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5 x, y, z f f(x, y, z) ( ) f(x x, y, z) f(x, y, z) lim x 0 x (1.1) f x 1. y, z., () y, z., (x, y, z),, xf (1.2)., 2,. 1, f(x x, y y, z z) ( ) f(x, y, z) x ( ) y ( ) z (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)
6 2 1 f., g, h f (df) gh (1.5)., g f h f (df) gh 0 (1.6). ( ) ( ) ( ) (df) (dx) (dy) (dz) ( ) (1.7) (dx), (dx) (df) (dx) ( ) (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) (dx), (du) ( ) ( ) (df) (dx) (du) (du) (dy) (dy) (du) ( ) (dz) (1.10) ( ) (dz), (du) (1.11), ( ) ( ) ( ) ( ) ( ) ( ) ( ) (1.12)
7 v, w, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (1.13) ( ) ( ) (1.14). (1.12), (1.13), (1.14),. (1.12), (1.13), (1.14) [x, y, z]/ [u, v, w] ( ) ( ) ( ) [x, y, z] ( ) ( ) ( ) [u, v, w] ( ) ( ) ( ) (1.15) [( ) ( ) ( ) ] [ ( ) ( ) ( ) ] [x, y, z] [u, v, w] (1.16).,, (x, y, z)/ (u, v, w)., ( ) ( ) ( ) (x, y, z) ( ) ( ) ( ) (u, v, w) (1.17) ( ) ( ) ( )
8 4 1. (1.16) f r, s, t, [ r, s, t ] [x, y, z] [x, y, z] [u, v, w] [ r, s, t ] [u, v, w] (1.18)., ( r, s, t ) (x, y, z) (x, y, z) (u, v, w) ( r, s, t ) (u, v, w) (1.19). 1.2 a b a b a b cos θ (1.20)., 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 (1.21)., a,, a a 1 e 1 a 2 e 2 a 3 e 3 (1.22) e i e j δ ij (1.23) e 1, e 2, e 3., (1.22) e i a i a i. a a 2 1 a 2 2 a 2 (1.24) 3
9 1.2. 5,,. a b a 1 b 1 a 2 b 2 a 3 b 3 (1.25) 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 (1.26)., (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. (1.26) (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 (1.27). a b e 1 e 2 e 3 a 1 a 2 a 3 b 1 b 2 b 3 (1.28). (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 (1.29)
10 6 1., (1.26) (1), (2), (3) (1.27)., (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 (1.30). 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 (1.31)., 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 (1.32), t ( a 2 a 3 c t b 2 b 3 e a 3 a 1 1 b 3 b 1 e 2 a 1 a 2 b 1 b 2 ) e 3 (1.33)., (2) c 2 a 2 b 2 sin 2 θ a 2 b 2 a 2 b 2 cos 2 θ (a 2 1 a 2 2 a 2 3)(b 2 1 b 2 2 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 2 b 2 b 3 b 3 b 1 b 1 b 2 (1.34), (1.33) 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 1 1 b 3 b 1 c a 1 a 2 2 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 (1.35)
11 1.2. 7, t 1., (1.26) (1), (2), (3) (1.27) a, b, c a (b c) (1.36) 3. 3 a 1 a 2 a 3 a (b c) b 1 b 2 b 3 c 1 c 2 c 3. (1.37) a (b c) b (c a) c (a b) (1.38)., 3 a, b, c., 3. a (b c), a, b, c, a (b c), a, b, c., a (b c) (1.39) 3. 3, a (b c) (a c)b (a b)c (1.40).,,, a (b c) (a b) c. 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 (1.41) e w P 13 e x P 23 e y P 33 e z
12 8 1., P ij P 11 P 12 P 13 P P 21 P 22 P 23 (1.42) P 31 P 32 P 33 e x, e y, e z e u, e v, e w. (1.41) [e u e v e w ] [e x e y e z ] P (1.43). e x, e y, e z, (1.41) e x, e y, e z P e x e u e x e v e x e w e y e u e y e v e y e w e z e u e z e v e z e w., e x, e y, e z e u, e v, e w (1.44) 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 (1.45) e z Q 13 e u Q 23 e v Q 33 e w Q ij Q (1.44) 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 (1.46) e w e x e w e y e w e z., a b b a, Q P t P., Q t P (1.47)., (1.45) [e x e y e z ] [e u e v e w ] Q (1.48)
13 1.2. 9, (1.43), Q P P 1.., (1.47) (1.49) Q P 1 (1.49) P 1 t P (1.50). (1.50),.,., t P P I det P 1 1. 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 (1.51) a z a w, a u a v P 1 a x a y (1.52) a w a z. (1.52).
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15 r f(r), f., r A(r), A., r., r r r dr dr. dr (x, y, z) dr dx e x dy e y dz e z (2.1)., r (x, y, z) 1 (u, v, w) x x(u, v, w), y y(u, v, w), z z(u, v, w) (2.2). x, y, z ( ) ( ) ( ) dx du dv ( ) ( ) ( ) dy du dv ( ) ( ) ( ) dz du dv dw dw dw (2.3)
16 12 2, (2.1) dr dr du r u dv r v dw r w (2.4). ( ) ( ) r r u ( ) ( ) r r v ( ) ( ) r r w ( ) e x ( ) e x ( ) e x ( ) e y ( ) e y ( ) e y e z e z e z (2.5)., 3 r u, r v, r w,, r α r β 0 (α β) (2.6), (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 (2.7), e u, e v, e w., h u, h v, h w h u r u, h v r v, h w r w., ( ) 2 ( ) 2 ( ) 2 h u ( ) 2 ( ) 2 ( ) 2 h v (2.8) h w ( ) 2 ( ) 2. (2.7), dr ( ) 2 dr h u du e u h v dv e v h w dw e w (2.9)
17 , (x, y, z), u x, v y, w z, h x 1, h y 1, h z 1. (2.5) (2.7) e u, e v, e w e x, e y, e z e u 1 ( ) e x 1 ( ) e y 1 ( ) h u h u h u e v 1 ( ) e x 1 ( ) e y 1 ( ) h v h v h v e w 1 ( ) e x 1 ( ) e y 1 ( ) h w h w h w. h u e z e z e z (2.10) [e u e v e w ] [e x e y e z ] P (2.11), e x, e y, e z e u, e v, e w P ( ) ( ) ( ) h u h v h w ( ) ( ) ( ) P h u h v h w (2.12) ( ) ( ) ( ) h v., e x, e y, e z e u, e v, e w, P., u, v, w x, y, z u u(x, y, z), v v(x, y, z), w w(x, y, z) (2.13), u, v, w ( ) ( ) ( ) du dx dy dz ( ) ( ) ( ) dv dx dy dz ( ) ( ) ( ) dw dx dy dz h w (2.14)
18 14 2. (2.9),, (2.1) ( ) ( ) ( ) e x h u e u h v e v h w e w ( ) ( ) ( ) e y h u e u h v e v h w e w (2.15) ( ) ( ) ( ) e z h u e u h v e v h w e w., (2.11) [e x e y e z ] [e u e v e w ] P 1 (2.16), ( ) h u ( ) P 1 h v ( ) h w ( ) h u ( ) h v ( ) h w ( ) h u ( ) h v ( ) h w (2.17)., P 1, (2.12) P, P P 1 t P P 1., P P 1. (1.15) P [x, y, z] [u, v, w] 1/h u /h v /h w (2.18) P 1 h u h v h w [u, v, w] [x, y, z] (2.19)
19 , (1.18) P P 1., (1.16) [ ( ) ( ) ( ) ] h u h v h w [ ( ) ( ) ( ) ] (2.20) P [ ( ) ( ) ( ) ] [ 1 h u. ( ) h v ( ) 1 1 h w ( ) ] (2.21) P 1 (u, v, w) 1 (r, s, t) u u(r, s, t), v v(r, s, t), w w(r, s, t) (2.22)., [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 (2.23) (2.24)
20 16 2, 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 h u h u h r r st h s s tr h t t (2.25) rs ( ) ( ) ( ) h v h v h v h r h w h r r ) ( r st st h s h w h s s ) ( s tr tr h t h w h t t ) ( t [e r e s e t ] [e u e v e w ] P (2.26) rs rs., t P P I h r, h s, h t ( ) 2 ( ) 2 ( ) 2 h r h 2 u h r 2 v h st r 2 w st r st ( ) 2 ( ) 2 ( ) 2 h s h 2 u h s 2 v h tr s 2 w tr s tr ( ) 2 ( ) 2 ( ) 2 h t h 2 u h t 2 v h rs t 2 w rs t rs (2.27). (u, v, w) e u, e v, e w α( u, v, w) α [e u e v e w ] [e u e v e w ] Γ α (2.28)
21 , 0 Γ u 1 ( ) hu h v 1 ( ) hu h w Γ v Γ w 1 h u 1 h u ( hv 1 h v ( ) hu 0 1 ( ) hv h u ) ( ) hv h w ( hw 1 h w h w ( ) hu 0 ( hv h u ( hw ( ) hw h v ) ( ) 1 hw 0 h v 0 ) ) (2.29). (2.28), (2.29)., (2.5) r u, r v, r w r α r β h 2 αδ αβ (2.30) r α γ r β. (2.30) γ γ (r α r β ) r β γ r α r α γ r β 2h α γ h α δ αβ (2.31)., α β r α γ r α h α γ h α (2.32)., 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 (2.33)
22 18 2., (2.31) r β α r α r α α r β 0 (2.34) r β α r α r α α r β h α β h α (2.35)., α β γ α.,, α r β β r α,.,, r β γ r α r α γ r β 0 (2.36) r α γ r β r β γ r α r β α r γ r γ α r β (2.37) r α γ r β r α β r γ r γ β r α r γ α r β (2.38) r α γ r β 0 (2.39). (2.32), (2.33), (2.35), (2.39), (2.28), (2.29)., α [e x e y e z ] 0 α [e u e v e w ] [e x e y e z ] P (2.40) 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 (2.41)., P 1 t P (2.28) Γ α Γ α t P α P (2.42)., t P P I α ( α t P ) P t P α P t Γ α Γ α 0 (2.43)
23 , 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 (2.44). (2.44) (2.33), (2.35), (2.39) (2.29). 2.3 (r, θ, ϕ) x r sin θ cos ϕ, y r sin θ sin ϕ, z r cos θ (2.45)., r r, θ r z, ϕ r x. (2.5)
24 20 2 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 (2.46) r ϕ r sin θ sin ϕ e x r sin θ cos ϕ e y. 3 r r, r θ, r ϕ,., (2.8) h r r r, h θ r θ, h ϕ r ϕ h r 1, h θ r, h ϕ r sin θ (2.47), dr dr e r rdθ e θ r sin θdϕ e ϕ (2.48)., 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 (2.49) e ϕ sin ϕ e x cos ϕ e y., (2.11) [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 (2.50) cos θ sin θ 0., (2.12)., 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 (2.51) sin ϕ cos ϕ 0., (2.16) [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 (2.52) e z cos θ e r sin θ e θ
25 , (2.20) ( ) ( ) ( ) ( ) sin θ cos ϕ sin θ sin ϕ cos θ r θϕ ( ) ( ) ( ) ( r cos θ cos ϕ r cos θ sin ϕ r sin θ θ ϕr ( ) ( ) ( ) r sin θ sin ϕ r sin θ cos ϕ ϕ rθ,, (2.21) ( ) ( ) sin θ cos ϕ r ( ) ( ) sin θ sin ϕ r ( ) ( ) cos θ r θϕ θϕ θϕ sin θ r cos θ cos ϕ r cos θ sin ϕ r ( ) θ ϕr ( ) θ ( ) θ ϕr ϕr sin ϕ r sin θ cos ϕ r sin θ ) (2.53) ( ) ϕ ( ) ϕ rθ rθ (2.54).,. (2.44), α [e r e θ e ϕ ] [e x e y e z ] Γ α Γ α Γ r (ρ, ϕ, z), Γ θ , Γ ϕ 0 0 sin θ 0 0 cos θ sin θ cos θ 0 (2.55) x ρ cos ϕ, y ρ sin ϕ, z z (2.56)., ρ r
26 22 2, ϕ x. (2.5) r ρ cos ϕ e x sin ϕ e y r ϕ ρ sin ϕ e x ρ cos ϕ e y (2.57) r z e z. 3 r ρ, r ϕ, r z,., (2.8) h ρ r ρ, h ϕ r ϕ, h z r z h ρ 1, h ϕ ρ, h z 1 (2.58), dr dρ e ρ ρdϕ e ϕ dz e z (2.59)., 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 (2.60) e z e z., (2.11) [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 (2.61) , (2.12)., P P 1 t P cos ϕ sin ϕ 0 P 1 sin ϕ cos ϕ 0 (2.62) , (2.16) [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 (2.63) e z e z
27 , (2.20) ( ) ( ) ( ) cos ϕ sin ϕ ρ ϕz ( ) ( ) ( ) ρ sin ϕ ρ cos ϕ ϕ zρ ( ) ( ) ρϕ (2.64),, (2.21) ( ) ( ) cos ϕ ρ ( ) ( ) sin ϕ ρ ( ) ( ) ρϕ ϕz ϕz sin ϕ ρ cos ϕ ρ ( ) ϕ ( ) ϕ zρ zρ (2.65).,. (2.44), α [e ρ e ϕ e z ] [e x e y e z ] Γ α Γ α Γ ρ , Γ ϕ , Γ z (2.66)
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29 Γ p u u(p), v v(p), w w(p) (3.1)., (u, v, w) Γ r., dr dp h du u dp e dv u h v dp e dw v h w dp e w (3.2) Γ r., dr dp, s., s ds. ds (h u du) 2 (h v dv) 2 (h w dw) 2 (3.3). s, (3.2) 1. t., t dr ds (3.4) Γ., Γ dr t ds dr t ds (3.5)
30 26 3. f Γ f ds (3.6) Γ. s 1, (u, v, w),, Γ r., s f, (3.6)., A Γ A dr (3.7) Γ. (3.5) A t ds (3.8) Γ., Γ, (3.7) (3.8) A Γ. s 1, (u, v, w),, Γ r., s A A u, A v, A w, (3.7). p Γ A dr Γ ( ) du A u h u dp A dv vh v dp A dw wh w dp (3.9) dp., Γ v, w,, u, p u., f ds fh u du (3.10) Γ Γ,, A dr A u h u du (3.11) Γ Γ.
31 Σ p, q u u(p, q), v v(p, q), w w(p, q) (3.12)., (u, v, w) Σ r., q p,, p q. p, q,, (dr) q, (dr) p, (dr) q h u (du) q e u h v (dv) q e v h w (dw) q e w r p (dp) q (dr) p h u (du) p e u h v (dv) p e v h w (dw) p e w r q (dq) p (3.13)., ( ) r p h u p ( ) r q h u q q p ( ) e u h v p ( ) e u h v q q p ( ) e v h w p q ( ) e v h w q p e w e w (3.14)., p, q p q,, r p r q 0. (dr) q, (dr) p Σ ds ds (dr) q (dr) p r p r q (dp) q (dq) p (3.15)., r p r q. { } (v, w) ds h v h w (p, q) e (w, u) u h w h u (p, q) e (u, v) v h u h v (p, q) e w (dp) q (dq) p (3.16)., (α, β) (p, q) ( ) α p ( ) β p q q ( ) α q ( ) β q p p (3.17)
32 28 3., ds, ds r p r q (dp) q (dq) p { h v h w (v, w) (p, q) } 2 { h w h u (w, u) (p, q). ds } 2 { } 2 (u, v) h u h v (dp) q (dq) p (p, q) (3.18) n r p r q r p r q (3.19) ds n ds (3.20). Σ u,,., p v,, q w,,, ds h v h w (dv) w (dw) v e u (3.21) ds h v h w (dv) w (dw) v (3.22).,. Σ Σ, Σ Σ., Σ Σ. f Σ f ds (3.23) Σ., A Σ A ds (3.24) Σ
33 p, q A ds Σ { } (v, w) A u h v h w Σ (p, q) A (w, v) vh w h v (p, q) A (u, v) wh u h v dp dq (p, q) (3.25)., (dp) q, (dq) p,, dp, dq. p, q 1, (u, v, w),, Σ r., p, q A A u, A v, A w, (3.25)., Σ, p v,, q w., f ds f h v h w dv dw (3.26),,. Σ Σ Σ A ds Σ A u h v h w dv dw (3.27) 3.3 (u, v, w) dv dv (dr) {(dr) (dr) } h u h v h w (du) (dv) (dw) (3.28)., α, β (dr) αβ. f Ω (3.28) f dv fh u h v h w du dv dw (3.29) Ω Ω., (du), (dv), (dw),, du, dv, dw.
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35 f f f dr df (4.1)., dr. dr t ds f t df ds (4.2). (4.2) 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 ) (4.3) r a r b r a r b., f f,., f (4.3) 0., A f. (u, v, w) f f 1 ( ) e u 1 ( ) h u h v e v 1 h w ( ) e w (4.4)
36 32 4. (4.4). f u, v, w u e u h u, v e v h v, w e w h w (4.5)., ( ) ( ) ( ) f u v w (4.6).,,, f f f ( ) e x ( ) e r 1 r θϕ r f ( ) e y ( ) e z (4.7) ( ) e θ 1 ( ) e ϕ (4.8) θ ϕr r sin θ ϕ rθ ( ) e ρ 1 ( ) e ϕ ρ ϕz ρ ϕ zρ ( ) e z (4.9) ρϕ., f (4.4). (u, v, w) f dr h u du f e u h v dv f e v h w dw f e w (4.10). f ( ) ( ) df du dv,, u f e u 1 ( ) h u ( ) dw (4.11) (4.12)., v, w, f (4.4).
37 A A ( A) S σ A dr (4.13)., σ, S. S n S ( A) n 1 A dr (4.14) S. (4.14) A n.,., A A,., A 0. (4.13), A f σ 0,, A 0.,,. σ ( f) 0 (4.15) (u, v, w) A A 1 {( ) ( ) Aw h w Av h v h v h w 1 {( ) ( ) Au h u Aw h w h w h u 1 {( ) ( ) Av h v Au h u h u h v } } e u e v } e w (4.16)
38 34 4. (4.16). e u e v e w h v h w h w h u h u h v A A u h u A v h v A w h w (4.17)., f (4.4) ( f) 0,,. A, f f(u, v, w) u u 0 A u h u (u, v 0, w 0 ) du v w A v h v (u, v, w 0 ) dv A w h w (u, v, w ) dw v 0 w 0 (4.18), A A f.,,, A { ( Az ) A { ( Ax ) { ( Ay ) ( ) Ay ( ) Az ( ) Ax } } } e x e y e z (4.19) { ( Aϕ ) ( ) } 1 r sin θ Aθ r A r 2 sin θ θ ϕr ϕ rθ { 1 ( Ar ) ( ) } Aϕ r sin θ e θ r sin θ ϕ rθ r θϕ { 1 ( Aθ ) ( ) } r Ar e ϕ r r θ θϕ ϕr e r (4.20)
39 { A 1 ( Az ) ( ) Aϕ ρ ρ ϕ zρ { ( Aρ ) ( ) } Az ρϕ ρ ϕz { 1 ( Aϕ ) ( ) ρ Aρ ρ ρ ϕ ϕz ρϕ } e ϕ zρ } e ρ e z (4.21)., A (4.16)., (4.13) σ γ 1 : (u 0, v 0, w 0 ) (u 0, v 0 v, w 0 ) γ 2 : (u 0, v 0 v, w 0 ) (u 0, v 0 v, w 0 w) γ 3 : (u 0, v 0 v, w 0 w) (u 0, v 0, w 0 w) γ 4 : (u 0, v 0, w 0 w) (u 0, v 0, w 0 ) (4.22) 4., σ σ γ 1 γ 4., γ 1, γ 3., γ 1 t e v, γ 3 t e v A dr A dr γ 1 γ 3 v0 v v 0 {a v (u 0, v, w 0 ) a v (u 0, v, w 0 w)} dv (4.23)., a v A v h v., a v (u 0, v, w 0 w) w 2 ( ) av a v (u 0, v, w 0 w) a v (u 0, v, w 0 ) w (4.24) 0 v0 v ( ) av A dr A dr w dv γ 1 γ 3 v 0 0 u 0 v ( ) ( ) av Av h v v w v w 0 u 0 v 0 0 u 0 v 0 (4.25) u 0 v
40 36 4., 2 v v v 0., γ 2, γ 4., γ 2 t e w, γ 4 t e w A dr A dr γ 2 γ 4 w0 w w 0 {a w (u 0, v 0 v, w) a w (u 0, v 0, w)} dw (4.26)., a w A w h w., a w (u 0, v 0 v, w) v 2 ( ) aw a w (u 0, v 0 v, w) a w (u 0, v 0, w) v (4.27) 0 0 w0 w ( ) aw A dr A dr v dw γ 2 γ 4 w 0 0 ( ) 0 ( ) aw Aw h w v w v w 0 w 0 u 0 0 w 0 u 0 (4.28)., 2 w w w 0., σ σ { ( Aw ) ( ) } h w Av h v A dr v w (4.29) 0 0 σ w 0 u 0., σ n e u, S h v h w v w, (4.14), { ( A) e u 1 ( Aw ) ( ) } h w Av h v (4.30) h v h w 0 0 w 0 u 0., ( A) e v, ( A) e w, u 0, v 0, w 0 u, v, w, A (4.16). u 0 v 0 u 0 v 0
41 A A A V ω A ds (4.31)., ω V, ω ω. A 1 A ds (4.32) V,.,., A 0. (u, v, w) A {( ) ( ) 1 Au h v h w Av h w h u A h u h v h w ω ( ) } Aw h u h v (4.33). (4.33). A (4.16) ( A) 0 (4.34).,.,,, A A 1 r 2 sin θ A ( ) Ax { ( Ar ) r 2 sin θ r θϕ ( ) Ay ( ) Az ( ) Aθ r sin θ θ ϕr ( ) Aϕ r ϕ (4.35) rθ } (4.36)
42 38 4 A 1 ρ { ( Aρ ) ρ ρ ϕz ( ) Aϕ ϕ zρ ( ) Az ρ ρϕ } (4.37)., A (4.33)., (4.31) ω σ 1 : (u 0 u, v, w ) σ 2 : (u 0, v, w ) σ 3 : ( u, v 0 v, w ) σ 4 : ( u, v 0, w ) σ 5 : ( u, v, w 0 w) σ 6 : ( u, v, w 0 ) u 0 u u 0 u, v 0 v v 0 v, w 0 w w 0 w (4.38) 6., ω ω σ 1 σ 6., σ 1 σ 2., σ 1 n e u, σ 2 n e u A ds A ds σ 1 σ 2 w0 w v0 v (4.39) {a u (u 0 u, v, w) a u (u 0, v, w)} dv dw w 0 v 0., a u A u h v h w., a u (u 0 u, v, w) u 2 ( ) au a u (u 0 u, v, w) a u (u 0, v, w) u (4.40) 0 w0 w v0 v ( ) au A ds A ds u dv dw σ 1 σ 2 w 0 v 0 0 ( ) au u v w 0 v 0 w ( 0 ) Au h v h w u v w 0 v 0 w 0 (4.41)
43 , 2 v, w v v 0, w w 0. σ 3 σ 4, σ 5 σ 6 A ds ω { ( Au ) ( ) ( ) } h v h w Av h w h u Aw h u h v u v w v 0 w 0 w 0 u 0 u 0 v 0 (4.42)., V h u h v h w u v w, u 0, v 0, w 0 u, v, w, A (4.33). 2 f 2 f ( f) (4.43) 2. (u, v, w) 2 f (4.4), (4.33) {( ) 2 f 1 h u h v h w h v h w h u ( h w h u h v ) ( ) } (4.44) h u h v h w.,,, 2 f ( ) 2 2 f f 2 ( ) 2 f 2 ( ) 2 f 2 (4.45) 2 f 1 r 2 ( r ) r2 1 ( ) ( ) 1 2 f sin θ r θϕ r 2 sin θ θ θ ϕr r 2 sin 2 θ ϕ 2 rθ (4.46)
44 f 1 ρ ( ) ρ ρ 1 ( ) 2 f ρ ϕz ρ 2 ϕ 2 zρ ( ) 2 f 2 ρϕ (4.47)., A. ( A) ( A) 2 A (4.48) 4.4,,. (1) (fg) ( f)g f g (2) (fa) f A f A (3) (fa) f A f A (4) (A B) B ( A) A ( B) (5) (A B) (B )A (A )B B ( A) A ( B) (6) (A B) (B )A (A )B A( B) B( A) (4.49)
45 f(x) b a df dx f(b) f(a) (5.1) dx..,. f, (4.3) r a r b f dr f(r b ) f(r a ) (5.2). (5.1), (5.2) A A ( A) S σ A dr (5.3), 2 3 σ 1, σ 2., σ 1 σ 2 1 γ,, σ 1 σ 2., σ 1 σ 2 ( A) 1 S 1 ( A) 2 S 2 A dr A dr (5.4) σ 1 σ 2
46 42 5, γ., σ 1, σ 2 γ t 1, t 2., ( A) 1 S 1 ( A) 2 S 2 A dr (5.5) (σ 1 σ 2 )., σ 1 σ 2 σ 1 σ 2., Σ 3 Σ i σ i, ( A) i S i «P A dr (5.6) i σ i i. (5.6) ( A) ds A dr (5.7) Σ. (5.7)., A., f (5.2),. Σ 5.3 A A A V ω A ds (5.8), 2 4 ω 1, ω 2., ω 1 ω 2 1 σ., ω 1 ω 2 ( A) 1 V 1 ( A) 2 V 2 A ds A ds (5.9) ω 1 ω 1
47 , σ., ω 1, ω 2 σ n 1, n 2., ( A) 1 V 1 ( A) 2 V 2 A ds (5.10) (ω 1 ω 2 )., ω 1 ω 2 ω 1 ω 2., Ω 4 Ω i ω i, ( A) i V i «P A ds (5.11) i ω i i. (5.11) A dv A ds (5.12). (5.12). Ω Ω 5.4. (1) ( f)g dr [ fg ] rb r a f g dr r a r b r a r b (2) ( f A) ds fa dr (f A) ds Σ Σ Σ (3) f A dv fa ds f A dv (5.13) Ω Ω Ω (4) B ( A) dv (A B) ds Ω Ω A ( B) dv, (3) A g f g dv f g ds f 2 g dv (5.14) Ω Ω Ω Ω
48 44 5., f g ( (f g g f) ds f 2 g g 2 f ) dv (5.15) Ω Ω. (5.14), (5.15).
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2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
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
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145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
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II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
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II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier
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β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
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1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2
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20010916 22;1017;23;20020108;15;20; 1 N = {1, 2, } Z + = {0, 1, 2, } Z = {0, ±1, ±2, } Q = { p p Z, q N} R = { lim a q n n a n Q, n N; sup a n < } R + = {x R x 0} n = {a + b 1 a, b R} u, v 1 R 2 2 R 3
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
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x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
x,, z v = (, b, c) v v 2 + b 2 + c 2 x,, z 1 i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) v 1 = ( 1, b 1, c 1 ), v 2 = ( 2, b 2, c 2 ) v
12 -- 1 4 2009 9 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 4-9 4-10 c 2011 1/(13) 4--1 2009 9 3 x,, z v = (, b, c) v v 2 + b 2 + c 2 x,, z 1 i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) v 1 = ( 1, b 1, c 1 ), v 2
( 12 ( ( ( ( Levi-Civita grad div rot ( ( = 4 : 6 3 1 1.1 f(x n f (n (x, d n f(x (1.1 dxn f (2 (x f (x 1.1 f(x = e x f (n (x = e x d dx (fg = f g + fg (1.2 d dx d 2 dx (fg = f g + 2f g + fg 2... d n n
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38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: t
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb
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r 1 r 2 r 1 r 2 2 Coulomb Gauss Coulomb 2.1 Coulomb 1 2 r 1 r 2 1 2 F 12 2 1 F 21 F 12 = F 21 = 1 4πε 0 1 2 r 1 r 2 2 r 1 r 2 r 1 r 2 (2.1) Coulomb ε 0 = 107 4πc 2 =8.854 187 817 10 12 C 2 N 1 m 2 (2.2)
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20 21 2 8 1 2 2 3 21 3 22 3 23 4 24 5 25 5 26 6 27 8 28 ( ) 9 3 10 31 10 32 ( ) 12 4 13 41 0 13 42 14 43 0 15 44 17 5 18 6 18 1 1 2 2 1 2 1 0 2 0 3 0 4 0 2 2 21 t (x(t) y(t)) 2 x(t) y(t) γ(t) (x(t) y(t))
r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B
1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n
d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r
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2.4 ( ) U(r) ( ) ( ) U F(r) = x, U y, U = U(r) (2.4.1) z 2 1 K = mv 2 /2 dk = d ( ) 1 2 mv2 = mv dv = v (ma) (2.4.2) ( ) U(r(t)) r(t) r(t) + dr(t) du du = U(r(t) + dr(t)) U(r(t)) = U x = U(r(t)) dr(t)
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K E N Z OU 11 1 1 1.1..................................... 1.1.1............................ 1.1..................................................................................... 4 1.........................................
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39 3 3.1 2 Ax 1,y 1 Bx 2,y 2 x y fx, y z fx, y x 1,y 1, 0 x 1,y 1,fx 1,y 1 x 2,y 2, 0 x 2,y 2,fx 2,y 2 A s I fx, yds lim fx i,y i Δs. 3.1.1 Δs 0 x i,y i N Δs 1 I lim Δx 2 +Δy 2 0 x 1 fx i,y i Δx i 2 +Δy
ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d
A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9
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51 5 5.1 5.1.1 P r P z θ P P P z e r e, z ) r, θ, ) 5.1 z r e θ,, z r, θ, = r sin θ cos = r sin θ sin 5.1) e θ e z = r cos θ r, θ, 5.1: 0 r
x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s
... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z
,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,
l µ l µ l 0 (1, x r, y r, z r ) 1 r (1, x r, y r, z r ) l µ g µν η µν 2ml µ l ν 1 2m r 2mx r 2 2my r 2 2mz r 2 2mx r 2 1 2mx2 2mxy 2mxz 2my r 2mz 2 r
2 1 (7a)(7b) λ i( w w ) + [ w + w ] 1 + w w l 2 0 Re(γ) α (7a)(7b) 2 γ 0, ( w) 2 1, w 1 γ (1) l µ, λ j γ l 2 0 Re(γ) α, λ w + w i( w w ) 1 + w w γ γ 1 w 1 r [x2 + y 2 + z 2 ] 1/2 ( w) 2 x2 + y 2 + z 2
2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =
1 1 1.1 I R 1.1.1 c : I R 2 (i) c C (ii) t I c (t) (0, 0) c (t) c(i) c c(t) 1.1.2 (1) (2) (3) (1) r > 0 c : R R 2 : t (r cos t, r sin t) (2) C f : I R c : I R 2 : t (t, f(t)) (3) y = x c : R R 2 : t (t,
50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ
5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1
4 1 1.1 ( ) 5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1 da n i n da n i n + 3 A ni n n=1 3 n=1
y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =
[ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =
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83 ( Intrinsic ( (1 V v i V {e 1,, e n } V v V v = v 1 e 1 + + v n e n = v i e i V V V V w i V {f 1,, f n } V w 1 V w = w 1 f 1 + + w n f n = w i f i V V V {e 1,, e n } V {e 1,, e n } e 1 (e 1 e n e n
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211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
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63 6 6.1 6.1.1 v = v 0 =v 0x,v 0y, 0) t =0 x 0,y 0, 0) t x x 0 + v 0x t v x v 0x = y = y 0 + v 0y t, v = v y = v 0y 6.1) z 0 0 v z yv z zv y zv x xv z xv y yv x = 0 0 x 0 v 0y y 0 v 0x 6.) 6.) 6.1) 6.)
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
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y x = α + β + ε =,, ε V( ε) = E( ε ) = σ α $ $ β w ( 0) σ = w σ σ y α x ε = + β + w w w w ε / w ( w y x α β ) = α$ $ W = yw βwxw $β = W ( W) ( W)( W) w x x w x x y y = = x W y W x y x y xw = y W = w w