difgeo1.dvi

1 http://matlab0.hwe.oita-u.ac.jp/ matsuo/difgeo.pdf ver.1 8//001 1 1.1 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

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 = 1 0 0 ; e = 0 1 0 ; e = f 1 ; f ; f x 0 0 1 x = y 1 f 1 + y f + y f = y j f j () x = y 1 y () f 1 = 1 0 0 ; f = y 0 1 0 ; f = (),() P Einstein a j b j = aj bj j=1 0 0 1

. 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

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 = 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ; e 1 Ω e = ; e Ω e = ; e Ω e = 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 ; 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 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1. 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

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

. 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 1 0 0 0 1 0 0 0 1 f j (x) =e j x; j =1; ; i i x x i x 8 8.

.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

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

[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

.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

.1 x 1 ;x ;x ffl (x 1 ;x ;x ) ( f1 (x 1 ;x ;x )=0 f (x 1 ;x ;x )=0 rank " @f1 @x 1 @f 1 @x @f 1 @x @f @x 1 @f @x @f @x # = ffl (x 1 ;x ;x ) f(x 1 ;x ;x )=0 rank h @f @x 1 @f @x @f @x i =1 rank 1 1.1 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

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 ).19 1 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

(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 f @u 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 f @u 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 Df = @f @x i dxi () 1

f μ f @f @x i dxi = @ μ f @u 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 )+h 1 @u1 @u1 @u1 + h + h @x1 @x @x ;u (x 1 ;x ;x )+h 1 @u @u @u ; +h + h @x1 @x @x ; u (x 1 ;x ;x )+h 1 @u @u @u + h + h @x1 @x! f(u μ 1 ;u ;u ) @x ~h j = @uj @x i hi () f = μf h 1! dx 1 ; h! dx ; h! dx ~h 1! du 1 ; ~ h! du ; ~ h! du du j = @uj @x i dxi () Df = @f @x i dxi = @ μ f @u 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 i @u j = μff j @x i dxi @u j (ff i μff j @x i )dxi =0 9 dxi @ @x i 1

ff i = μff j @u j @x 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 = @u 1 @x 1 @u 1 @x @u 1 @x @u @x 1 @u @x @u @x @u @x 1 @u @x @u @x dx 1 ^ dx ^ dx. d i i +1 ffl 0 f = f(x 1 ;x ;x ) df df = @f @x i dxi () 1

ffl 1! = f i dx i d! @fk d! = @x i = dx i ^ dx k = @f k @x i dxi ^ dx k d! = df i ^ dx i () @f @x @f 1 dx 1 ^ dx @f + 1 @x @x @f dx ^ dx @f1 + @x @x @f dx ^ dx 1 @x 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 (8) @h1 d = @x + @h @x + @h dx 1 ^ dx ^ dx 1 @x ffl ο dο ffl ff; ffl f;g 0 dο =0 (9) d(ff! + ) =ffd! + d (0) d(fg)=gdf + fdg (1) d(fg)= @(fg) dx i = g @f @x i @x i dxi + f @g @x 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

ffl! d(d!) =0 () 0 f dxi ψ! @f @ f d(df) = d = @x i @x 1 @x @ f dx 1 ^ dx @x @x ψ! 1 ψ! @ f + @x @x @ f @ dx ^ dx f + @x @x @x 1 @x @ f dx 1 ^ dx =0 @x @x 1 1! = fdx 1 d(d!) = d = @f @x dx ^ dx 1 + @f @x dx ^ dx 1 @ f @x @x dx ^ dx ^ dx 1 + @ f @x @x 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 df = @f @x i dxi = @ μ f @u i dui = d μ f () @f @x = @ f μ @u j i @u j @x i du j = @uj @x i dxi 1 @f @x i dxi = @ f μ @u j @u j @x i dxi = @ f μ @u 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

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 i = @ui @x j dxj (9) f i = μ f j @u j @x 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

. 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 1 = @x1 @r dx = @x @r @x1 dr + d =cos dr r sin d @ @x dr + d =sin dr + r cos d @ 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 dx dx @f(x) @x 1 @f(x) @x @f(x) @x (1)

f(x) grad df = @f(x) dx i = gradf @x 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

.. 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 df = @f @x i dxi = gradf dx (8) gradf = @f @x 1 @f @x @f @x ffl 1! = g dx (9) d(g dx) =rotg ds (0) d! = dg i ^ dx i @g = @x @g dx ^ dx @g1 + @x @x @g dx ^ dx 1 @g + @x 1 @x @g 1 dx 1 ^ dx 1 @x rotg rotg = @g @g @x @x @g 1 @g @x @x 1 @g @g @x 1 1 @x ffl = h ds (1) d(h ds) = divh dv () = h 1 dx ^ dx + h dx ^ dx 1 + h dx ^ dx 1 1

@h1 d = @x + @h 1 @x + @h dx 1 ^ dx ^ dx @x divh divh = @h 1 @x 1 + @h @x + @h @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 + ()

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 ()

.. 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 11 dx = @x @t 1 1 + @x @t (80) @x, @x 1 ; @t 1 @t ds 0 =( @x @t 1 1 ) ( @x @t )=( @x @t 1 ) ( @x @t )1 (81) ds 0 S ds 0 = @(x ;x ) @(t 1 ;t ) 1 @(x ;x 1 ) @(t 1 ;t ) 1 @(x 1 ;x ) @(t 1 ;t ) 1 @(x ;x ) @(t 1 ;t ) @(x ;x 1 ) @(t 1 ;t ) @(x 1 ;x ) @(t 1 ;t ) = = = @x @x @t 1 @t @x @x @t 1 @t @x @x @t 1 @t @x 1 @x 1 @t 1 @t @x 1 @x 1 @t 1 @t @x @x @t 1 @t 10 : 11

ds 0 @x @t @x @t 1 x O ds = x (t 1 ;t ) dx ^ dx dx ^ dx 1 dx 1 ^ dx dx i = @x i @t j j dx ^ dx = @(x ;x ) @(t 1 ;t ) 1 ^ dx ^ dx 1 = @(x ;x 1 ) @(t 1 ;t ) 1 ^ dx 1 ^ dx = @(x 1;x ) @(t 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 )) s G @x @t 1 ; @x @t 1 (8)

r G @x ; @x 1 @t 1 @t ds ds S D jdsj = ds = D s G @x @t 1 ; @x @t 1 (8) = h ds S h ds = h ds 0 @(x ;x ) = h 1 @(t 1 ;t ) + h @(x ;x 1 ) @(t 1 ;t ) + h @(x 1 ;x ) @(t 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

.8.8.1 p +1 V @V p! d! p +1 V d! = @V! (88) <d!;v >=<!;@V > (89)!! d! V! @V 1.8. S @S S rotg ds = d(g dx) = g dx (90) S @S.8.! = g 1 dx 1 + g dx d! = d(g 1 dx 1 + g dx )= @g @x @g 1 dx 1 ^ dx = 1 rotg ds @x rotg = @g @x @g 1 1 @x ds = dx 1 ^ dx 1 S @g @x @g 1 dx 1 dx = 1 @x @S g 1 dx 1 + g dx (91).8. V @S V divh dv = V d(h ds) = 1 1 @S h ds (9)

.8. h =(h 1 ;h ) divh = @h 1 + @h @x 1 @x S divh ds = @S S @S h dx (9) d(h dx) = d(h 1 dx h dx 1 )=dh 1 ^ dx dh ^ dx 1 = @h 1 @x 1 dx1 ^ dx @h @x dx ^ dx 1 = @h1 @x + @h dx 1 ^ dx = divh ds 1 @x.9 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 )( @x i @u 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) 0 1 1 ψ Λ (d!) =d(ψ Λ!) (98) 1 dο d(ψ Λ ο)=0 8

ffl 0 f ψ Λ (df) =ψ Λ @f! dx @x i = ψ Λ @f @xi du j i @x i ψ @u j @f(x1 ψ! (u 1 ;u );x (u 1 ;u );x (u 1 ;u )) @xi @(ψ Λ f) = du j = @x i @u j @x i @(ψ Λ f) ψ! @u = k @xi du j = @(ψλ f) f kj du j = @(ψλ f) du j @u k @x i @u j @u k @u 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! = D @D ψ @xi @u j du j! ψ Λ (d!) (99) ψ Λ! (100) @D ψ(@d) ψ Λ! (101)! (10).0.0.1 1! = g dx! = df f @f @x 1 = g 1 ;! =0 (10) @f @x = g ; @f @x = 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 (t) = @H @p i ; i =1; ;n (10) = @H @x i ; i =1; ;n (10) 9

(x 1 (t); ;x n (t);p 1 (t); ;p n (t)) H x i (t);p i (t) dh = dh = @H @x i dx i + @H @p i dp i (10) dx i = dx i @H = (108) @p i dp i = dp i = @H (109) @x i @H @p i @H @x i + @H @x i @H @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 ) @f = 1 @r @x i r @x i E = @f @x 1 @f @x @f @x = x i r = 1 r x 1 x x = x r V 0 @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

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 0 @ f @x i = 1 r + r x i r = 0 dive ψ! @ f dive = div(gradf) = + @ f + @ f =0 @x 1 @x @x V O S V E nds = divedv =0 S 0 E n 0 ds =ß rmdive 1

. 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) 1 @E x @y y; E x(x; y)+ 1 @E x @y y AB CD E x (x; y) 1 y @E x x; @y E x (x; y)+ 1 y @E x ( x) @y BC DA E y E y (x; y)+ 1 @E y @x x; E y(x; y) 1 @E y @x x BC DA E y (x; y)+ 1 x @E y y; @x E y (x; y) 1 x @E y ( y) @x C C @Ey E ds ß @x @E x @y @Ey x y = @x @E x @y S rot E

. dx = @U(x) @x (11) U(x) x x 1 = x; x = _x dx 1 dx = x (11) = @U(x 1) @x 1 (11) (x 1 ;x ) dx 1 =0; dx =0 @U(x 1 ) @x 1 = 0 (11) x = 0 (11) x t dx 1 = dx 1 = x (118) dx = dx = @U(x) (119) @x @U(x 1 ) @x 1 dx 1 + x dx = @U(x 1) @x 1 x x @U(x 1 ) @x 1 =0 (10) f(x 1 ;x ) df = @f(x 1;x ) @x 1 dx 1 + @f(x 1;x ) @x dx = @U(x 1) @x 1 dx 1 + x dx =0 (11) f(x 1 ;x ) @f(x 1 ;x ) = @U(x 1) (1) @x 1 @x 1 @f(x 1 ;x ) = x (1) @x f(x 1 ;x )=U(x 1 )+ 1 x (1)

(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 1 + 1 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

S x dx dx = @x @t 1 1 + @x @t (19) @x, @x 1 ; @t 1 @t S (tangent space)t (S) (,cotangent space)t Λ (S) ffl T (S) : @ @ ; @t 1 @t ffl T Λ (S) : 1 ( ); ( ) v i @ @t j = f i j (10) @ v = v i (11) @t i! 1! = f i i (1) i t i i j (v) = j @ v i = v j (1) @t (19) i.. [1] (1998). [] 0 (1989). [] (19). [] (198).

[] 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] http://matlab0.hwe.oita-u.ac.jp/~matsuo/robot.pdf (000).