( x, y, z 0 + z) σ 6 : ( x, y, z 0 ) x 0 x x 0 + x, y 0 y y 0 + y, z 0 z z 0 + z (3.66) 6., ω ω σ σ 6., σ 1 σ 2., σ 1 n e x, σ 2 n e x V ds + V ds σ 1 σ 2 z0 + z y0 + y z 0 y 0 {V x (x 0 + x, y, z) V x (x 0, y, z)} dy dz (3.67)

33 , V x (x 0 + x, y, z) x 2 ( ) Vx V x (x 0 + x, y, z) V x (x 0, y, z) + x (3.68) x 0 yz z0 + z y0 + y ( ) Vx V ds + V ds σ 1 σ 2 z 0 y 0 x 0 ( ) Vx x y z x 0 y 0 z 0 yz x dy dz (3.69)., 2 y, z y y 0, z z 0. σ 3 σ 4, σ 5 σ 6 V ds ω { ( Vx ) ( ) ( ) } (3.70) Vy Vz x y z x 0 y 0 z 0 y 0 z 0 + z 0 x 0 +., V x y z, x 0, y 0, z 0 x, y, z, V (3.64). x 0 y 0 ( ) ( ) ( ) 2 2 f 2 f 2 f f + + x 2 y 2 z 2 yz zx xy (3.71)., V. ( V ) ( V ) 2 V (3.72)

34 ,,. (1) (fg) ( f)g + f g (2) (fv ) f V + f V (3) (fv ) f V + f V (4) (V U) U ( V ) V ( U) (5) (V U) (U )V + (V )U + U ( V ) + V ( U) (6) (V U) (U )V (V )U + V ( U) U( V ) (3.73)

35 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 θφ, φr, rθ,. 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., 4.2 (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 (4.1). h r, h θ, h φ h r 1, h θ r, h φ r sin θ (4.2). r, θ, φ, h r (r, θ, φ), h θ (r, θ, φ), h φ (r, θ, φ)

36 32 4 z r θ r er eϕ eθ θ ϕ x 0 ϕ y 4.1:. 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 4.2:., h r, h θ, h φ. (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 (4.3)

37 z r sinθ dϕ eϕ r dθ eθ ds dθ θ r x 0 ϕ r sinθ dϕ r sinθ dϕ 2 ds r dθ eθ x r sinθ dϕ eϕ r sinθ dθ dϕ er y 4.3: (θφ ). 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 4.4:.

38 34 4 ds dr 1 dr 2 h θ h φ dθ 1 dφ 1 dθ 2 dφ 2 e r + h φ h r dφ 1 dr 1 dφ 2 dr 2 e θ + h r h θ dr 1 dθ 1 dr 2 dθ 2 e φ (4.4).. θφ,., θφ r. dθ, dφ dr 1 h θ dθ e θ r dθ e θ dr 2 h φ dφ e φ r sin θ dφ e φ (4.5). 2 θφ 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 (4.6)., ds n ds n e r, ds h θ h φ dθ dφ r 2 sin θ dθ dφ., φr, 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 (4.7),, rθ, 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 (4.8)., dr, dθ, dφ. 4.4, 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 (4.9) dr 3 h φ dφ e φ r sin θ dφ e φ

39 , 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 (4.10). 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 (4.11)., Γ {(r, θ, φ) a r b, θ θ 0, φ φ 0 } (4.12)., 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 (4.13)., 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 (4.14), Γ V b V dr V r (r, θ 0, φ 0 )h r (r, θ 0, φ 0 ) dr Γ a b a V r (r, θ 0, φ 0 ) dr., (4.15) V r cos θ cos φ e r + r 2 sin θ sin φ e θ + r 3 tan θ tan φ e φ (4.16)

40 36 4 V dr Γ b a r cos θ 0 cos φ 0 dr 1 2 (b2 a 2 ) cos θ 0 cos φ 0 (4.17). r, θ Γ {(r, θ, φ) r r 0, a θ b, φ φ 0 } (4.18) 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 (4.19)., φ Γ {(r, θ, φ) r r 0, θ θ 0, a φ b} (4.20) 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 (4.21).., Γ., p r r(p), θ θ(p), φ φ(p) (4.22) Γ (r(p), θ(p), φ(p)). V V V r (p)e r + V θ (p)e θ + V φ (p)e φ (4.23)

41 , V r (r(p), θ(p), φ(p)) V r (p)., dr dr dp dp, dθ dθ dp dp, dφ dφ dp (4.24) 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 (4.25)., 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 (4.26) 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 Γ dp + V θ(p)r(p) dθ dp + V φ(p)r(p) sin θ(p) dφ } dp dp (4.27).. θφ. V V V r (r, θ, φ)e r + V θ (r, θ, φ)e θ + V φ (r, θ, φ)e φ (4.28)., Σ {(r, θ, φ) r r 0, a θ b, c φ d} (4.29)., Σ r. 4.1 (4.6), θφ ds h θ h φ dθ dφ e r (4.30)

42 38 4., 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 (4.31), Σ V d b V ds V r (r 0, θ, φ)h θ (r 0, θ, φ)h φ (r 0, θ, φ) dθ dφ Σ c a d b c a V r (r 0, θ, φ)r 2 0 sin θ dθ dφ., (4.32) V r cos θ cos φ e r + r 2 sin θ sin φ e θ + r 3 tan θ tan φ e φ (4.33) V ds d b Σ 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) (4.34). θφ, φr Σ {(r, θ, φ) c r d, θ θ 0, a φ b} (4.35) d b V ds V θ (r, θ 0, φ)h φ (r, θ 0, φ)h r (r, θ 0, φ) dφ dr Σ c a d b c a V θ (r, θ 0, φ)r sin θ 0 dφ dr., rθ (4.36) Σ {(r, θ, φ) a r b, c θ d, φ φ 0 } (4.37) d b V ds V φ (r, θ, φ 0 )h r (r, θ, φ 0 )h θ (r, θ, φ 0 ) dr dθ Σ c a d b c a V φ (r, θ, φ 0 )r dr dθ (4.38)

43 , Σ., p, q r r(p, q), θ θ(p, q), φ φ(p, q) (4.39) 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 (4.40)., V r (r(p, q), θ(p, q), φ(p, q)) V r (p, q)., 4.1 (4.4) ( ) ( ) ( ) r θ φ dr 1 dp, dθ 1 dp, dφ 1 dp dr 2 p ) ( r q q dq, dθ 2 p p ) ( θ q q dq, dφ 2 p p ) ( φ q q dq p (4.41), 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 (4.42)., h r (r(p, q), θ(p, q), φ(p, q)) h r (p, q)., ( ) ( ) α α (α, β) (p, q) p q q p ( ) ( ) β β (4.43) p q q p

44 40 4., 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) (4.44) (p, q) } (r, θ) +V φ (p, q)h r (p, q)h θ (p, q) dp dq (p, q), 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) (4.45). Ω f. 4.1, (4.10)., f dv f(r, θ, φ) h r (r, θ, φ)h θ (r, θ, φ)h φ (r, θ, φ) dr dθ dφ Ω Ω f(r, θ, φ) r 2 sin θ dr dθ dφ. Ω (4.46)

45 (r, θ, φ) f f 1 ( ) e r + 1 ( ) h r r h θ θ f θφ ( ) e r + 1 r θφ r φr e θ + 1 ( ) e φ (4.47) h φ φ rθ ( ) e θ + 1 ( ) e φ (4.48) θ φr r sin θ φ rθ. (4.47). f r, θ, φ r e r h r, θ e θ h θ, φ e φ h φ (4.49)., ( ) ( ) ( ) f r + θ + φ (4.50) r θφ θ φr φ rθ., f (4.47). (r, θ, φ) f dr h r dr f e r + h θ dθ f e θ + h φ dφ f e φ (4.51). f ( ) ( ) df dr + r θ θφ φr dθ + ( ) dφ (4.52) φ rθ,, r f e r 1 h r ( r ) θφ (4.53)., θ, φ, f (4.47).

46 42 4 (r, θ, φ) V { V 1 ( Vφ ) ( ) } h φ Vθ h θ h θ h φ θ φr φ rθ { + 1 ( Vr ) ( ) } h r Vφ h φ h φ h r φ rθ r θφ { + 1 ( Vθ ) ( ) } h θ Vr h r h r h θ r θ { ( Vφ ) ( ) } 1 r sin θ Vθ r V r 2 sin θ θ φr φ rθ { + 1 ( Vr ) ( ) } Vφ r sin θ e θ r sin θ φ rθ r θφ { + 1 ( Vθ ) ( ) } r Vr e φ r r θ. (4.54). e r e θ e φ h θ h φ h φ h r h r h θ V r θ φ V r h r V θ h θ V φ h φ θφ θφ φr φr e r e θ e φ e r (4.54) (4.55) (4.56)., f (4.47) ( f) 0,,. V, f f(r, θ, φ) + r θ r 0 V r (r, θ 0, φ 0 )h r (r, θ 0, φ 0 ) dr θ 0 V θ (r, θ, φ 0 )h θ (r, θ, φ 0 ) dθ + φ φ 0 V φ (r, θ, φ )h φ (r, θ, φ ) dφ (4.57), V V f., V (4.54)., (2.11) σ

47 γ 1 : A B, γ 2 : B C, γ 3 : C D, γ 4 : D A (4.58) 4., σ σ γ 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 (4.59)., 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 (4.60) φ 0 θ0 + θ ( ) vθ V dr + V dr φ dθ γ 1 γ 3 θ 0 φ 0 r 0 θ ( ) ( ) vθ Vθ h θ θ φ θ φ φ 0 r 0 θ 0 φ 0 r 0 θ 0 (4.61)., 2 θ θ θ 0., γ 2, γ 4., γ 2 t e φ, γ 4 t e φ V dr + V dr γ 2 γ 4 φ0 + φ φ 0 r 0 θ {v φ (r 0, θ 0 + θ, φ) v φ (r 0, θ 0, φ)} dφ (4.62)

48 44 4., 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 (4.63) θ 0 φr 0 φ0 + φ ( ) vφ V dr + V dr θ dφ γ 2 γ 4 φ 0 θ 0 φr ( ) 0 ( ) vφ Vφ h φ θ φ θ φ θ 0 φ 0 r 0 θ 0 φ 0 r 0 (4.64)., 2 φ φ φ 0., σ σ { ( Vφ ) ( ) } h φ Vθ h θ V dr θ φ (4.65) θ 0 φ 0 σ φ 0 r 0., σ n e r, S h θ h φ θ φ, (2.12), ( V ) e r 1 h θ h φ { ( Vφ ) h φ θ 0 φ 0 r 0 r 0 θ 0 ( ) Vθ h θ φ 0 r 0 θ 0 } (4.66)., ( V ) e θ, ( V ) e φ, r 0, θ 0, φ 0 r, θ, φ, V (4.54). (r, θ, φ) V { ( Vr ) ( ) 1 h θ h φ Vθ h φ h r V + h r h θ h φ r θ θφ φr ( ) Vφ h r h θ + φ rθ } (4.67)

49 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) 4.5: V r. θ0 ϕ r0 y z θ θ0 r0 r σ2 x σ1 0 ϕ0 ϕ σ4 σ3 σ6 σ5 y 4.6: V.

50 46 4 V 1 r 2 sin θ { ( Vr ) r 2 sin θ r θφ ( ) Vθ r sin θ + θ φr + ( ) } Vφ r φ rθ (4.68). (4.67). V (4.54) ( V ) 0 (4.69).,., V (4.67)., (2.14) ω 4.6 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 (4.70) 6., ω ω σ 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 (4.71)., 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 (4.72) r 0 θφ

51 φ0 + φ θ0 + θ ( ) vr V ds + V ds r dθ dφ σ 1 σ 2 φ 0 θ 0 r 0 θφ ( ) vr r θ φ r 0 θ 0 φ ( 0 ) Vr h θ h φ r θ φ r 0 θ 0 φ 0 (4.73)., 2 θ, φ θ θ 0, φ φ 0. σ 3 σ 4, σ 5 σ 6 V ds ω { ( Vr ) ( ) ( ) } h θ h φ Vθ h φ h r Vφ h r h θ r θ φ r 0 θ 0 φ 0 θ 0 φ 0 + φ 0 r 0 + r 0 θ 0 (4.74)., V h r h θ h φ r θ φ, r 0, θ 0, φ 0 r, θ, φ, V (4.67). { ( ) 2 1 h θ h φ f h r h θ h φ r h r r θφ ( ) ( h φ h r h r h θ + + θ h θ θ φr φ h φ 1 ( ) r 2 r2 + 1 ( ) sin θ + r r θφ r 2 sin θ θ θ φr. ) } φ rθ 1 r 2 sin 2 θ ( ) 2 f φ 2 rθ (4.75)

52

53 ρ, φ, z 1 (ρ, φ, z)., ρ, φ, z 0 ρ <, 0 φ < 2π, < z <., φ, z, z, ρ, ρ, φ ρ, φ, z,., ρ, φ, z φz, zρ, ρφ,. ρ, φ, z ρ, φ, z, 3 (ρ, φ, z)., ρ, φ, z e ρ, e φ, e z., e ρ, e φ, e z e ρ, e φ, e z.,., (ρ, φ, z) e ρ (ρ, φ, z), e φ (ρ, φ, z), e z (ρ, φ, z), (ρ, φ, z) e ρ, e φ, e z., 5.2 (ρ, φ, z) dr (ρ, φ, z) (ρ + dρ, φ + dφ, z + dz), dr h ρ dρ e ρ + h φ dφ e φ + h z dz e z (5.1). h ρ, h φ, h z h ρ 1, h φ ρ, h z 1 (5.2)

54 50 5 z z z ez ρ eϕ eρ ϕ x 0 ϕ ρ y 5.1:. z dz z dz ez dρ eρ ρ dρ dr ρ dϕ eϕ x 0 ϕ dϕ ρ dϕ y dr dρ eρ + ρ dϕ eϕ + dz ez 5.2:.. ρ, φ, z, h ρ (ρ, φ, z), h φ (ρ, φ, z), h z (ρ, φ, z), h ρ, h φ, h z. (ρ, φ, z) 2 dr 1 h ρ dρ 1 e ρ + h φ dφ 1 e φ + h z dz 1 e z dr 2 h ρ dρ 2 e ρ + h φ dφ 2 e φ + h z dz 2 e z (5.3)

55 z dz z dz ez ds ρ ρ dϕ eϕ x 0 ϕ dϕ ρ dϕ y ds ρ dϕ eϕ x dz ez ρ dϕ dz eρ 5.3: (φz ). z dz z dz ez ρ ρ dϕ eϕ dρ eρ x 0 ϕ dϕ dv ρ dρ dϕ dz y 5.4:.

56 52 5 ds dr 1 dr 2 h φ h z dφ 1 dz 1 dφ 2 dz 2 e ρ + h z h ρ dz 1 dρ 1 dz 2 dρ 2 e φ + h ρ h φ dρ 1 dφ 1 dρ 2 dφ 2 e z (5.4).. φz,., φz ρ. dφ, dz dr 1 h φ dφ e φ ρ dφ e φ dr 2 h z dz e z dz e z (5.5). 2 φz ds h φ dφ e φ h z dz e z h φ h z dφ dz e ρ ρ dφ dz e ρ (5.6)., ds n ds n e ρ, ds h φ h z dφ dz ρ dφ dz., zρ, φ, ds h z dz e z h ρ dρ e ρ h z h ρ dz dρ e φ dz dρ e φ (5.7),, ρφ, z, ds h ρ dρ e ρ h φ dφ e φ h ρ h φ dρ dφ e z ρ dρ dφ e z (5.8)., dρ, dφ, dz. 5.4, dρ, dφ, dz (ρ, φ, z) 3 dr 1 h ρ dρ e ρ dρ e ρ dr 2 h φ dφ e φ ρ dφ e φ (5.9) dr 3 h z dz e z dz e z

57 , dv dr 1 (dr 2 dr 3 ) h ρ dρ e ρ (h φ dφ e φ h z dz e z ) h ρ h φ h z dρ dφ dz ρ dρ dφ dz (5.10). 5.2., ρ. V V V ρ (ρ, φ, z)e ρ + V φ (ρ, φ, z)e φ + V z (ρ, φ, z)e z (5.11)., Γ {(ρ, φ, z) a ρ b, φ φ 0, z z 0 } (5.12)., Γ (a, φ 0, z 0 ) (b, φ 0, z 0 ). Γ φ, z,, dφ 0, dz 0, Γ dr h ρ dρ e ρ (5.13)., Γ (ρ, φ 0, z 0 ) V dr V dr V ρ (ρ, φ 0, z 0 )h ρ (ρ, φ 0, z 0 ) dρ (5.14), Γ V b V dr V ρ (ρ, φ 0, z 0 )h ρ (ρ, φ 0, z 0 ) dρ Γ a b a V ρ (ρ, φ 0, z 0 ) dρ., (5.15) V ρz cos φ e ρ + ρ 2 z 2 sin φ e φ + ρ 3 z 3 tan φ e z (5.16)

58 54 5 V dr Γ b a ρz 0 cos φ 0 dρ 1 2 (b2 a 2 )z 0 cos φ 0 (5.17). ρ, φ Γ {(ρ, φ, z) ρ ρ 0, a φ b, z z 0 } (5.18) V dr Γ b a b a V φ (ρ 0, φ, z 0 )h φ (ρ 0, φ, z 0 ) dφ V φ (ρ 0, φ, z 0 )ρ 0 dφ (5.19)., z Γ {(ρ, φ, z) ρ ρ 0, φ φ 0, a z b} (5.20) V dr Γ b a b a V z (ρ 0, φ 0, z)h z (ρ 0, φ 0, z) dz V z (ρ 0, φ 0, z) dz (5.21).., Γ., p ρ ρ(p), φ φ(p), z z(p) (5.22) Γ (ρ(p), φ(p), z(p)). V V V ρ (p)e ρ + V φ (p)e φ + V z (p)e z (5.23)

59 , V ρ (ρ(p), φ(p), z(p)) V ρ (p)., dρ dρ dp dp, dφ dφ dp dp, dz dz dp (5.24) dp, Γ { dr h ρ (p) dρ dp e ρ + h φ (p) dφ dp e φ + h z (p) dz } dp e z dp (5.25)., h ρ (ρ(p), φ(p), z(p)) h ρ (p)., Γ (ρ(p), φ(p), z(p)) V dr V dr { V ρ (p)h ρ (p) dρ dp + V φ(p)h φ (p) dφ dp + V z(p)h z (p) dz } dp (5.26) dp, Γ V { V dr V ρ (p)h ρ (p) dρ Γ Γ dp + V φ(p)h φ (p) dφ dp + V z(p)h z (p) dz } dp dp { V ρ (p) dρ Γ dp + V φ(p)ρ(p) dφ dp + V z(p) dz } dp dp (5.27).. φz. V V V ρ (ρ, φ, z)e ρ + V φ (ρ, φ, z)e φ + V z (ρ, φ, z)e z (5.28)., Σ {(ρ, φ, z) ρ ρ 0, a φ b, c z d} (5.29)., Σ ρ. 5.1 (5.6), φz ds h φ h z dφ dz e ρ (5.30)

60 56 5., Σ (ρ 0, φ, z) V ds V ds V ρ (ρ 0, φ, z)h φ (ρ 0, φ, z)h z (ρ 0, φ, z) dφ dz (5.31), Σ V d b V ds V ρ (ρ 0, φ, z)h φ (ρ 0, φ, z)h z (ρ 0, φ, z) dφ dz Σ c a d b c a V ρ (ρ 0, φ, z)ρ 0 dφ dz., (5.32) V ρz cos φ e ρ + ρ 2 z 2 sin φ e φ + ρ 3 z 3 tan φ e z (5.33) V ds d b Σ c a ρ 2 0z cos φ dφ dz 1 2 ρ2 0(d 2 c 2 )(sin b sin a) (5.34). φz, zρ Σ {(ρ, φ, z) c ρ d, φ φ 0, a z b} (5.35) d b V ds V φ (ρ, φ 0, z)h z (ρ, φ 0, z)h ρ (ρ, φ 0, z) dz dρ Σ c a d b c a V φ (ρ, φ 0, z) dz dρ., ρφ (5.36) Σ {(ρ, φ, z) a ρ b, c φ d, z z 0 } (5.37) d b V ds V z (ρ, φ, z 0 )h ρ (ρ, φ, z 0 )h φ (ρ, φ, z 0 ) dρ dφ Σ c a d b c a V z (ρ, φ, z 0 )ρ dρ dφ (5.38)

61 , Σ., p, q ρ ρ(p, q), φ φ(p, q), z z(p, q) (5.39) Σ (ρ(p, q), φ(p, q), z(p, q)). V V V ρ (p, q)e ρ + V φ (p, q)e φ + V z (p, q)e z (5.40)., V ρ (ρ(p, q), φ(p, q), z(p, q)) V ρ (p, q)., 5.1 (5.4) ( ) ( ) ( ) ρ φ z dρ 1 dp, dφ 1 dp, dz 1 dp dρ 2 p ) ( ρ q q dq, dφ 2 p p ) ( φ q q dq, dz 2 p p ) ( z q q dq p (5.41), Σ { (φ, z) ds h φ (p, q)h z (p, q) (p, q) e (z, ρ) ρ + h z (p, q)h ρ (p, q) (p, q) e φ } (ρ, φ) +h ρ (p, q)h φ (p, q) (p, q) e z dp dq (5.42)., h ρ (ρ(p, q), φ(p, q), z(p, q)) h ρ (p, q)., ( ) ( ) α α (α, β) (p, q) p q q p ( ) ( ) β β (5.43) p q q p

62 58 5., Σ (ρ(p, q), φ(p, q), z(p, q)) V ds { (φ, z) V ds V ρ (p, q)h φ (p, q)h z (p, q) (p, q) (z, ρ) + V φ (p, q)h z (p, q)h ρ (p, q) (5.44) (p, q) } (ρ, φ) +V z (p, q)h ρ (p, q)h φ (p, q) dp dq (p, q), V Σ { (φ, z) V ds V ρ (p, q)h φ (p, q)h z (p, q) Σ Σ (p, q) (z, ρ) + V φ (p, q)h z (p, q)h ρ (p, q) (p, q) } (ρ, φ) +V z (p, q)h ρ (p, q)h φ (p, q) dp dq (p, q) { (φ, z) V ρ (p, q)ρ(p, q) Σ (p, q) (z, ρ) + V φ (p, q) (p, q) } (ρ, φ) +V z (p, q)ρ(p, q) dp dq (p, q) (5.45). Ω f. 5.1, (5.10)., f dv f(ρ, φ, z) h ρ (ρ, φ, z)h φ (ρ, φ, z)h z (ρ, φ, z) dρ dφ dz Ω Ω f(ρ, φ, z) ρ dρ dφ dz. Ω (5.46)

63 (ρ, φ, z) f ) ) f 1 h ρ ( ρ φz e ρ + 1 h φ ( φ zρ e φ + 1 ( ) e z (5.47) h z z ρφ f ( ) e ρ + 1 ( ) e φ + ρ φz ρ φ zρ ( ) e z (5.48) z ρφ. (5.47). f ρ, φ, z ρ e ρ h ρ, φ e φ h φ, z e z h z (5.49)., ( ) ( ) f ρ + ρ φ φz φ + zρ ( ) z (5.50) z ρφ., f (5.47). (ρ, φ, z) f dr h ρ dρ f e ρ + h φ dφ f e φ + h z dz f e z (5.51). f ( ) ( df dρ + ρ φ φz ) dφ + zρ ( ) dz (5.52) z ρφ,, ρ f e ρ 1 ( ) h ρ ρ φz (5.53)., φ, z, f (5.47).

64 60 5 z D S z C z0 ρ0 S ρ ϕ z eρ A x B 0 ϕ0 ϕ A( ρ0, ϕ0, z0 ) B( ρ0, ϕ0+ ϕ, z0 ) C( ρ0, ϕ0+ ϕ, z0+ z) D( ρ0, ϕ0, z0+ z) 5.5: V ρ. y (ρ, φ, z) V { V 1 ( Vz ) ( h z Vφ h φ h φ h z φ zρ z { ( Vρ ) ( h ρ Vz h z + 1 h z h ρ + 1 h ρ h φ z { ( Vφ h φ ρ ) ρφ φz ρ ( Vρ h ρ φ ) ) ) ρφ φz zρ } } } e ρ e φ e z (5.54) { V 1 ( Vz ) ( ) Vφ ρ ρ φ zρ z { ( Vρ ) ( ) } Vz + z ρφ ρ φz { + 1 ( Vφ ) ( ) ρ Vρ ρ ρ φ φz ρφ } e φ zρ } e ρ e z (5.55)

65 (5.54). e ρ e φ e z h φ h z h z h ρ h ρ h φ V ρ φ z V ρ h ρ V φ h φ V z h z (5.56)., f (5.47) ( f) 0,,. V, f f(ρ, φ, z) + ρ ρ 0 V ρ (ρ, φ 0, z 0 )h ρ (ρ, φ 0, z 0 ) dρ φ φ 0 V φ (ρ, φ, z 0 )h φ (ρ, φ, z 0 ) dφ + z z 0 V z (ρ, φ, z )h z (ρ, φ, z ) dz (5.57), V V f., V (5.54)., (2.11) σ 5.5 γ 1 : A B, γ 2 : B C, γ 3 : C D, γ 4 : D A (5.58) 4., σ σ γ γ 4., γ 1, γ 3., γ 1 t e φ, γ 3 t e φ V dr + V dr γ 1 γ 3 φ0 + φ φ 0 {v φ (ρ 0, φ, z 0 ) v φ (ρ 0, φ, z 0 + z)} dφ (5.59)., v φ V φ h φ., v φ (ρ 0, φ, z 0 + z) z 2 ( ) vφ v φ (ρ 0, φ, z 0 + z) v φ (ρ 0, φ, z 0 ) + z (5.60) z 0 ρ 0 φ

66 62 5 φ0 + φ ( ) vφ V dr + V dr z dφ γ 1 γ 3 φ 0 z 0 ρ 0 φ ( ) ( ) vφ Vφ h φ φ z φ z z 0 ρ 0 φ 0 z 0 ρ 0 φ 0 (5.61)., 2 φ φ φ 0., γ 2, γ 4., γ 2 t e z, γ 4 t e z V dr + V dr γ 2 γ 4 z0 + z z 0 {v z (ρ 0, φ 0 + φ, z) v z (ρ 0, φ 0, z)} dz (5.62)., v z V z h z., v z (ρ 0, φ 0 + φ, z) φ 2 ( ) vz v z (ρ 0, φ 0 + φ, z) v z (ρ 0, φ 0, z) + φ (5.63) φ 0 zρ 0 z0 + z ( ) vz V dr + V dr φ dz γ 2 γ 4 z 0 φ 0 zρ ( ) 0 ( ) vz Vz h z φ z φ z φ 0 z 0 ρ 0 φ 0 z 0 ρ 0 (5.64)., 2 z z z 0., σ σ { ( Vz ) ( ) } h z Vφ h φ V dr φ z (5.65) φ 0 z 0 σ z 0 ρ 0 ρ 0 φ 0

67 z z z0 ρ0 ρ x σ2 0 ϕ0 ϕ σ5 y σ4 σ3 σ1 σ6 5.6: V.., σ n e ρ, S h φ h z φ z, (2.12), ( V ) e ρ 1 h φ h z { ( Vz ) h z φ 0 z 0 ρ 0 ( ) Vφ h φ z 0 ρ 0 φ 0 } (5.66)., ( V ) e φ, ( V ) e z, ρ 0, φ 0, z 0 ρ, φ, z, V (5.54). (ρ, φ, z) V { ( Vρ ) ( ) 1 h φ h z Vφ h z h ρ V + h ρ h φ h z ρ φ φz zρ ( ) Vz h ρ h φ + z ρφ } (5.67) V 1 ρ { ( Vρ ) ρ ρ φz + ( ) Vφ + φ zρ ( ) Vz ρ z ρφ } (5.68)

68 64 5. (5.67). V (5.54) ( V ) 0 (5.69).,., V (5.67)., (2.14) ω 5.6 σ 1 : (ρ 0 + ρ, φ, z ) σ 2 : (ρ 0, φ, z ) σ 3 : ( ρ, φ 0 + φ, z ) σ 4 : ( ρ, φ 0, z ) σ 5 : ( ρ, φ, z 0 + z) σ 6 : ( ρ, φ, z 0 ) ρ 0 ρ ρ 0 + ρ, φ 0 φ φ 0 + φ, z 0 z z 0 + z (5.70) 6., ω ω σ σ 6., σ 1 σ 2., σ 1 n e ρ, σ 2 n e ρ V ds + V ds σ 1 σ 2 z0 + z φ0 + φ z 0 φ 0 {v ρ (ρ 0 + ρ, φ, z) v ρ (ρ 0, φ, z)} dφ dz (5.71)., v ρ V ρ h φ h z., v ρ (ρ 0 + ρ, φ, z) ρ 2 ( ) vρ v ρ (ρ 0 + ρ, φ, z) v ρ (ρ 0, φ, z) + ρ (5.72) ρ 0 φz z0 + z φ0 + φ ( ) vρ V ds + V ds ρ dφ dz σ 1 σ 2 z 0 φ 0 ρ 0 φz ( ) vρ ρ φ z ρ 0 φ 0 z ( 0 ) Vρ h φ h z ρ φ z ρ 0 φ 0 z 0 (5.73)

69 , 2 φ, z φ φ 0, z z 0. σ 3 σ 4, σ 5 σ 6 V ds ω { ( Vρ ) ( ) ( ) } h φ h z Vφ h z h ρ Vz h ρ h φ ρ φ z ρ 0 φ 0 z 0 φ 0 z 0 + z 0 ρ 0 + ρ 0 φ 0 (5.74)., V h ρ h φ h z ρ φ z, ρ 0, φ 0, z 0 ρ, φ, z, V (5.67). { ( ) 2 1 h φ h z f h ρ h φ h z ρ h ρ ρ φz ( ) ( ) } h z h ρ h ρ h φ + + φ h φ φ zρ z h z z ρφ 1 ( ) ρ ρ ρ + 1 ( ) ( ) 2 f 2 f + ρ φz ρ 2 φ 2 zρ z 2 ρφ (5.75).

70

71 f(x) b a df dx f(b) f(a) (6.1) dx..,. f, 2 (2.10) r a r b f dr f(r b ) f(r a ) (6.2). (6.1), (6.2) V V ( V ) S σ V dr (6.3), 2 3 σ 1, σ 2., σ 1 σ 2 1 γ,, σ 1 σ 2., σ 1 σ 2 ( V ) 1 S 1 + ( V ) 2 S 2 V dr + σ 1 V dr σ 2 (6.4)

72 68 6 S1 γ S2 σ1 t1 t2 σ2 6.1:., γ., σ 1, σ 2 γ t 1, t 2., ( V ) 1 S 1 + ( V ) 2 S 2 V dr (6.5) (σ 1 +σ 2 )., σ 1 + σ 2 σ 1 σ 2., Σ 3 Σ i σ i, ( V ) i S i «P V dr (6.6) i σ i i. (6.6) ( V ) ds V dr (6.7) Σ. (6.7)., V., f (6.2),. Σ

73 V1 V2 ω1 σ n1 n2 ω :. V V V V ω V ds (6.8), 2 4 ω 1, ω 2., ω 1 ω 2 1 σ., ω 1 ω 2 ( V ) 1 V 1 + ( V ) 2 V 2 V ds + V ds (6.9) ω 1 ω 1, σ., ω 1, ω 2 σ n 1, n 2., ( V ) 1 V 1 + ( V ) 2 V 2 V ds (6.10) (ω 1 +ω 2 )., ω 1 + ω 2 ω 1 ω 2., Ω 4 Ω i ω i, ( V ) i V i «P V ds (6.11) i ω i i

74 70 6. (6.11) V dv V ds (6.12) Ω Ω. (6.12) (1) ( f)g dr [ fg ] rb r a f g dr r a r b r a r b (2) ( f V ) ds fv dr (f V ) ds Σ Σ Σ (3) f V dv fv ds f V dv (6.13) Ω Ω Ω (4) U ( V ) dv (V U) ds Ω Ω + V ( U) dv, (3) V g f g dv f g ds f 2 g dv (6.14) Ω Ω Ω., f g ( (f g g f) ds f 2 g g 2 f ) dv (6.15) Ω Ω. (6.14), (6.15). Ω

75 71 A A.1 (1.12), (1.13), (1.14) [x, y, z]/ [u, v, w] ( ) ( ) ( ) x x x u vw v wu w [x, y, z] ( ) ( ) ( ) uv [u, v, w] y y y u vw v wu w uv ( ) ( ) ( ) z z z u v w vw wu uv (A.1) [( ) u vw ( ) ( ) ] v wu w uv [ ( ) x yz ( ) y zx ( ) ] z xy [x, y, z] [u, v, w] (A.2).,, (x, y, z)/ (u, v, w)., ( ) ( ) ( ) x x x u vw v wu w (x, y, z) ( ) ( ) ( ) uv (u, v, w) y y y u vw v wu w (A.3) uv ( ) ( ) ( ) z z z u v w. (A.2) f r, s, t, vw wu uv [ r, s, t ] [x, y, z] [x, y, z] [u, v, w] [ r, s, t ] [u, v, w] (A.4)

76 72 A., ( r, s, t ) (x, y, z) (x, y, z) (u, v, w) ( r, s, t ) (u, v, w) (A.5). A.2 R 3 R 3 2 e x, e y, e z e u, e v, e w, e u, e v, e w e x, e y, e z e u P 11 e x + P 21 e y + P 31 e z e v P 12 e x + P 22 e y + P 32 e z (A.6) e w P 13 e x + P 23 e y + P 33 e z., P ij P 11 P 12 P 13 P P 21 P 22 P 23 P 31 P 32 P 33 (A.7) e x, e y, e z e u, e v, e w. (A.6) [e u e v e w ] [e x e y e z ] P (A.8). e x, e y, e z, (A.6) e x, e y, e z e x e u e x e v e x e w P e y e u e y e v e y e w (A.9) e z e u e z e v e z e w., e x, e y, e z e u, e v, e w e x Q 11 e u + Q 21 e v + Q 31 e w e y Q 12 e u + Q 22 e v + Q 32 e w (A.10) e z Q 13 e u + Q 23 e v + Q 33 e w

77 A.2. R 3 73 Q ij Q (A.9) e u e x e u e y e u e z Q e v e x e v e y e v e z (A.11) e w e x e w e y e w e z., a b b a, Q P t P., Q t P (A.12)., (A.10) [e x e y e z ] [e u e v e w ] Q (A.13), (A.8), Q P P 1. Q P 1 (A.14)., (A.12) (A.14) P 1 t P (A.15). (A.15),.,., t P P I det P e x, e y, e z e u, e v, e w det P +1,, det P 1..,. a e x, e y, e z e u, e v, e w a [e x e y e z ] a x a y [e u e v e w ] a u a v (A.16) a z a w

78 74 A, a u a v P 1 a x a y (A.17) a w a z. (A.17). A.3 dr (x, y, z) dr dx e x + dy e y + dz e z (A.18)., r (x, y, z) 1 (u, v, w) x x(u, v, w), y y(u, v, w), z z(u, v, w) (A.19). x, y, z ( ) ( ) ( ) x x x dx du + dv + dw u vw v wu w uv ( ) ( ) ( ) y y y dy du + dv + dw u vw v wu w uv ( ) ( ) ( ) z z z dz du + dv + dw u v w vw, (A.18) dr. ( ) ( ) r x r u u vw u ( ) ( ) r x r v v wu v ( ) ( ) r x r w w w uv wu dr du r u + dv r v + dw r w vw wu uv ( ) y e x + u ( ) y e x + v ( ) y e x + w vw wu uv uv ( ) z e y + u ( ) z e y + v ( ) z e y + w e z vw e z wu e z uv (A.20) (A.21) (A.22)

79 A , 3 r u, r v, r w,, r α r β 0 (α β) (A.23), (u, v, w). (u, v, w) r u, r v, r w e u r u /h u, e v r v /h v, e w r w /h w (A.24), e u, e v, e w., h u, h v, h w h u r u, h v r v, h w r w., ( x ) 2 ( ) 2 ( ) 2 y z h u + + u vw u vw u vw ( x ) 2 ( ) 2 ( ) 2 y z h v + + (A.25) h w v ( x w wu ) 2 uv + v ( y w wu ) 2. (A.24), dr uv + v ( z w wu ) 2 uv dr h u du e u + h v dv e v + h w dw e w (A.26)., (x, y, z), u x, v y, w z, h x 1, h y 1, h z 1. (A.22) (A.24) e u, e v, e w e x, e y, e z e u 1 ( ) x e x + 1 ( ) y e y + 1 ( ) z h u u vw h u u vw h u u e v 1 ( ) x e x + 1 ( ) y e y + 1 ( ) z h v v wu h v v wu h v v e w 1 ( ) x e x + 1 ( ) y e y + 1 ( ) z h w w h w w h w w uv uv e z vw e z wu e z uv (A.27)

80 76 A. h u [e u e v e w ] [e x e y e z ] P (A.28), e x, e y, e z e u, e v, e w P ( ) ( ) ( ) 1 x 1 x 1 x h u u vw h v v wu h w w ( ) ( ) ( ) uv 1 y 1 y 1 y P h u u vw h v v wu h w w (A.29) uv ( ) ( ) ( ) 1 z 1 z 1 z u v w vw h v., e x, e y, e z e u, e v, e w, P., u, v, w x, y, z wu u u(x, y, z), v v(x, y, z), w w(x, y, z) (A.30) h w uv, u, v, w ( ) ( ) u u du dx + x yz y ( ) ( ) v v dv dx + x yz y ( ) ( ) w w dw dx + x y yz dy + zx zx ( ) u dz z xy ( ) v dy + dz z xy ( ) w dy + dz z zx xy (A.31). (A.26),, (A.18) ( ) ( ) ( ) u v w e x h u e u + h v e v + h w e w x yz x yz x yz ( ) ( ) ( ) u v w e y h u e u + h v e v + h w e w (A.32) y zx y zx y zx ( ) ( ) ( ) u v w e z h u e u + h v e v + h w e w z z z xy xy xy

81 A , (A.28), ( ) u h u x ( ) P 1 v h v x ( ) w h w x [e x e y e z ] [e u e v e w ] P 1 (A.33) yz yz yz ( ) u h u y ( ) v h v y ( ) w h w y zx zx zx ( ) u h u z ( ) v h v z ( ) w h w z xy xy xy (A.34)., P 1, (A.29) P, P P 1 t P P 1., P P 1. (A.1) P [x, y, z] [u, v, w] P 1 1/h u /h v /h w h u h v h w [u, v, w] [x, y, z] (A.35) (A.36)., (A.4) P P 1., (A.2) [ ( ) ( ) ( ) ] h u u vw h v v wu h w w uv [ ( ) ( ) ( ) ] (A.37) P x y z [( ) x yz h u yz ( ) ( ) ] y zx z xy [ ( ) 1 u 1 vw h v zx ( ) v wu xy 1 h w ( ) ] (A.38) P 1 w uv

82 78 A. (u, v, w) 1 (r, s, t) u u(r, s, t), v v(r, s, t), w w(r, s, t) (A.39)., [e u e v e w ] [e x e y e z ] [e r e s e t ] [e x e y e z ] [x, y, z] [u, v, w] [x, y, z] [ r, s, t ] 1/h u /h v /h w 1/h r /h s /h t (A.40) (A.41), e u, e v, e w e r, e s, e t h u 0 0 1/h r 0 0 [u, v, w] P 0 h v 0 0 1/h s 0 [ r, s, t ] 0 0 h w 0 0 1/h t ( ) ( ) ( ) h u u h u u h u u h r r st h s s tr h t t (A.42) rs ( ) ( ) ( ) h v v h v v h v v h r h w h r r ) ( w r st st h s h w h s s ) ( w s tr tr h t h w h t t ) ( w t [e r e s e t ] [e u e v e w ] P (A.43) rs rs

83 A , t P P I h r, h s, h t ( ) 2 ( ) 2 ( ) 2 u v w h r h 2 u + h r 2 v + h st r 2 w st r st ( ) 2 ( ) 2 ( ) 2 u v w h s h 2 u + h s 2 v + h tr s 2 w tr s tr ( ) 2 ( ) 2 ( ) 2 u v w h t h 2 u + h t 2 v + h rs t 2 w rs t rs. (A.44) (u, v, w) e u, e v, e w α( u, v, w)., 0 Γ u 1 ( ) hu h v v 1 ( ) hu h w w Γ v Γ w 1 h u 1 h u ( hv u α [e u e v e w ] [e u e v e w ] Γ α wu uv 1 h v ( ) hu v wu 0 1 ( ) hv h u u ) 0 vw 0 1 ( ) hv h w w ( hw u 1 h w vw uv 1 h w ( ) hu w uv 0 ( hv w h u ( hw u ( ) hw h v v ) ( ) 1 hw 0 vw h v v wu 0 ) ) uv vw wu (A.45) (A.46)

84 80 A. (A.45), (A.46)., (A.22) r u, r v, r w r α r β h 2 αδ αβ (A.47) r α γ r β. (A.47) γ γ (r α r β ) r β γ r α + r α γ r β 2h α γ h α δ αβ (A.48)., α β r α γ r α h α γ h α (A.49)., α β γ α γ β γ α γ β., γ α β. α r β β r α r α α r β r α β r α h α β h α (A.50)., (A.48) r β α r α + r α α r β 0 (A.51) r β α r α r α α r β h α β h α (A.52)., α β γ α., r β γ r α + r α γ r β 0 (A.53), α r β β r α, r α γ r β r β γ r α r β α r γ r γ α r β (A.54)., r α γ r β r α β r γ r γ β r α r γ α r β (A.55), r α γ r β 0 (A.56)

85 A (A.49), (A.50), (A.52), (A.56), (A.45), (A.46)., α [e x e y e z ] 0 α [e u e v e w ] [e x e y e z ] P (A.57) α [e u e v e w ] [e x e y e z ] α P [e u e v e w ] P 1 α P (A.58)., P 1 t P (A.45) Γ α Γ α t P α P (A.59)., t P P I α ( α t P ) P + t P α P t Γ α + Γ α 0 (A.60), t Γ α Γ α,, Γ α., [x, y, z]/ [u, v, w] [r u r v r w ], Γ α Γ α t P α P 1/h u /h v /h w α r u α r u h u h u r v α r u h v h u r w α r u h w h u 0 r u α r w h u h w t r u t r v t r w [ ] r u r v r w r u α r v h u h v r v α r v h v h v r w α r v h w h v r u α r v h u h v r u α r v h u h v 0 r u α r w h u h w r v α r w h v h w r w α r w h w h w 1/h u /h v /h w α h u 0 0 h u α h v 0 0 h v r u α r w h u h w r v α r w h v h w r v α r w h v h w α h w h w (A.61)

86 82 A. (A.61) (A.50), (A.52), (A.56) (A.46). (r, θ, φ) x r sin θ cos φ, y r sin θ sin φ, z r cos θ (A.62)., r r, θ r z, φ r xy x. (A.22) r r sin θ cos φ e x + sin θ sin φ e y + cos θ e z r θ r cos θ cos φ e x + r cos θ sin φ e y r sin θ e z (A.63) r φ r sin θ sin φ e x + r sin θ cos φ e y. 3 r r, r θ, r φ,., (A.25) h r r r, h θ r θ, h φ r φ h r 1, h θ r, h φ r sin θ (A.64), dr dr e r + rdθ e θ + r sin θdφ e φ (A.65)., e r r r /h r, e θ r θ /h θ, e φ r φ /h φ, e r sin θ cos φ e x + sin θ sin φ e y + cos θ e z e θ cos θ cos φ e x + cos θ sin φ e y sin θ e z (A.66) e φ sin φ e x + cos φ e y., (A.28) [e r e θ e φ ] [e x e y e z ] P, P sin θ cos φ cos θ cos φ sin φ P sin θ sin φ cos θ sin φ cos φ (A.67) cos θ sin θ 0

87 A , (A.29)., P P 1 t P sin θ cos φ sin θ sin φ cos θ P 1 cos θ cos φ cos θ sin φ sin θ (A.68) sin φ cos φ 0., (A.33) [e x e y e z ] [e r e θ e φ ] P 1, e x sin θ cos φ e r + cos θ cos φ e θ sin φ e φ e y sin θ sin φ e r + cos θ sin φ e θ + cos φ e φ (A.69) e z cos θ e r sin θ e θ., (A.37) ( ) ( ) ( ) ( ) sin θ cos φ + sin θ sin φ + cos θ r θφ x yz y zx z xy ( ) ( ) ( ) ( r cos θ cos φ + r cos θ sin φ r sin θ θ φr x yz y zx z ( ) ( ) ( ) r sin θ sin φ + r sin θ cos φ φ x y rθ,, (A.38) ( ) ( ) sin θ cos φ x yz r ( ) ( ) sin θ sin φ y zx r ( ) ( ) cos θ z r xy θφ θφ θφ yz + sin θ r cos θ cos φ r cos θ sin φ + r ( ) θ φr ( ) θ ( ) θ φr φr zx sin φ r sin θ + cos φ r sin θ ) (A.70) ( ) φ ( ) φ rθ rθ (A.71).,. (A.61), α [e r e θ e φ ] [e x e y e z ] Γ α xy

88 84 A Γ α Γ r (ρ, φ, z), Γ θ , Γ φ 0 0 sin θ 0 0 cos θ sin θ cos θ 0 (A.72) x ρ cos φ, y ρ sin φ, z z (A.73)., ρ r xy, φ x. (A.22) r ρ cos φ e x + sin φ e y r φ ρ sin φ e x + ρ cos φ e y (A.74) r z e z. 3 r ρ, r φ, r z,., (A.25) h ρ r ρ, h φ r φ, h z r z h ρ 1, h φ ρ, h z 1 (A.75), dr dρ e ρ + ρdφ e φ + dz e z (A.76)., e ρ r ρ /h ρ, e φ r φ /h φ, e z r z /h z, e ρ cos φ e x + sin φ e y e φ sin φ e x + cos φ e y (A.77) e z e z

89 A , (A.28) [e ρ e φ e z ] [e x e y e z ] P, P cos φ sin φ 0 P sin φ cos φ 0 (A.78) , (A.29)., P P 1 t P cos φ sin φ 0 P 1 sin φ cos φ 0 (A.79) , (A.33) [e x e y e z ] [e ρ e φ e z ] P 1, e x cos φ e ρ sin φ e φ e y sin φ e ρ + cos φ e φ (A.80) e z e z., (A.37) ( ) ( ) ( ) cos φ + sin φ ρ φz x yz y zx ( ) ( ) ( ) ρ sin φ + ρ cos φ φ zρ x yz y ( ) ( ) z ρφ z xy zx (A.81),, (A.38) ( ) ( ) cos φ x yz ρ ( ) ( ) sin φ y zx ρ ( ) ( ) z xy z ρφ φz φz sin φ ρ + cos φ ρ ( ) φ ( ) φ zρ zρ (A.82)

90 86 A.,. (A.61), α [e ρ e φ e z ] [e x e y e z ] Γ α Γ α Γ ρ , Γ φ , Γ z (A.83)

91 , , 76, , , 49 31, , 84 31, , 84 31,

92 , 36, 54 22, 39, ,