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1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1

1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [ 1] 1 1.1.1.2 2 2

1.1. v1.1 2011/04/10 1. 1.1.1.3 e 1, e 2, e 3 v = v i e i (7) e 1, e 2, e 3 T T T (v) = v i T (e i ) (8) v = v i e i = v ie i (9) v i = R ij v j (10) e i = e j (R 1 ) ji = 3 v i e i v ie i = i,j,k=1 R ij v j R ik e k = R ij e j (11) j,k=1 δ jk v j e k = R v j e j (12) 1.1.1.4 [ ] f(x) = 1 2π f(k)e ikx dk (13) f(x) e ikx / 2π f(k) 3

1.1. v1.1 2011/04/10 1. 1.1.2 1.1.2.1 q p (14) q p grad q = K p (15) µ K µ K µ 1.1.2.2 q = K(f) (16) f = 1 p (17) µ K K(λf) = λk(f) (18) K(f 1 + f 2 ) = K(f 1 + f 2 ) (19) x y K q f q = f = q i e i (20) f i e i (21) 4

1.1. v1.1 2011/04/10 1. q i e i = K( f i e i ) (22) K q i e i = f i K(e i ) (23) K(e i ) K(e i ) = e j K ji (24) q i e i = e j K ji f i (25) i, q i = K ij f j (26) K ij 1.1.2.3 T v = T (u) (27) T (e i ) = e j T ji (28) T ij = e i T (e j ) (29) 1.1.2.1 v i = T ij u j (30) I(u) = u (31) 5

1.1. v1.1 2011/04/10 1. 1 0 0 0 1 0 = (δ ij ) (32) 0 0 1 1.1.2.4 u, v s = T (u, v) (33) T (u 1 + u 2, v) = T (u 1, v) + T (u 2, v) (34) T (u, v 1 + v 2 ) = T (u, v 1 ) + T (u, v 2 ) (35) T (λu, v) = T (u, λv) = λt (u, v) (36) u = v = u i e i (37) v i e i (38) s = u i T (e i, e j )v j (39) i, T ij = T (e i, e j ) (40) s = u i T ij v j (41) i, (33) (27) (33) T T (u) = T (e i, u)e i (42) T T ij = e i T (e j ) = e i T (e k, e j )e k = T (e i, e j ) = T ij (43) k=1 6

1.1. v1.1 2011/04/10 1. T (27) T T (u, v) = u T (v) (44) T T ij = T (e i, e j ) = e i T (e j ) = T ij (45) T (33) (27) 1.1.2.5 v v T, T, T, T v T T 1.1.2.6 (27) (T + S)(u) = T (u) + S(u) (46) (T S)(u) = T (u) S(u) (47) (λt )(u) = λt (u) (48) (T + S) ij = T ij + S ij (49) (T S) ij = T ij S ij (50) (λt ) ij = λt ij (51) u O(u) = 0 (52) O 0 7

1.1. v1.1 2011/04/10 1. 1.1.2.7 q i = K ij f j (53) q i = f i = R ij q j (54) R ij f j (55) q i = K ij f j (56) q i = K ijf j (57) (54) (55) (57) R ij q j = K ijr jk f k (58) j,k=1 (56) R ij K jk f k = K ijr jk f k (59) j,k=1 j,k=1 f R ij K jk = K ijr jk (60) (R 1 ) kl = R lk k K il = R ij R lk K jk (61) j,k=1 [ 2] z 180 8

1.1. v1.1 2011/04/10 1. 1.1.2.8 (v i ) (i = 1, 2, 3) R ij 1.1.1.1 v i = R ij v j (62) (T ij ) (i, j = 1, 2, 3) T ij = R ik R jl T kl (63) k,l=1 (T ijk ) (i, j, k = 1, 2, 3) T ijk = R il R jm R kn T lmk (64) l,m.n=1 [ ] (R ij ) [ 3] (40) (63) [ 4] (x, y) x (σ xx = σ, σ ij = 0) z 3 x θ 9

1.1. v1.1 2011/04/10 1. 3 σ n τ θ 45 1.1.2.9 1.1.2.9.1 T ij = T ji (65) (33) T (u, v) = T (v, u) (66) T Maxwell 1.1.2.9.2 T ij = T ji (67) (33) T (u, v) = T (v, u) (68) T 10

1.1. v1.1 2011/04/10 1. Levi-Civita 1 ((i, j, k) = (1, 2, 3), (2, 3, 1), (3, 1, 2)) ɛ ijk = 1 ((i, j, k) = (1, 3, 2), (3, 2, 1), (2, 1, 3)) 0 (otherwise) ɛ ijk Levi-Civita Eddington ɛ ijk (69) ɛ ijk ɛ lmk = δ il δ jm δ im δ jl (70) k=1 j,k=1 i,j,k=1 [ 5] ɛ ijk ɛ ljk = 2δ il (71) ɛ ijk ɛ ijk = 6 (72) 1.1.2.9.3 0 A 3 A 2 A = A 3 0 A 1 (73) A 2 A 1 0 A ( A) i = 1 2 j,k=1 A ( A) ij = ɛ ijk A jk (74) ɛ ijk A k (75) k=1 A A A = A (76) A = (aɛ ijk ) (77) 11

1.2. v1.1 2011/04/10 1. A A = 1 ɛ ijk A ijk (78) 3! i,j,k=1 A ( A) ijk = ɛ ijk A (79) A A A = A (80) [ 6] 1.2 1.2.1 1.2.1.1 1.2.1.1.1 a, b a b = a 1 b 1 + a 2 b 2 + a 3 b 3 (81) 1 0 0 0 1 0 0 0 1 (82) e 1 e 1 = e 2 e 2 = e 3 e 3 = 1 (83) e 1 e 2 = e 2 e 3 = e 3 e 1 = 0 (84) e 2 e 1 = e 3 e 2 = e 1 e 3 = 0 (85) 12

1.2. v1.1 2011/04/10 1. a b = b a (86) a (b + c) = a b + a c (87) (a + b) c = a c + b c (88) (λa) b = a (λb) = λ(a b) (89) a = a a (90) a a 1.2.1.1.2 [ ] (x 0, x 1, x 2, x 3 ) = (ct, x, y, z) a b = a 0 b 0 + a 1 b 1 + a 2 b 2 + a 3 b 3 (91) Minkowski 4 a b = g ij a i b j (92) i,j=0 g ij f(x), g(x) f g = f(x)g(x)dx (93) 1.2.1.1.3 a b θ a b = a b cos θ (94) 0 a, b a b a b 0 (95) a b a b (96) 13

1.2. v1.1 2011/04/10 1. 4 cos 0 a = (a 1, a 2, a 3 ) x y z α, β, γ (l, m, n) l = cos α (97) m = cos α (98) n = cos γ (99) l = a e 1 a m = a e 2 a n = a e 3 a = = = a 1 a 2 1 + a 2 2 + a2 3 a 2 a 2 1 + a 2 2 + a2 3 a 3 a 2 1 + a 2 2 + a2 3 (100) (101) (102) l 2 + m 2 + n 2 = 1 (103) 4 r A n A a t r a = tn (104) 14

1.2. v1.1 2011/04/10 1. A n A a n (r a) = 0 (105) A R A a (r a) (r a) = R 2 (106) 1.2.1.1.4 W = F r (107) F r 1.2.1.2 1.2.1.2.1 A B A : B = i, A ij B ij (108) : A = 1 2 A : A (109) A = A : A (110) 1/2 A 12 (= A 21 ) A A 12 (111) 15

1.2. v1.1 2011/04/10 1. 1.2.1.2.2 : (112) τ ė ė = 1 ( τ ) n 1 τ (113) 2µB G n 3.5 µ G B τ ij ė ij ė ij = 1 2µ τ ij (114) ė ij = 1 ( τ ) n 1 τij (115) 2µB G τ (109) 1.2.1.3 1.1.2.3 (27) v = T (u) (116) v i = T ij u j (117) v = T u (118) v = T u (119) 16

1.2. v1.1 2011/04/10 1. (27) u T S v = S(T (u)) (120) v i = S ij T jk u k (121) j,k=1,3 T S S T (122) 1.2.1.4 S = (S ij ), T = (T ijk ), (123) p q p + q S T = (S ij T klm ), (124) ST ρu u (125) ρuu Maxwell (SI ) 1 µ 0 ( B B 1 ) 2 B2 I (126) (33) a b (a b)(u, v) = (a u)(b v) (127) (a b) ij = (a b)(e i, e j ) = (a e i )(b e j ) = a i b j (128) 17

1.2. v1.1 2011/04/10 1. T = i, T ij e i e j (129) e i e j (33) T T (u, v) = (41) 1.2.1.5 i, T ij (e i e j )(u, v) = i, 1.2.1.5.1 a, b (a b) i = j,k=1 u i T ij v j (130) ɛ ijk a j b k (131) Levi-Civita ɛ ijk a b = e 1 e 2 e 3 a 1 a 2 a 3 b 1 b 2 b 3 (132) a b = b a (133) a (b + c) = a b + a c (134) (a + b) c = a c + b c (135) (λa) b = a (λb) = λ(a b) (136) (133) b = a a a = 0 (137) e 1 e 1 = e 2 e 2 = e 3 e 3 = 0 (138) e 1 e 2 = e 3, e 2 e 3 = e 1, e 3 e 1 = e 2 (139) 18

1.2. v1.1 2011/04/10 1. 5 1.2.1.5.2 a b θ (0 θ π) 5 a b a b a b sin θ a b a b a b a b a b [ 7] a b θ = 0 π a b a b = 0 (140) 1.2.1.5.3 r p L = r p (141) 1.2.1.6 1.2.1.6.1 a b c a b c = a (b b) = b (c a) = c (a c) (142) a b c a 1 a 2 a 3 = b 1 b 2 b 3 c 1 c 2 c 3 = a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 (143) a, b, c 6 a, b, c 6 b 19

1.2. v1.1 2011/04/10 1. 6 c b c a cos θ V V = a b c cos θ = a (b c) (144) 1.2.1.6.2 a, b, c a (b c) (a b) c a (b c) (a b) c a (b c) = (a c)b (a b)c (145) (a b) c = (a c)b (b c)a (146) 1.2.1.7 1.2.1.7.1 a, b a b = a b b a (147) 0 a 1 b 2 b 1 a 2 a 1 b 3 b 1 a 3 a b = (a i b j b i a j )e i e j = a 2 b 1 b 2 a 1 0 a 2 b 3 b 2 a 3 (148) i, a 3 b 1 b 3 a 1 a 3 b 2 b 3 a 2 0 20

1.2. v1.1 2011/04/10 1. e 1 e 1 = e 2 e 2 = e 3 e 3 = 0 (149) e 2 e 3 = e 3 e 2 = 0 0 0 0 0 1 (150) 0 1 0 e 3 e 1 = e 1 e 3 = 0 0 1 0 0 0 (151) 1 0 0 0 1 0 e 1 e 2 = e 2 e 1 = 1 0 0 (152) 0 0 0 (148) 1.1.2.9.3 ( (a b)) ij = k,l,m=1 ɛ ijk ɛ klm a l b m = l,m=1 (δ il δ jm δ im δ jl )a l b m = a i b j a j b i = (a b) ij (153) (a b) = a b (154) ( (a b)) i = 1 2 ɛ ijk (a j b k b j a k ) = ɛ ijk a j b k = (a b) i (155) j,k=1 j,k=1 (a b) = a b (156) a b 0 a b (157) [ 8] (a 1 a 2 ) : (b 1 b 2 ) = 2 a 1 b 1 a 1 b 2 a 2 b 1 a 2 b 2 (158) (a 1 a 2 ) (b 1 b 2 ) = a 1 b 1 a 1 b 2 a 2 b 1 a 2 b 2 (159) 21

1.2. v1.1 2011/04/10 1. 1.2.1.7.2 a b c = a b c + b c a + c a b a c b b a c c b a (160) (a b c) ijk = a i b j c k + b i c j a k + c i a j b k a i c j b k b i a j c k c i b j a k (161) a i a j a k = b i b j b k (162) c i c j c k a 1 a 2 a 3 = ɛ ijk b 1 b 2 b 3 (163) c 1 c 2 c 3 e 1 e 2 e 3 = e 2 e 3 e 1 = e 3 e 1 e 2 (164) = e 1 e 3 e 2 = e 2 e 1 e 3 = e 3 e 2 e 1 (165) = (ɛ ijk ) (166) 1.1.2.9.3 ( ( a b c a 1 a 2 a 3 )) ijk = ɛ ijk b 1 b 2 b 3 c 1 c 2 c 3 = (a b c) ijk (167) ( a b c ) = a b c (168) (a b c) = 1 3! ɛ ijk ɛ ijk i,j,k=1 a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = a b c (169) a b c 0 a, b, c (170) 22

1.2. v1.1 2011/04/10 1. 1.2.2 1.2.2.1 σ n σ i (n) = σ ij n j (171) (σ ij ) σ ij [ ] σ i (n) = n j σ ji (172) (171) σ ij = R ik R jl σ kl (173) k,l=1 σ = R σ R 1 (174) σ R 1.2.2.2 23

1.2. v1.1 2011/04/10 1. [ 9] T = 3 1 1 1 2 0 1 0 2 (175) R A 0 p A p = λp (176) λ (A λi) p = p i (A λi) e i = 0 (177) (A λi) e i, (i = 1, 2, 3) [(A λi) e 1 ] [(A λi) e 2 ] [(A λi) e 3 ] = 0 (178) λ 3 (e 1 e 2 e 3 ) λ 2 [(A e 1 ) e 2 e 3 + e 1 (A e 2 ) e 3 + e 1 e 2 (A e 3 )] + λ[e 1 (A e 2 ) (A e 3 ) + (A e 1 ) e 2 (A e 3 ) + (A e 1 ) (A e 2 ) e 3 ] [(A e 1 ) (A e 2 ) (A e 3 )] = 0 (179) (e 1 e 2 e 3 ) (A e 1 ) e 2 e 3 + e 1 (A e 2 ) e 3 + e 1 e 2 (A e 3 ) = I A [e 1 e 2 e 3 ] (180) e 1 (A e 2 ) (A e 3 ) + (A e 1 ) e 2 (A e 3 ) + (A e 1 ) (A e 2 ) e 3 = II A [e 1 e 2 e 3 ] (181) (A e 1 ) (A e 2 ) (A e 3 ) = III A [e 1 e 2 e 3 ] (182) I A, II A, III A A (179) λ 3 I A λ 2 + II A λ III A = 0 (183) 24

1.2. v1.1 2011/04/10 1. I A, II A, III A 1.2.2.3 A = A ij e i e j (184) i, (A e 1 ) e 2 e 3 + e 1 (A e 2 ) e 3 + e 1 e 2 (A e 3 ) = ( A i1 e i ) e 2 e 3 + e 1 ( A i2 e i ) e 3 + e 1 e 2 ( A i3 e i ) = A 11 (e 1 e 2 e 3 ) + A 22 (e 1 e 2 e 3 ) + A 33 (e 1 e 2 e 3 ) = (A 11 + A 22 + A 33 )(e 1 e 2 e 3 ) (185) I A = A 11 + A 22 + A 33 = A ii = tra (186) A e 1 (A e 2 ) (A e 3 ) + (A e 1 ) e 2 (A e 3 ) + (A e 1 ) (A e 2 ) e 3 = e 1 ( A i2 e i ) ( A j3 e j ) + ( A j1 e j ) e 2 ( A i3 e i ) +( A i1 e i ) ( A j2 e j ) e 3 = (A 22 A 33 A 32 A 23 )(e 1 e 2 e 3 ) + (A 11 A 33 A 31 A 13 )(e 1 e 2 e 3 ) +(A 11 A 22 A 21 A 12 )(e 1 e 2 e 3 ) = (A 11 A 22 + A 22 A 33 + A 33 A 11 A 12 A 21 A 23 A 32 A 31 A 13 )(e 1 e 2 e 3 ) (187) II A = A 11 A 22 + A 22 A 33 + A 33 A 11 A 12 A 21 A 23 A 32 A 31 A 13 ( ) 2 = 1 A ii A ij A ji 2 = 1 I A 2 2 i, i, A ij A ji (188) 25

1.2. v1.1 2011/04/10 1. (A e 1 ) (A e 2 ) (A e 3 ) = ( A i1 e i ) ( A j2 e j ) ( A k3 e k ) = i,j,k=1 k=1 ɛ ijk A i1 A j2 A k3 (e 1 e 2 e 3 ) (189) III A = ɛ ijk A i1 A j2 A k3 = det A (190) i,j,k=1 A I A = A ii = II A = 1 I A 2 2 = 1 I A 2 2 = 1 I A 2 2 = 1 2 [ = II A I 2 A i,k,l=1 i, A ij = k,l=1 R ik R il A kl = A ija ji i,j,k,l,m,n=1 k,l,m,n=1 m,n=1 R ik R jl A kl (191) k,l=1 δ kl A kl = A kk = I A (192) k=1 R ik R jl A kl R jm R in A mn δ kn δ lm A kl A mn A nm A mn ] (193) det(a B) = (det A)(det B) (194) A = R A R 1 (195) 26

1.2. v1.1 2011/04/10 1. III A = det A = det R A R 1 = det A = III A (196) A = λ 1 0 0 0 λ 2 0 0 0 λ 3 (197) I A = λ 1 + λ 2 + λ 3 (198) II A = λ 1 λ 2 + λ 2 λ 3 + λ 3 λ 1 (199) III A = λ 1 λ 2 λ 3 (200) 0 τ I τ = 0 (201) II τ = 1 τ ij τ ij = τ 2 2 (202) i, III A = det τ (203) 1.2.2.4 A J A1 = tra = A ii (204) J A2 = 1 2 tra2 = 1 2 J A3 = 1 3 tra3 = 1 3 A ij A ji (205) i, i,j,k=1 A ij A jk A ki (206) A J A1, J A2, J A3 I A = J A1 (207) II A = 1 2 (J A1 2 J A2 ) (208) III A = 1 6 J A1 3 J A1 J A2 + J A3 (209) [ 10] (209) 27

1.2. v1.1 2011/04/10 1. 1.2.2.5 web pages H. (1967) P. (1979) (1981) 23 (1987) 53 (1997) Introduction to Elasticity/Tensors, http://en.wikiversity.org/wiki/introduction_to_elasticity/tensors http://www.thefullwiki.org/introduction_to_elasticity/tensors 4 27 28

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