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1 1 matsuo/difgeo.pdf ver.1 8// a A. O 1 e 1 ; e ; e e 1 ; e ; e x 1 ;x ;x e 1 ; e ; e X x x x 1 ;x ;x X (x 1 ;x ;x ) 1 1
2 x x X e e 1 O e x x 1 x x = x 1 e 1 + x e + x e = X j=1 x j e j = x j e j (1) x 1 ;x ;x x x x x = x 1 x x () e 1 = ; e = ; e = f 1 ; f ; f x x = y 1 f 1 + y f + y f = y j f j () x = y 1 y () f 1 = ; f = y ; f = (),() P Einstein a j b j = aj bj j=
3 . ffl Kronecker :f ij ;f j i f ij = f j i = ( 0; (i = j) 1; (i = j) ffl :e ijk e ijk = 8 >< >: 0; (i; j; k ) 1; ((i; j; k) (1; ; ) ) 1; ((i; j; k) (1; ; ) ) e 11 = e 1 = e 1 = =0 e 1 = e 1 = e 1 =1 e 1 = e 1 = e 1 = 1 a = a 1 a ; b = b 1 b () a b a b ffl : a b = a i b i a i b j f ij = X X i=1 j=1 a i b j f ij e i e j = f ij ffl : a b = a i b j e ij1 = a b a b a i b j e ij = a b 1 a 1 b a i b j e ij = a 1 b a b 1
4 e 1 e 1 =0; e e =0; e e =0 e 1 e = e ; e e = e 1 ; e e 1 = e e e 1 = e ; e e = e 1 ; e 1 e = e ffl : a Ω b = a 1 b 1 a 1 b a 1 b a b 1 a b a b a b 1 a b a b e 1 ; e ; e e 1 Ω e 1 = e Ω e 1 = e Ω e 1 = ; e 1 Ω e = ; e Ω e = ; e Ω e = ; e 1 Ω e = ; e Ω e = ; e Ω e = (a Ω b)c = a(b c) =(b c)a =(c b)a = a(c b) =(a Ω c)b a b A A b = Aa () A(a + b) =Aa + Ab A(ka) =k(aa) k a; b a = a 1 a ; b = b 1 b a b
5 A A ij b 1 b b = A 11 A 1 A 1 A 1 A A A 1 A A a 1 a a () Einstein b i = A ij a j = X j=1 A ij a j. a,a [a] [A] i a i (a) i i j A ij (A) ij e 1 ; e ; e A =[A] =(A) ij = A = A ij (e i Ω e j ) (8) A 11 A 1 A 1 A 1 A A A 1 A A (9). () B c = Bb = B(Aa) =(BA)a (10) BA A B B 11 B 1 B 1 B 1 B B B 1 B B b 1 b b = B 11 B 1 B 1 B 1 B B B 1 B B A 11 A 1 A 1 A 1 A A A 1 A A Einstein a 1 a a (11) B ki b i = B ki A ij a j (1) Einstein (BA) k;j = B ki A ij (1) i
6 . f x f(x) x; y ff; f(ffx + y) = fff(x)+ f(y) f; ψ x f + ψ fff x (f + ψ)(x) =f(x)+ψ(x) (fff)(x) =fff(x) f i (e j )=f i j; i =1; ; ; j =1; ; x = x i e i f j (x) =f j (x i e i )=x i f j (e i )=x i f j = i xj f i ( ) f f(e i )=a i f f( ) =a j f j ( ) f 1 ( );f ( );f ( ) f 1 (x) = f (x) = f (x) = h h h f j (x) =e j x; j =1; ; i i x x i x 8 8.
7 .8 e 1 ; e ; e f 1 ( );f ( );f ( ) e i Ω e j (1) e i Ω f j ( ) (1) f i ( ) Ω e j (1) f i ( ) Ω f j ( ) (1) (1) (1) (1),(1) e i Ω e j Ω e k ; e i Ω e j Ω f k ( ); e i Ω f j ( ) Ω f k ( ); f i ( ) Ω f j ( ) Ω f k ( ).9 ( ) e i ^ e j, e i ^ e j ^ e k e i Ω e j, e i Ω e j Ω e k A ij e i ^ e j = 1 A ij (e i Ω e j e j Ω e i ) (18) B ijk e i ^ e j ^ e k = 1 B ijk (e i Ω e j Ω e k + e j Ω e k Ω e i + e k Ω e i Ω e j e i Ω e k Ω e j e k Ω e j Ω e i e j Ω e i Ω e k ) (19) e i ^ g j = e j ^ e i e i ^ e i =0 (e i + e j ) ^ e k = e i ^ e k + e j ^ e k (e i ^ e j ) ^ e k = e i ^ (e j ^ e k )=e i ^ e j ^ e k e 1 ^ e 1 (ff 1 e 1 + ff e ) ^ ( 1 e 1 + e )= ff 1 ff (ff 1 e 1 + ff e + ff e ) ^ ( 1 e 1 + e + e ) ^ (fl 1 e 1 + fl e + fl e ) = ff 1 ff ff 1 fl 1 fl fl e 1 ^ e ^ e j j
8 Λ( ) Λ : e 1 ^ e! e Λ : e ^ e! e 1 Λ : e ^ e 1! e e 1 e = Λ(e 1 ^ e ) e e = Λ(e ^ e ) e e 1 = Λ(e ^ e 1 ).10 e 0 1 ; e0 ; e0 e 1; e ; e e 0 k = M kj e j ; k =1; ; (0) e 1 ; e ; e e 0 1 ; e0 ; e0 e k = Mkj μ e 0 j ; k =1; ; (1) [M] [ M] μ μm kl M lj = f kj M ij μ Mjl = f il.11 e 0 1 ; e0 ; e0 e 1; e ; e e 0 k = M kj e j ; k =1; ; () a a = a i e i = a 0 k e0 k () () () a a i = a 0 k M ki () [a] = [M] T [a 0 ] () 8
9 [a] = [M] = a 1 a a ; 0Λ a = a 0 1 a 0 a 0 M 11 M 1 M 1 M 1 M M M 1 M M () ().1 e 1 ; e ; e! e 0 1 ; e0 ; e0 f 1 ( );f ( );f ( )! f 01 ( );f 0 ( );f 0 ( ) f 0j ( ) = Mij μ f i ( ) (8) e 0 i = M ij e j f 0j ( ) f 0j (e 0 k)=f j = k f jk μm ij f 0i (e 0 k)= Mij μ f 0i (M kl e l )= Mij μ M kl f 0i (e l )= Mij μ M kl f i l = Mij μ M ki = M kimij μ = f kj.1 A = A ij e i Ω e j = A 0 ij e0 i Ω e 0 j (9) e 0 i = M ik e 0 k e 0 j = M jl e 0 l (9) A ij e i Ω e j = A 0 ij M ikm jl e k Ω e l = A 0 kl M kim lj e i Ω e j A ij A 0 ij A ij = A 0 kl M kim lj (0) 9
10 .1 t x = a + bt x = x 1 x ; a = a 1 a ; b = b 1 b x a b b 1 b a tb x O t 1 ;t x = a + bt 1 + ct x = x 1 x ; a = a 1 a ; b = b 1 b ; c = c 1 c x a b c b; c 1 rank [b; c] = b; c 1.1 x 1 ;x ;x ffl (x 1 ;x ;x ) h rank a 0 + a 1 x 1 + a x + a x =0 a 1 a a i =1 ffl (x 1 ;x ;x ) ( a 0 + a 1 x 1 + a x + a x =0 b 0 + b 1 x 1 + b x + b x =0 " # a 1 a a rank = b 1 b b 10
11 .1 x 1 ;x ;x ffl (x 1 ;x ;x ) ( f1 (x 1 ;x ;x )=0 f (x 1 ;x ;x )=0 1 @x # = ffl (x 1 ;x ;x ) f(x 1 ;x ;x )=0 @x i =1 rank x 1 ;x ;x ffl (x 1 ;x ;x ) t t x 1 x x = x 1 (t) x (t) x (t) ffl x y h dx(t) y = x(t)+ dx(t) h (1) ffl (x 1 ;x ;x ) t 1 ;t t 1 ;t x 1 x x = x 1 (t 1 ;t ) x (t 1 ;t ) x (t 1 ;t ) 11
12 dx(t) x(t) x(t + h) O.18 D x D h(x) h(x); x D D x D f(x) = f 1 (x) f (x) f (x) () f(x); x D 1 (C 1 ) y = f(x) y = f(x) y df (x) dx y 0 + dy y 0 x x 0 x 0 + dx dy x x 0 dx f(x) y = y 0 y 0 = f(x 0 ) dy = df(x 0) dx () dx 1
13 (dx; dy) (x 0 ;y 0 ) f(x) x 0.0 (x 1 ;x ;x ) f(x 1 ;x ;x ) x 1 ;x ;x ψ ψ 8 >< >: u 1 = u 1 (x 1 ;x ;x ) u = u (x 1 ;x ;x ) u = u (x 1 ;x ;x ) ψ 1 ψ 8 >< >: x 1 = x 1 (u 1 ;u ;u ) x = x (u 1 ;u ;u ) x = x (u 1 ;u ;u ) f(x 1 ;x ;x ) u 1 ;u ;u f(u μ 1 ;u ;u ) f(x 1 ;x ;x ) = μ f(u 1 (x 1 ;x ;x );u (x 1 ;x ;x );u (x 1 ;x ;x )) μf(u 1 ;u ;u ) = f(x 1 (u 1 ;u ;u );x (u 1 ;u ;u );x (u 1 ;u ;u )) C 1.1 μf(u 1 ;u ;u ) D μ f D μ f μ i dui () (u 1 ;u ;u ) (u 1 + h ~ 1 ;u + h ~ ;u + h ~ ) f(u μ 1 ;u ;u ) μf Taylor μf = μ f(u 1 + ~ h 1 ;u + ~ h ;u + ~ h ) μ f(u 1 ;u ;u μ i μ h i + ( ~ h 1 ; ~ h ; ~ h ) h ~ i du i (x 1 ;x ;x ) (x 1 + h 1 ;x + h ;x + h ) f(x 1 ;x ;x ) f h i dx i Df i dxi () 1
14 f i dxi μ i dui dx i du i f(x 1 + h 1 ;x + h ;x + h ) f(x 1 ;x ;x ) = μ f(u 1 (x 1 + h 1 ;x + h ;x + h );u (x 1 + h 1 ;x + h ;x + h );u (x 1 + h 1 ;x + h ;x + h )) ß μ f(u 1 ;u ;u ) μ f ψ u 1 (x 1 ;x ;x @u1 + h @x ;u (x 1 ;x ;x @u ; +h @x ; u (x 1 ;x ;x @u + h f(u μ 1 ;u ;u ~h i hi () f = μf h 1! dx 1 ; h! dx ; h! dx ~h 1! du 1 ; ~ h! du ; ~ h! du du i dxi () i dxi μ i dui = D μ f (8) dx 1 ;dx ;dx du 1 ;du ;du 9 (x 1 ;x ;x ) (u 1 ;u ;u ) () ο dx 1 ;dx ;dx du 1 ;du ;du () ο = ff i dx i = μff j du j (9) ff i dx j = μff i j (ff i μff i )dxi =0 i 1
15 ff i = μff i (0). ffl 0 f(x 1 ;x ;x ) ffl 1 1! (x 1 ;x ;x )! = g i (x 1 ;x ;x )dx i (1) g i (x 1 ;x ;x ) C 1 (u 1 ;u ;u )! =μg i (u 1 ;u ;u )du i () ffl (x 1 ;x ;x ) dx 1 ^ dx,dx ^ dx,dx ^ dx 1 = h 1 (x 1 ;x ;x )dx 1 ^ dx + h (x 1 ;x ;x )dx ^ dx + h (x 1 ;x ;x )dx ^ dx 1 () (u 1 ;u ;u ) = μ h 1 (u 1 ;u ;u )du 1 ^ du + h μ (u 1 ;u ;u )du ^ du + h μ (u 1 ;u ;u )du ^ du 1 () ffl 1 dx 1 ^ dx ^ dx du 1 ^ du ^ du = k(x 1 ;x ;x )dx 1 ^ dx ^ dx = μ k(u 1 ;u ;u )du 1 ^ du ^ du () () du 1 ^ du ^ du dx 1 ^ dx ^ dx. d i i +1 ffl 0 f = f(x 1 ;x ;x ) df i dxi () 1
16 ffl 1! = f i dx i d! i = dx i ^ dx k i dxi ^ dx k d! = df i ^ dx @f 1 dx 1 ^ @f dx ^ dx ^ dx 1 ffl = h 1 dx 1 ^ dx + h dx ^ dx + h dx ^ dx 1 d d = dh 1 ^ dx 1 ^ dx + dh ^ dx ^ dx + dh ^ dx ^ dx 1 d + dx 1 ^ dx ^ dx ffl ο dο ffl ff; ffl f;g 0 dο =0 (9) d(ff! + ) =ffd! + d (0) d(fg)=gdf + fdg (1) dx i i dxi i dxi = gdf + fdg ffl! i (i =0; 1; )!; 1 d(! ^ ) =d! ^ +( 1) i! ^ d ()! = f 1 dx 1 ; = g dx! ^ =(f 1 dx 1 ) ^ (g dx )=(f 1 g )dx 1 ^ dx d(! ^ ) = d(f 1 g ) ^ dx 1 ^ dx =(g df 1 + f 1 dg ) ^ dx 1 ^ dx = (df 1 ^ dx 1 ) ^ (g dx )+dg ^ (f 1 dx 1 ) ^ dx = d! ^ (f 1 dx 1 ) ^ (dg ^ dx )=d! ^! ^ d 1
17 ffl! d(d!) =0 () 0 f f d(df) = d @ f dx 1 ψ! 1 @ dx ^ dx f dx 1 ^ 1 1! = fdx 1 d(d!) = dx ^ dx dx ^ dx f dx ^ dx ^ dx 1 f dx ^ dx ^ dx 1 =0. f (x 1 ;x ;x ) f(x 1 ;x ;x ) (u 1 ;u ;u ) f(u μ 1 ;u ;u ) 0 i dxi μ i dui = d μ f j i du i i dxi f i dxi f j duj! = f i dx i = μ f i du i d! = df i ^ dx i = d μ f i ^ du i () f i (x 1 ;x ;x )= μ f(u 1 (x 1 ;x ;x );u (x 1 ;x ;x );u (x 1 ;x ;x )) 1
18 ffl f;g 0 d(fg)=gdf + fdg () ffl f 0! 1 d(f!)=df ^! + fd! () ffl f 0 d(df) =0 (8) (),(0) du j dxj (9) f i = μ f i (0) (9) u i = u i (x 1 ;x ;x ) () f i (x 1 ;x ;x )dx 1 = μ f i (u 1 ;u ;u )du i = μ f i (u 1 (x 1 ;x ;x );u (x 1 ;x ;x );u (x 1 ;x ;x ))du i (x 1 ;x ;x ) x 1 ;x ;x df i (x 1 ;x ;x ) ^ dx i = d( μ f i (u 1 ;u ;u )du i ) d( μ f i (u 1 ;u ;u )du i )=d μ f i ^ du i + μ f i d(du i )=d μ f i ^ du i d μ f i (u 1 (x 1 ;x ;x );u (x 1 ;x ;x );u (x 1 ;x ;x )) = d μ f i (u 1 ;u ;u ) x 1 ;x ;x df μ u 1 ;u ;u d( μ f i du i )=d μ f i ^ du i 1 df i ^ dx i = d μ f i ^ du i 18
19 . ffl x 1 ;x u 1 = r;u = : x 1 (r; )=r cos ; x (r; )=r sin x 1 ;x r; 1 dx 1 ;dx dx @x1 dr + d =cos dr r dr + d =sin dr + r cos dx 1 ^ dx dx 1 ^ dx = r cos dr ^ d r sin d ^ dr = rdr ^ d ffl x 1 ;x u 1 = x 1 x 1 ;u = x 1 1 x : du 1 ^ du =(x 1 x 1 dx1 x 1 x dx ) ^ ( x 1 x dx 1 +x 1 1 x dx )=dx 1 ^ dx. grad,div,rot grad,div,rot..1 grad 1 1! = g i dx i! = g dx g = dx = f(x) gradient gradf(x) = 19 g 1 g g dx 1 @x (1)
20 f(x) grad df dx i = i dx.. 1! 1 = g dx;! = h dx g = g 1 g ; h = h 1 h g h! 1 ^! =(g h g h )dx ^ dx +(g h 1 g 1 h )dx ^ dx 1 +(g 1 h g h 1 )dx 1 ^ dx g h g h = g h g h g h 1 g 1 h g 1 h g h 1 ds ds = dx ^ dx dx ^ dx 1 dx 1 ^ dx ()! 1 ^!! 1 ^! =(g dx) ^ (h dx) =(g h) ds () = g ds ().. ο = gdx 1 ^ dx ^ dx dv dv = dx 1 ^ dx ^ dx () ο = gdv 0
21 .. 1! = g dx = h ds! ^ =(g dx) ^ (h ds) =(g h)dv () 1! 1 ;! ;!! 1 = g dx;! = h dx;! = k dx! 1 ^! ^! = g 1 g g h 1 h h k 1 k k dv ().. grad,div,rot ffl 0 grad gradf i dxi = gradf dx (8) @x ffl 1! = g dx (9) d(g dx) =rotg ds (0) d! = dg i ^ dx = dx ^ dx ^ dx @g 1 dx 1 ^ dx rotg @x 1 ffl = h ds (1) d(h ds) = divh dv () = h 1 dx ^ dx + h dx ^ dx 1 + h dx ^ dx 1 1
22 @h1 d dx 1 ^ dx ^ divh divh @x () d(d!) =0 ffl rot(gradf) =0 0 f df = gradf dx d(df) =d(gradf dx) =rot(gradf) ds =0 ffl div(rotg) =0 1! = g dx d! = rotg ds d(d!) =d(rotg ds) =div(rotg)dv =0...1 x f(x) =f(x 1 ;x ;x ) C C t x i = x i (t); i =1; ; ; ff» t» x(t) = C X i f(x) s i = X i x 1 (t) x (t) x (t) f(x i ) q ; ff» t» ( x 1 i ) +( x i ) +( x i ) x i s i i C f(x) R C f(x)ds s dx 1 f(x)ds = f(x(t)) C ff dx + dx + ()
23 ds C s dx 1 dx dx ds = + + x g(x) C g(x) g(x) = g 1 (x) g (x) g (x) C x dx(t) = g(x) g dx(t) C dx 1 (t) dx (t) dx (t) dx(t) = ; ff» t» dx 1 (t) dx (t) dx (t) () C g(x) R C g(x) dx g(x) dx = C g i dx i = ff ψ g i dx i (t)! ().. C 1! = g i (x)dx i! = g(x) dx 1 C C! = C g i (x)dx i = ff dx i g i ()
24 .. a b ja bj a b 10 a b ja bj = a b a b + a b a b + a b = a a b a a b b b (8) G(a; b) G(a; b) = a a b a a b b b (9) S t 1 ;t x = x(t 1 ;t )= x 1 (t 1 ;t ) x (t 1 ;t ) x (t 1 ;t ) S x dx @x 1 ds @t 1 )1 (81) ds 0 S ds 0 ;x 1 ;t ) ;x 1 1 ;t ) 1 ;x 1 ;t ) ;x 1 ;t ;x 1 1 ;t 1 ;x 1 ;t ) = @x @x 10 : 11
25 1 x O ds = x (t 1 ;t ) dx ^ dx dx ^ dx 1 dx 1 ^ dx dx i j j dx ^ dx ;x 1 ;t ) 1 ^ dx ^ dx 1 ;x 1 1 ;t ) 1 ^ dx 1 ^ dx 1;x 1 ;t ) 1 ^ (8) ^.. f(x) S = fx = x(t 1 ;t ):(t 1 ;t ) Dg S f(x)ds = D f(x(t 1 ;t )) 1 (8)
26 r ds ds S D jdsj = ds = D 1 (8) = h ds S h ds = h ds ;x ) = h 1 ;t ) + ;x 1 1 ;t ) + 1 ;x 1 ;t ) h S 1 (8).. h(x) h(x) = " h1 (x 1 ;x ) h (x 1 ;x ) h dx = h 1 dx 1 # (8) h dx C = fxjx = x(t);ff» t» g dx h dx = h 1 h dx 1 (8) C ff " # dx Λ dx = ; dx 1 dx Λ = " dx dx 1 # dx Λ dx =0; dxλ h dx dx Λ s = dx dx1 = = jhj dx cos dx =0 dx + h C C h dx = C jhj dx C cos = jhj cos ds = C h 1 dx h dx 1
27 p +1 p! d! p +1 V d! (88) <d!;v >=<!;@V > (89)!! d! 1.8. S rotg ds = d(g dx) = g dx (90) = g 1 dx 1 + g dx d! = d(g 1 dx 1 + g @g 1 dx 1 ^ dx = 1 rotg @g 1 ds = dx 1 ^ dx @g 1 dx 1 dx g 1 dx 1 + g dx (91).8. V divh dv = V d(h ds) = 1 h ds (9)
28 .8. h =(h 1 ;h ) divh S divh ds h dx (9) d(h dx) = d(h 1 dx h dx 1 )=dh 1 ^ dx dh ^ dx 1 1 dx1 dx ^ dx dx 1 ^ dx = divh ds u 1 ;u (x 1 ;x ;x ) ψ ψ :(u 1 ;u )! (x 1 ;x ;x ) (9) x 1 = x 1 (u 1 ;u ) x = x (u 1 ;u ) x = x (u 1 ;u ) (u 1 ;u ) D C ψ (x 1 ;x ;x ) S C ~ C 1 ffl : S f(x 1 ;x ;x ) ψ ψ Λ f ffl 1 : (ψ Λ f)(u 1 ;u )=f(x 1 (u 1 ;u );x (u 1 ;u );x (u 1 ;u )) (9)! = f i dx i ψ Λ! =(ψ Λ f i j du j =(ψ Λ f i )d(ψ Λ x i ) (9) ffl : 1 = gdx 1 ^ dx ψ Λ =(ψ Λ g)d(ψ Λ x 1 ) ^ d(ψ Λ x ) (9) ψ Λ (d!) =d(ψ Λ!) (98) 1 dο d(ψ Λ ο)=0 8
29 ffl 0 f ψ Λ (df) =ψ i = du j i ψ! (u 1 ;u );x (u 1 ;u );x (u 1 Λ f) = du j Λ f) = du j f) f kj du j f) du j ffl 1! = f i dx i ψ Λ (d!) =ψ Λ (d(f i dx i )) = ψ Λ (df i ^ dx i )) = d(ψ Λ f i ) ^ d(ψ Λ x i ) d(ψ Λ!)=d(ψ Λ (f i dx i )) = d ((ψ Λ f i )d(ψ Λ x i )) = d(ψ Λ f i ) ^ d(ψ Λ x i ) ψ(d) ψ(@d) D d! =! = ψ Λ (d!) = ψ(d) d! = ψ j du j! ψ Λ (d!) (99) ψ Λ! ψ(@d) ψ Λ! (101)! (10) ! = g dx! = 1 = g 1 ;! = = g (10)! f (10) f =.0. H = H(x; p) =H(x 1 ; ;x n ;p 1 ; ;p n ) H dx i (t) dp i i ; i =1; ;n i ; i =1; ;n (10) 9
30 (x 1 (t); ;x n (t);p 1 (t); ;p n (t)) H x i (t);p i (t) dh = i dx i dp i (10) dx i = dx = i dp i = dp i @p i =0 (110) H(x; p) = (111).1 div 1 f 1 (x 1 ;x ;x ) f(x 1 ;x ;x )= 1 r(x 1 ;x ;x ) r q r = x + 1 x + x E E = gradf(x 1 ;x ;x i i @x = x i r = 1 r x 1 x x = x r V 0 S 0 n 0 ds = n 0 ds V 0 divedv = V 0 d(e n 0 ds) = 0 S 0 E n 0 ds
31 n 0 = r r E n 0 = r r n 0 = 1 r S0 E n 0 ds = 1 r S0 ds = ßr r =ß S S 0 O S S 0 V V S S 0 n; n 0 n 0 V S 0 V S O n 0 n divedv = E nds E n 0 ds V S S i = 1 r + r x i r = 0 dive f dive = div(gradf) = f f 1 V O S V E nds = divedv =0 S 0 E n 0 ds =ß rmdive 1
32 . rot rot C C S n E rot 1 rote n = lim S!0 S C E ds (11) AB CD E x y y + 1 y D C y y 1 y A B O x 1 x x x + 1 x x E x (x; y) y; E x(x; y)+ y AB CD E x (x; y) 1 x E x (x; y)+ 1 x ( BC DA E y E y (x; y)+ x; E y(x; y) x BC DA E y (x; y)+ 1 y E y (x; y) 1 y ( C E @Ey x S rot E
33 . (11) U(x) x x 1 = x; x = _x dx 1 dx = x (11) 1 (11) (x 1 ;x ) dx 1 =0; dx 1 1 = 0 (11) x = 0 (11) x t dx 1 = dx 1 = x (118) dx = dx (119) 1 1 dx 1 + x dx 1 x 1 1 =0 (10) f(x 1 ;x ) df 1;x 1 dx 1 1;x dx 1 dx 1 + x dx =0 (11) f(x 1 ;x 1 ;x ) 1) 1 ;x ) = x f(x 1 ;x )=U(x 1 )+ 1 x (1)
34 (11) df =0 f(x 1 ;x )=U(x 1 )+ 1 x = C (1) C 1 (x 1 ;x ) 1 U(x 1 ) U(x 1 )= 1 x 1 (1) d x 1 = x 1 (1) 1 x x = C (18) x x 1... S t 1 ;t x = x(t 1 ;t )= x 1 (t 1 ;t ) x (t 1 ;t ) x (t 1 ;t ) 1 x _x
35 S x dx 1 @x 1 S (tangent space)t (S) (,cotangent space)t Λ (S) ffl T ffl T Λ (S) : 1 ( ); ( ) j = f i j v = v i i! 1! = f i i (1) i t i i j (v) = v i = v j (19) i.. [1] (1998). [] 0 (1989). [] (19). [] (198).
36 [] W. (199). [] (190). [] (199). [8] (199). [9] (199). [10] (19). [11] J.L.Stynge and A.Schild : Tensor Calculus, Dover Pub. Inc. (198). [1] H. (19). [1] (1998). [1] (000). [1] (001). [1] R.W.R ( ) (000). [1] (198). [18] (000).
( 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
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
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II 2 II 2005 yugami@cc.utsunomiya-u.ac.jp 2005 4 1 1 2 5 2.1.................................... 5 2.2................................. 6 2.3............................. 6 2.4.................................
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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II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F
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1. x { e 1,..., e n } x = x1 e1 + + x n en = (x 1,..., x n ) X, Y [X, Y ] Intrinsic ( ) Intrinsic M m P M C P P M P M v 3 v : C P R 1 f, g C P, λ R (1) v(f + g) = v(f) + v(g) (2) v(λf) = λv(f) (3) v(fg)
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I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345
F S S S S S S S 32 S S S 32: S S rot F ds = F d l (63) S S S 0 F rot F ds = 0 S (63) S rot F S S S S S rot F F (63)
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 )
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v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i
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1 1 1.1 (Isaac Newton, 1642 1727) 1. : 2. ( ) F = ma 3. ; F a 2 t x(t) v(t) = x (t) v (t) = x (t) F 3 3 3 3 3 3 6 1 2 6 12 1 3 1 2 m 2 1 x 1.1: v mg x (t) = v(t) mv (t) = mg 0 x(0) = x 0 v(0) = v 0 x(t)
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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
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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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