211 12 1 19 2.9 F 32 32: rot F d = F d l (63) F rot F d = 2.9.1 (63) rot F rot F F (63)
12 2 F F F (63) 33 33: (63) rot 2.9.2 (63) I = [, 1] [, 1]
12 3 34: = 1 2 1 2 1 1 = C 1 + C C 2 2 2 = C 2 + ( C ) C C = C 1 + C 2 1 2 rot F d = rot F d + rot F d 1 2 F d l = F d l + F d l = F d l + F d l + C 1 C 2 C 1 C = F d l + F d l 1 2 1 2 rot F d = F d l rot F d = 1 2 1 C 2 2 F d l F d l C F d l 2 rot F d = F d l I = [, 1] [, 1] 2.9.3 xyz x y z = T (s, t) = ξ(s, t) η(s, t) ζ(s, t) s 1 t 1
12 4 4 C 1 = {T (s, ) s 1} C 2 = {T (1, t) t 1} C 3 = {T (s, 1) s 1} C 4 = {T (, t) t 1} = C 1 + C 2 + ( C 3 ) + ( C 4 ) 35 1 t C 3 T z C 3 C 4 C 4 I C 2 y C 1 C 2 C 1 1 s x 35: C 1, C 2, C 3, C 4 F F (x, y, z) = F (x, y, z) G(x, y, z) H(x, y, z) (63) I st [, 1] [, 1] (s, t) rot F d = (( F ) T ) (T s T t )dsdt I H y T G z T η s ζ t ζ s η t = F z T H x T ζ s ξ t ξ s ζ t dsdt I G x T F y T ξ s η t η s ξ t { = (H y T )η s ζ t (H y T )ζ s η t (G z T )η s ζ t + (G z T )ζ s η t I + (F z T )ζ s ξ t (F z T )ξ s ζ t (H x T )ζ s ξ t + (H x T )ξ s ζ t + (G x T )ξ s η t (G x T )η s ξ t (F y T )ξ s η t + (F y T )η s ξ t }dsdt (F x T )ξ s ξ t (F x T )ξ t ξ s (G y T )η s η t (G y T )η t η s (H z T )ζ s ζ t (H z T )ζ t ζ s
12 5 { (Fx T )ξ s + (F y T )η s + (F z T )ζ s } ξt + { (G x T )ξ s + (G y T )η s + (G z T )ζ s } ηt + { (H x T )ξ s + (H y T )η s + (H z T )ζ s } ζt { (F x T )ξ t + (F y T )η t + (F z T )ζ t } ξs (64) { (G x T )ξ t + (G y T )η t + (G z T )ζ t } ηs { (H x T )ξ t + (H y T )η t + (H z T )ζ t } ζs (64) (F x T )ξ s + (F y T )η s + (F z T )ζ s = (F T ) s =(F T ) s ξ t + (G T ) s η t + (H T ) s ζ t (F T ) t ξ s (G T ) t η s (H T ) t ζ s (F T ) s ξ t (F T ) t ξ s = (G T ) s η t (G T ) t η s (65) (H T ) s ζ t (H T ) t ζ s s, t 2 f(s, t) g(s, t) f(s, t) = F T G T H T ξ t η t ζ t g(s, t) = F T G T H T ξ s η s ζ s ξ st = ξ ts (F T ) s f s g t = (G T ) s (H T ) s ξ t η t ζ t + F T G T H T ξ ts η ts ζ ts (F T ) t (G T ) t (H T ) t ξ s η s ζ s + F T G T H T ξ st η st ζ st = (F T ) s (G T ) s ξ t η t (F T ) t (G T ) t ξ s η s (H T ) s ζ t (H T ) t ζ s (65) rot F d = (f t (s, t) g s (s, t))dsdt (66) I
12 6 (66) f s (s, t) f(s, t) t s 1 f s (s, t)dsdt = g t (s, t)dsdt = I I 1 1 1 f s (s, t)dt = f(1, t) f(, t) ( 1 ( 1 ) f s (s, t)ds dt = ) g t (s, t)dt ds = (66) rot F d = 1 f(1, t)dt 1 f(, t)dt 1 1 1 f(1, t)dt g(s, 1)ds g(s, 1)ds + 1 1 1 f(, t)dt g(s, )ds g(s, )ds (67) f(s, t) g(s, t) 1 f(1, t)dt = 1 F (T (1, t)) ξ t (1, t) G(T (1, t)) η t (1, t) dt = H(T (1, t)) ζ t (1, t) 1 F (T (1, t)) T t (1, t)dt = C 2 F d l 35 1 1 f(, t)dt = F d l g(s, 1)ds = C 4 C 3 F d l 1 g(s, )ds = C 1 F d l (67) rot F d = F d l F d l F d l + C 2 C 4 C 3 = F d l = F d l C 1 +C 2 +( C 3 )+( C 4 ) C 1 F d l 2.9.4 F grad φ = F φ rot F = rot(grad φ) = 11 28 rot F = F
12 7 F rot F = 2.3.2 F C C F d l = C F d l = rot F d C rot F = C F d l = F C C F F 1 x 2 + y 2 11 27 y rot F = C xy C F d l = 2π x sin θ sin θ cos θ cos θ dθ = 2π 1dθ = 2π F 8 23 F z C z F 1. X X C C X 36 36 3
12 8 F F rot F = 34. xyz (1) (y + 1)(z 2 1) (x + 1)(z 2 + 1) (x + 1)(y + 1) F rot F (2) z = {(x, y, z) x 2 + y 2 + z 2 = 1, z } (1) F rot F d 2.1 1 2.4 1 1 1 1 2.1.1 1 p θ q ω θ ω p + q 11. P p + q v 1,..., v p+q (θ ω) P ( v 1,..., v p+q ) = 1 p!q! σ p+q sgn(σ)θ P ( v σ(1),..., v σ(p) )ω P ( v σ(p+1),..., v σ(p+q) ) (68)
12 9 p+q p + q sgn(σ) σ 1 1 p q p q φ (φ ω) P ( v 1,..., v q ) = φ(p )ω P ( v 1,..., v q ) φ ω φω p = q = 1 (θ ω) P ( v, w) = θ P ( v)ω P ( w) θ P ( w)ω P ( v) (69) 2 p = 1, q = 2 (θ ω) P ( u, v, w) = θ P ( u)ω P ( v, w) + θ P ( v)ω P ( w, u) + θ P ( w)ω P ( u, v) (7) 35. (69) (7) (68) 36. 1 θ, σ, τ P u, v, w θ P ( u) θ P ( v) θ P ( w) (θ σ τ) P ( u, v, w) = det σ P ( u) σ P ( v) σ P ( w) τ P ( u) τ P ( v) τ P ( w) 2.1.2 1.9.4 1.9.5 x, y, z 3 1 1 O.K. θ 1 P θ P R (a, b, c) R 3 R 1 3 1 ( ) f(x, y, z) g(x, y, z) h(x, y, z) 3 ( ) ( ) 1 3 ( 1 1) 1 dx 1 dy 1 dz θ = f(x, y, z)dx + g(x, y, z)dy + h(x, y, z)dz
12 1 ω 2 1.9.4 (25) 2 ω 3 f(x, y, z), g(x, y, z), h(x, y, z) P (a, b, c) v, w v 1 v 2, w 1 w 2 ( ω P ( v, w) = ) v 1 v 2 v 3 v 3 w 3 h(a, b, c) g(a, b, c) h(a, b, c) f(a, b, c) g(a, b, c) f(a, b, c) w 1 w 2 w 3 2 dx dy P, v, w (dx dy) P ( v, w) = d P x( v)d P y( w) d P x( w)d P y( v) = v 1 w 2 w 1 v 2 ( ) 1 w 1 = v 1 v 2 v 3 1 w 2 w 3 ( (dy dz) P ( v, w) = v 2 w 3 w 2 v 3 = ( (dz dx) P ( v, w) = v 3 w 1 w 3 v 1 = ) v 1 v 2 v 3 ) v 1 v 2 v 3 1 1 1 1 w 1 w 2 w 3 w 1 w 2 w 3 ω = f(x, y, z)dy dz + g(x, y, z)dz dx + h(x, y, z)dx dz µ 3 e 1, e 2, e 3 xyz e 1 1 u, v, w u 1 u 2 v 1 v 2 w 1 w 2 u 3 v 3 w 3 u = u 1 e 1 + u 2 e 2 + u 3 e 3 v = v 1 e 1 + v 2 e 2 + v 3 e 3 w = w 1 e 1 + w 2 e 2 + w 3 e 3 P µ 3 3 3 3 µ P ( u, v, w) = u i v j w k µ P ( e i, e j, e k ) i=1 j=1 k=1 = u 1 v 2 w 3 µ P ( e 1, e 2, e 3 ) + u 1 v 3 w 2 µ P ( e 1, e 3, e 2 ) + u 2 v 1 w 3 µ P ( e 2, e 1, e 3 ) + u 2 v 3 w 1 µ P ( e 2, e 3, e 1 ) + u 3 v 1 w 2 µ P ( e 3, e 1, e 2 ) + u 3 v 2 w 1 µ P ( e 3, e 2, e 1 )
12 11 = (u 1 v 2 w 3 u 1 v 3 w 2 u 2 v 1 w 3 + u 2 v 3 w 1 + u 3 v 1 w 2 u 3 v 2 w 1 )µ P ( e 1, e 2, e 3 ) u 1 v 1 w 1 = det u 2 v 2 w 2 µ P ( e 1, e 2, e 3 ) u 3 v 3 w 3 P, u, v, w 36 d P x( u) d P x( v) d P x( w) (dx dy dz) P ( u, v, w) = det d P y( u) d P y( v) d P y( w) d P z( u) d P z( v) d P z( w) = det u 1 v 1 w 1 u 2 v 2 w 2 u 3 w 3 w 3 f(x, y, z) = µ P ( e 1, e 2, e 3 ) (x, y, z) P 3 f(x, y, z) µ = f(x, y, z)dx dy dz n k ω n x 1 x 2 x n n C k n f i1 i 2 i k (x 1, x 2,..., x n ) 1 i 1 < i 2 < < i k n ω = f i1 i k dx i1 dx i2 dx ik (71) 1 i 1< <i k n 2.1.3 1 1 θ = f(x, y, z)dx + g(x, y, z)dy + h(x, y, z)dz f(x, y, z)dx d(fdx) = df dx d(dx) = f df = df d(fdx) = f x dx dx + f y dy dx + f z dz dx = f z f dz dx dx dy y 35 dx dx =, dy dx = dx dy d(gdy) d(hdz) dθ = d(fdx) + d(gdy) + d(hdz) ( h = y g ) dy dz + z ( f z h x ) ( g dz dx + y f ) dx dy y
12 12 xyz f(x, y, z) g(x, y, z) h(x, y, z) h y g z f z h x g x f y 1 f fdx + gdy + hdz g h 2 fdy dz + gdz dx + hdx dy f g h 1 d rot 2.1.4 2 2 ω 3 ω = fdy dz + gdz dx + hdx dy dω 1 dω = df dy dz + dg dz dx + dh dx dy df = f f f dx + dy + x y z dz dz dx = dx dz ( f dω = x + g y + h ) dx dy dz z xyz f g h f x + g y + h z
12 13 2 f fdy dz + gdz dx + hdx dy g 3 h fdx dy dz f 2 d div 2.1.5 k n k x 1 x 2 x n (71) k ω dω dω = df i1 i 2...i k (x 1,..., x n ) dx i1 dx i2 dx ik 1 i 1<i 2<...<i k n 2.1.6. φ P v φ P v v P (φ) F F (φ) P F (P ) P (φ) G G( F (φ)) F G F ( G(φ)) [ F, G](φ) := F ( G(φ)) G( F (φ)) φ F G [ F, G] F G
12 14 ω p dω (dω)( F 1, F 2,..., F p+1 ) p+1 = ( 1) k+1 Fk (ω( F 1,..., F k 1, F k+1,..., F ) p+1 ) k=1 + ( 1) k+l ω([ F k, F l ], F 1,..., F k 1, F k+1,..., F l 1, F l+1,..., F p+1 ) 1 k<l p+1 2.1.7 1 θ 1 dθ = θ (72) xyz 2 ω F fdy dz + gdz dx + hdx dy ω = F d f g h (73) T : E (s, t) (ξ(s, t), η(s, t), ζ(s, t)) U ω = (f(t )dy dz(t s, T t ) + g(t )dz dx(t s, T t ) + h(t )dx dy(t s, T t )) dsdt E = (f(t )(dy(t s )dz(t t ) dy(t t )dz(t s )) = = E E E +g(t )(dz(t s )dx(t t ) dz(t t )dx(t s )) +h(t )(dx(t s )dy(t t ) dx(t t )dy(t s ))) dsdt f T η s ζ t η t ζ s g T ζ s ξ t ζ t ξ s dsdt h T ξ s η t ξ t η s f T ξ s ξ t g T dsdt = F d h T η s ζ s η t ζ t
12 15 2.1.3 1 2.4.3 (35) 1 2 1 (73) (72) rot F d = F d l (72) 2.1.8 2 ω 2 dω = ω (74) xyz 3 µ φ f(x, y, z)dx dy dz f(x, y, z) (75) µ = φdv 3 3 Id: (x, y, z) (x, y, z) U µ µ = = = f(id)dx dy dz(id x, Id y, Id z )dxdydz dx(id x ) dx(id y ) dx(id z ) f(x, y, z) det dy(id x ) dy(id y ) dy(id z ) dxdydz dz(id x ) dz(id y ) dz(id z ) 1 f(x, y, z) det 1 dxdydz = f(x, y, z)dxdydz = 1 φdv
12 16 2.1.4 2 (73) 2 3 2 (75) (74) div F dv = F d (74)
12 17 34 (1) F = (y + 1)(z 2 1) G = (x + 1)(z 2 + 1) H = (x + 1)(y + 1) rot F H y G z F z H x = G x F y (x + 1) 2z(x + 1) 2z(y + 1) (y + 1) (z 2 + 1) (z 2 1) = (x + 1)(1 2z) (y + 1)(2z 1) 2 (2) xy T T = { (x, y, z) x 2 + y 2 1, z = } z T T xy T rot F d = F d l = T F d l = T rot F d x = s y = t z = E = { (s, t) s 2 + t 2 1 } s + 1 rot F d = t 1 dsdt = 2 T E 2 1 E 1dsdt = 2π 35 ( v, w) ( v, w) ( w, v) 1! = 1 (θ ω) P ( v, w) = θ P ( v)ω P ( w) θ P ( w)ω P ( v) ( ) θ P ( v) θ P ( w) det ω P ( v) ω P ( w) ( u, v, w) ( u, w, v), ( v, u, w), ( w, v, u)
12 18 ( u, v, w), ( v, w, u), ( w, u, v) ω (θ ω) P ( u, v, w) = 1 1!2! ( θ P ( u)ω P ( w, v) θ P ( v)ω P ( u, w) θ P ( w)ω P ( v, u) +θ P ( u)ω P ( v, w) + θ P ( v)ω P ( w, u) + θ P ( w)ω P ( u, v)) = 1 2 (θ P ( u)ω P ( v, w) + θ P ( v)ω P ( w, u) + θ P ( w)ω P ( u, v) +θ P ( u)ω P ( v, w) + θ P ( v)ω P ( w, u) + θ P ( w)ω P ( u, v)) = θ P ( u)ω P ( v, w) + θ P ( v)ω P ( w, u) + θ P ( w)ω P ( u, v) 36 (7) ω σ τ (69) (θ σ τ) P ( u, v, w) = θ P ( u)(σ τ) P ( v, w) + θ P ( v)(σ τ) P ( w, u) + θ P ( w)(σ τ) P ( u, v) = θ P ( u)(σ P ( v)τ P ( w) σ P ( w)τ P ( v)) + θ P ( v)(σ P ( w)τ P ( u) σ P ( u)τ P ( w)) = θ P ( u)σ P ( v)τ P ( w) + θ P ( v)σ P ( w)τ P ( u) + θ P ( w)σ P ( u)τ P ( v) + θ P ( w)(σ P ( u)τ P ( v) σ P ( v)τ P ( u)) θ P ( u)σ P ( w)τ P ( v) θ P ( v)σ P ( u)τ P ( w) θ P ( w)σ P ( v)τ P ( u) θ P ( u) θ P ( v) θ P ( w) = det σ P ( u) σ P ( v) σ P ( w) τ P ( u) τ P ( v) τ P ( w)