0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ). f ( ). x i : M R.,,

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1 ,,,.,,.,.,,. 2?.,,. 1,, 1. (θ, φ), θ, φ (0, π),, (0, 2π). 1

2 0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ). f ( ). x i : M R.,, ψ ϕ 1.. (x 1,, x m ) ( x 1,, x m ),, ( x 1 (x 1,, x m ),, x m (x 1,, x m ) ) E 2, (x, y) R 2., (r, θ)., r (0, ), θ (0, 2π),. 2

3 3. r = x 2 + y 2 (1) θ = tan 1 y x (2).. x = r cos θ (3) y = r sin θ (4) 3... v. T M. T M.,, 3.,. 3

4 4,.,. (x 1,, x m ) M., f, ( / x i ). ( / x i ) f. ( ) ( ) x i f := x i f(x1,, x m ). (5),,. ( / x i ) x i.. v = m i=1 ( ) v i x i, Einstein,. ( ) v = v i x i (6) m M T M. M N f. M (x 1,, x m ), N (y 1,, y n ), f, i = 1,, n, y i = f i (x 1,, x m ). f, N g, 4

5 f g = g f, M. f g g. M T M, N T f() N.. v T M f v, N. (f v)g = v(f g) (7) v f g., f v. M (x 1,, x m ), N (y 1,, y n ). v = v i ( / x i )., ( ) v i x i g ( f 1 (x),, f n (x) ) = { v i ( f j x i ) ( ) } y j g (8) f()., ( ) f f v = v i j x i ( ) y j f() (9)., f v, v f Jacobi., f v f. 4 T M. T M,. m V,.,. k (v 1,, v k ) V V R k., T, (a, b R). T (, av + bw, ) = a T (, v, ) + b T (, w, ) (10) 1. V. V. 5

6 k S l T, S T. S T (v 1,, v k, v k+1,, v k+l ) = S(v 1,, v k ) T (v k+1,, v k+l ) (11). (S T ) U = S (T U) (12) (a S + b T ) U = a S U + b T U (13) U (a S + b T ) = a U S + b U T (14) ( S, T, U,, a, b R. ) e i V. V e i (i = 1,, n). e i (e j ) = δ i j (15) δ i j i = j 1, 0 Kronecker. k, T i1 i k = T (e i1,, e ik ) (16)., v i = v i j e j V, T i1 i k e i 1 e i k i1 i (v 1,, v k ) = v 1 v k k T i1 i k i1 i = v 1 v k k T (e i1,, e ik ) i = T (v 1 i 1 e i1,, v k k e ik ) = T (v 1,, v k ) (17), T = T i1 i k e i 1 e i k (18)., e i 1 e i k k., k k V. T i1 i k T i 1 i k. k V m k. 6

e i (e j ) = δ i j (15) δ i j i = j 1, 0 Kronecker. k, T i1 i k = T (e i1,, e ik ) (16).

7 k T 2,,., T i1 i k T (, v,, w, ) = T (, w,, v, ) (19) 2 i,.,., T,. A k (T ) = T [i1 i k ]e i 1 e i k = 1 k! δj 1 j k i 1 i k T j1 j k e i 1 e i k (20), T [i1 i k ] = 1 k! δj 1 j k i 1 i k T j1 j k (21), Kronecker δ j 1 j k i 1 i k, (i 1,, i k ) (j 1,, j k ) 1, 1, 0., 2 S, T,. A k+l (A k (S) A l (T )) = A k+l (S T ) (22), , 2, [ ] 0 1 [T ij ] = (23) (22). 7

8 k T, w, w T., w T w T (v 1,, v k 1 ) = T (w, v 1,, v k 1 ) (24).,, k 1. wedge k α l β k + l α β *1. α β = (k + l)! A k+l (α β). (25) k! l! wedge. (25). (k + l)! α β = α [i1 i k! l! k β ik+1 i k+l ] e i 1 e i k+l = 1 k! l! δj 1 j k+l i 1 i k α j1 j k β jk+1 j k+l e i 1 e i k+l (k + l)! = α j1 j k! l! k β jk+1 j k+l e [j 1 e jk+l]. (26) k α l β, m γ,, a, b R, wedge. 2. e i e j. α β = ( 1) kl β α, (27) α (a β + b γ) = a α β + b α γ, (28) α (β γ) = (α β) γ. (29) *1. 8

= α j1 j k! l! k β jk+1 j k+l e [j 1 e jk+l]. (26) k α l β, m γ,, a, b R, wedge. 2. e i e j.

9 ω. ω = ω i1 i k e i 1 e i k = ω [i1 i k ]e i 1 e i k, e i 1 e i k = = ω i1 i k e [i 1 e i k]. (30) k! (k 1)! A ( k e i 1 (e 2 e k ) ) = k! A k (e i 1 e i k ) = k! e [i 1 e i k]. (31), e [i 1 e i k] = 1 k! ei 1 e i k. (32) (32) (30), ω = 1 k! ω i 1 i k e i 1 e i k = ω i1 i k e i1 e ik (33) i 1 <i 2 < <i k., {e i 1 e i k i 1 < < i k } k., k k V. k V m C k (V m ). e i 1 e i k (32), v 1,, v k, e i 1 e i k (v 1,, v k ) = δ i 1 i k j 1 j k e j 1 (v 1 ) e j k (v k ) e i 1 (v 1 ) e i k (v 1 ) = det e i 2 (v 2 ) e i 2 (v 2 ). (34) e i k (v k ) e i k (v k ), e i 1 e i k (e i1,, e ik ) = 1.., e i 1 e i k (e i1,, e ik ) k!. 9

ω i 1 i k e i 1 e i k = ω i1 i k e i1 e ik (33) i 1 <i 2 < <i k., {e i 1 e i k i 1 < < i k } k., k k V. k V m C k (V m ).

10 ., (e i ) (ē i ) ē i = e j L j i (35)., e i = R i jē j (36), R i j., (35) e j. L j i = e j (ē i ) = R j kē k (ē i ) = R j kδi k = R j i.,, ē i = (L 1 ) i j ej (37)., L 1 [L j i]., v i = ē i (v j e j ) = (L 1 ) i k ek (v j e j ) = (L 1 ) i k δk j v j = (L 1 ) i j vj, v i = (L 1 ) i j vj (38) L 1.,., ᾱ i = α j e j (ē i ) = α j e j (e k L k i) = α j δ j k Lk i = α j L j i 10

11 ., ᾱ i = L j iα j (39), L.,. T i1 i k = L j 1 i1 L j k ik T j1 j k (40).,, L 1, L., LL 1 = I,. Einstein. V = T M, L. M (x i ) ( x i ). e i = / x i, ē i = / x i., ( ) ( ) x j x i = x i ( ) x j (41)., ( ) x L j j i = x i. (42)., ( ) x (L 1 ) j j i = x i (43). v, ( ) x v i i = x j v j (44)., Jacobi,.. α ( ) x j ᾱ i = x i.. 11 α j (45)

12 3. 2 Euclid E 2. xy, (r, θ) L, L 1. 5, M T M.,., v., v = v i (x 1,, x m ) x i (46). v i (x 1,, x m ). v. M k k. 0. e i = / x i. / x i dx i. k ω. ω = 1 k! ω i 1 i k (x 1,, x k ) dx i 1 dx i k. (47), ω i1 i k (x 1,, x k ). M k k T M. 12

13 6 ω k T M, dω k+1 T M. dω = 1 k! = ω i1 i k x j dx j dx i 1 dx i k i 1 <i 2 < <i k ω i1 ik x j dx j dx i 1 dx i k. (48), x ( ).,.,. d x i = xi x j dxj. (49) a, b R, k ω, l τ, d. d(a ω + b τ) = a dω + b dτ, (50) d(ω τ) = (dω) τ + ( 1) k ω dτ. (51) d(dω) = 0 (52) 4. 2 Euclid E 2 f = x 2 + y 2 = r 2 df,, df. 13

14 5. 3 Euclid E 3 (x, y, z) 0, 1, 2 grad, curl, div. 7 M N f, f 1, M N f., f (. ). k T f() N k T k T M, v i T M. (f T )(v 1,, v k ) = T (f v 1,, f v k ). (53) M (x i ), N (y j ), f y j = f j (x 1,, x m ),, (( ) ( ) ) (f T ) i1 i k =(f T ) x i,, 1 x i k ( ( f j 1 ) ( ) ( f j k ) ( ) ) =T x i 1 y j,, 1 f() x i k y j k f() ( f j 1 ) ( f j k ) = x i 1 x i T j1 j k k (54) ( ). N,, M., 14

15 ,.,. wedge, M N f, f. ω, λ k T N, τ l T N,. f (ω + λ) = f (ω) + f (λ), (55) f (ω τ) = f (ω) f (τ). (56), 0 ω = g, (56) f (gτ) = f gf (τ).,,., ω k T M, df ω = f dω (57).,,,. 6. (s, t) E 2, (x, y, z) E 3 f, x = s 2 + t 2, y = st, z = s + t., dx + dy + x dz. 8 Stokes f(x)dx.,. 2 Euclid E 2 ω = fdx dy = fd x dȳ (58) 15

16 ., f E 2 x = y, ȳ = x. E 2 ω. ω := E 2 dx dy = 1 R 2 (59).,, := E 2 d x dȳ = 1 R 2 (60).,.. M 2 (x 1,, x n ), ( x 1,, x n ), det[ x j / x i ] > 0, (x 1,, x n ) ( x 1,, x n ). M, M., 2.,, M.,,,. (x i ), ( / x i ). x i = xj x i x j (61),, 2.,., 1,, 1 o. o (e 1,, e n )., o. o *2. o. m m ω. ω 0 (x i )., ω. M ω := ω 1 m (x 1,, x m )dx 1 dx 2 dx m (62) *2, o n T M/R >., R > = {x R x > 0} 16

o *2. o. m m ω. ω 0 (x i )., ω. M ω := ω 1 m (x 1,, x m )dx 1 dx 2 dx m (62) *2, o n T M/R >.

17 (, ω 1 m (x 1,, x m ) 0. ), ω 0, ω,.. 7. S 1 = {(x, y) E 2 x 2 + y 2 = 1}, θ. S 1 dθ. S 1 E 3 S 2,.. p S, M (x 1,, x m ), (x 1,, x p, 0,, 0). ι : S M. m M, p < m, ω p T M. M p S, ω. ω := S S ι ω. (63) Stokes 5, R m 1 R M., V = [0, ). V., Stokes. 1. (Stokes [2]) n M p 1 α, M ( 17

18 ) p V. dα = α. (64) V V. V (y 1,, y n 1 ), (t, y 1,, y n 1 ) V, V., t t V ( ).. [dt] o V = o V (65) V g.,, g(u, v) = 0 u T M, v = 0. g ( ), i = 1,, s g ii = 1, i = s + 1,, m g ii = 1, g ij = 0 (Sylvester )., (s, (m s)). Euclid 18

19 s = 0. s = 1, m s = 3. g(v, v) v.,. v, v g g. g V V. g, e i, g(e i ) = g ij e j (66)., g ij g., v v i g ij v j. det g 0, g. g 1 g ij., g ij., g ij g jk = δi k (67). g ij,. 10 o, p ω o 1., ω o = ω o (68), ω twisted p. n Ω twisted n form ω. twisted form. ω := ω o (69) Ω Ω o Ω o, o Ω.., Ω, (69)., well-defined. p α, n M p S. ι : S M., twisted form ι.,. 19

10 o, p ω o 1., ω o = ω o (68), ω twisted p. n Ω twisted n form ω. twisted form.

20 V, W. V. v, w W, v w v w W. V/W := V/. V/W, W. S, x T x S x. o T S, twisted form α, (o S ). (ι α) o = ι (α ot o) (70),., o 1 ( x ) > 0, o 2 ( y, z ) > 0, ( x, y, z ), o 1 o 2., M twisted p form S twisted p form. twisted p form p S,, S., S, twisted p form., form dual S twisted form dual S., form, Faraday-Schouten ( 6). Stokes 2. ( Stokes [2]) n M p 1 α, M ( ) V. dα = α. (71) V V *3. o V,T = o V,T [dt] (72) V *3. 20

, M twisted p form S twisted p form. twisted p form p S,, S., S, twisted p form.

21 6 3 form Faraday-Schouten 11 n., n. o. o E a. vol o = E 1 E n (73). E a = e i L i a (74),,, g ab = L i al j bg ij. (75) ±1 = (det[l i a]) 2 det[g ij ] (76) det[l i a] = o(e 1,, e n ) det[g ij ]. (77) 21

22 ., e i, E a., det[l i a] = 1. (73) vol o = det[l i a]e 1 e n = e 1 e n., (73)., e i. o,. vol o = o(e 1,, e n ) det g ij e 1 e n. (78) 12 Hodge m V Hodge. { (e α 1 e α p )} o = 1 (n p)! (vol o) α 1 α p β1 β m p (e β 1 e β m p ). (79), vol p. 1 = vol (80).,,., p ω. s. ω = ( 1) p(m p)+s ω (81) 1 (m = 3, s = 0 ) p ( 1) p(m p)+s p ω, 1 ϕ, (ω ϕ) = g 1 (φ) ω. (82). 22

23 2 (m = 4, s = 1 ) p ( 1) p(m p)+s p ω, η, ω η = 1 p! ω α 1 α p η α 1 α p vol (83)., ω η = η ω (84)., p (ω, η) Ω. (ω, η) Ω = ω η. (85) 8. 4 (t, x, y, z) g ij = diag( 1, 1, 1, 1)., dt, dt dx, dx dy 4 Hodge. Ω V, 3 ρ, 2 D. D = V V ρ. (86) 23

24 , dd = ρ (87). 2 j S D, 1 H, H = S S j + t S D (88)., dh = j + D t. (89) 2 B 1 E., S, E = S t S B (90)., de = B t. (91) B, S, B = 0. (92) S, E, D db = 0. (93) D = ε 0 E (94). H, B B = µ 0 H (95). Maxwell.. 24

25 4 4 2 F = E dt + B (96). 4. (3 d (3). ) df = d (3) E dt + d (3) B + dt B ( t = d (3) E B ) dt + d (3) B t = 0. (97) (91), (93)., df = 0 (91), (93) 2 Maxwell , 4 3. H = H dt + D, (98) J = j dt + ρ (99) dh = d (3) H dt + d (3) D + dt D ( t = d (3) H + D ) dt + d (3) D t = j dt + ρ = J (100)., (87),(89) 1., H = Y 0 F (101). 4, (96), (98), (99)., F, H, J 4. 25

26 9. (t, x, y, z), (101)., g ij = diag( 1, 1, 1, 1). 14 [1].,,,.,, Maxwell. [2],,. [3] 3.,.., (1965).,,.,,,., [4].,,, (= ).. [5]. 26

27 A α β = a k,l A k+l (α β). (102), a k,l. (29), a k,l+m a l,m = a k+l,m a k,l (103), e 1 e k = k!a k (e 1 e k ) (104) *4, a k 1,1 a k 2,1 a 2,1 a 1,1 = k!. (105), a k,1 = (k + 1)!. (106) (103), m = 1, a k,l+1 = k + l + 1 a k,l. (107) l + 1, a k,l = (k + l)! k! l! (108). *4 e 1 e k = A k (e 1 e k ).,. 27

28 [1] F. W. Hehl and Y. N. Obukhov, Foundations of Classical Electrodynamics: Charge, Flux, and Metric, Birkhäuser (2003). [2] The Geometry of hysics: An Introduction, 2nd. ed., Theodore Frankel, Cambridge (2004). [3],, (1988). [4],, (2005). [5] :,, (2009). 28

24.15章.微分方程式

24.15章.微分方程式 m d y dt = F m d y = mg dt V y = dy dt d y dt = d dy dt dt = dv y dt dv y dt = g dv y dt = g dt dt dv y = g dt V y ( t) = gt + C V y ( ) = V y ( ) = C = V y t ( ) = gt V y ( t) = dy dt = gt dy = g t dt