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1 0.1 1 vector.tex ;20;23;25; ;09 19; ;22;23;1017;1127;1204; ;15; ; grd rot Green div Guss grd, div, rot Guss Green Stokes grd div rot A 46 A A A A A B

2 (1) 2, 3 ( ) (2) (1) (2) (3) ( ) , 4 5 grd 6 7 rot Green 8 div Guss Guss Stokes [5] A [5] 0.2! ( V), b zdz= 1 2 (b2 2 ) zdz =0,b =1+ 1, 1 (y =0,0 x 1) b 2 (x =0,0 y 1) b zdz= zdz= 1 xdx ydy+ (1 1y) dy =1+ 1, 1 (x 1) dx =1 1,

3 0.2 3 ( ) f f(z) dz =0 b 2 i, i =1, 2, = 1 2 f(z) dz = 1 f(z) dz =0, 2 f(z) dz = 1 f(z) dz 2 R 2 ( ) ( ) R n R 2 y 2 x 2 +1 (x 2 + y 2 +1) 2 dx 2xy (x 2 + y 2 dy. +1) 2 (1) ( 1, 2 ), (b 1,b 2 ) R 2 1 b 1 b b (2) x V (x, y) = x 2 + y V : R2 R (1) = V V (x, y) dx + (x, y) dy x y (x(s),y(s)), s [0, 1], (x(0),y(0)) = ( 1, 2 ), (x(1),y(1)) = (b 1,b 2 ) 1 ( V = x (x(s),y(s)) x (s)+ V ) y (x, y) y (s) ds 0 = 1 0 d V (x(s),y(s)) ds ds V (x, y) (x, y) =(x(s),y(s)) s 2 = V (x(s),y(s)) 1 0 = V (x(1),y(1)) V (x(0),y(0)) = V (b 1,b 2 ) V ( 1, 2 ) (2) (2 R 2 R 2 R 2 ) ( V 2 V : R 2 R x, V ) y 2 R 2 (f 1,f 2 ): R 2 R 2 1 2

4 1.1 4 ( ) 0 S = S f 1 (x, y) dx + f 2 (x, y) dy = 0 (?) 3 S f 1 (x, y) dx + f 2 (x, y) dy = S S f 2 x (x, y) f 1 (x, y) dx dy. y f : R 2 R 2 (Stokes Guss ) R 2 R 3 R n ( ) R 2 R 3 Stokes Guss ( ) ( ) ( ) ( ) ( ) R 2 (u 1,u 2 R) (u 1,u 2 ) R 2 (u 1,u 2 ) R 2 R 2 ( ) A.4.1 u u u u R 2 u + v cu ( ) R 2 4 R 2 u 1 u 2 u = u u ( ) u, v θ u v = u v cos θ f : R 2 R, g : R 2 R 2 2 R 2 2 (f,g) : R 2 R 2 R 2 (x, y) R (f(x, y),g(x, y)) R 3? (f 1,f 2 )

5 1.2 5 R 2 2 ( 2 ) ( 2 ) R 2 P P = (x, y) R 2 u(p ) = (f(x, y),g(x, y)) u : R 2 R 2 P u(p ) R 2 ( ) u(p ) P ( ) ( ) u(p ) 1 (1) u(x, y) =(x, 0) (2) u(x, y) =(y, x) (3) u(x, y) =( y, ( x) ) x (4) u(x, y) = x 2 + y 2, y x 2 + y 2 ( ) y (5) u(x, y) = x 2 + y 2, x x 2 + y 2 ( y 2 x 2 ) +1 (6) u(x, y) = (x 2 + y 2 +1) 2, 2xy (x 2 + y 2 +1) 2 ( 0.2 ) ( ) f : R 2 R, g : R 2 R, u : R 2 R 2, v : R 2 R 2, fu + gv : R 2 R 2 (fu + gv)(p )= f(p )u(p )+g(p)v(p), P R 2, u v : R 2 R R 2 (u v)(p )= u(p ) v(p ) R ) cf. ) b f(x) dx ( [, b] R 2 ( ) 6, b R 7, b 5 u v R 2, R 3 6 [5, 1.7] 7 s = tn t b = b R

6 1.2 6 ( ), b ( ) r (r r ) r u : R R 2 ( ) d u dt (t) = du1 dt (t), du 2 dt (t) 1:1 8 (R 2 ) b (, b )1 : 1 Jordn R 2 2u(t) =(t 2,t 3 ), t R, 1 [5, 1.24] t [2, 40] 1 9 d u (t) (0, 0), t [, b], (3) dt 1 (3) 1:1 1 x y 1 :1 1:1 1 1 (x y) x y (3) r 1:1 x y r 1:1. u(t) =(x(t),y(t)) (3) x (t 0 ) 0 y (t 0 ) 0 x (t 0 ) 0 t = t 0 x 0 = x(t 0 ) t = x 1 (x) 38 r u(x 1 (x)) = (x, y(x 1 (x)) 1:1 r (3) ( t x 1 r r 2 (0, 0) :1 1:1 1:1 9 [5, 1.23] 2 1:1 [5, 1.23] 1 [5, 1.3] 1:1

7 1.2 7 ( )1 : 1 3 = {(x, y) y 2 = x 2 (x +1)} ( ) u =(u 1,u 2 ): [, b] R 2 L() L() = sup [, b] n u(t i ) u(t i 1 ) (4) i=1 2 1:1 L() ( ) u : [, b] R 2 ( 1:1 ) L() = lim 0 i=1 0 b L() = d u dt (t) b dt = n u(t i ) u(t i 1 ) (5) u 1 (t)2 + u 2 (t)2 dt (6) c b t [, c], t [c, b], 1, 2 L() =L( 1 )+L( 2 ). u ( 1 ) ( 34) u(t i ) u(t i 1 )= d u dt (ξ i)(t i t i 1 ), t i 1 <ξ i <t i, (7) ξ i t i 1 <t i (4) n L() = sup d u dt (ξ i) (t i t i 1 ) i=1 ξ i = ξ,i u 1 d u dt (t) 40 L() = lim 0 i=1 n d u dt (ξ i) (t i t i 1 ), (7) (5) 41 (6) 1:1 1 2 u 1, u 2 1 :1 h : [, b] [, b] u 2 (t) =u 1 (h(t)), t [, b] h (5) b d u 2 dt (t) n n dt = lim u 2 (t i ) u 2 (t i 1 ) = lim u 1 (h(t i )) u 1 (h(t i 1 )). 0 0 i=1 h : [, b] [, b] ={t i } [, b] = {h(t i )} h [, b] ( 32) 0 0 (5) b d u 2 dt (t) b dt = d u 1 dt (t) dt i=1

8 (1) u u2 2 =1 (2) u u2 2 =1, b b R 2 f : R 2 R R 2 R 2 u : [, b] R 2 f : R ={t i } t i 1 <ξ i <t i, i =1,,n, n S,ξ = f(u(ξ i )) u(t i ) u(t i 1 ) (8) i=1 0 S = lim S,ξ S = 0 fds f f(u) ds(u) = 3 f 1:1 u : [, b] R 2 ( 1:1 ) f(u) ds(u) = b f(u(t)) d u dt (t) dt. 2 d u dt (ξ i) f(u(ξ i)) d u dt (ξ i) f 1 ds = L() ds 2 [2, 41] u =(x, y) : [, b] R 2 f : R ={t i } t i 1 <ξ i <t i, i =1,,n, n S,ξ = f(x(ξ i ),y(ξ i ))(x(t i ) x(t i 1 )) (9) i=1 10

9 S = lim S,ξ S = f(x, y)dx 0 S = f(x, y)dy S,ξ y S = f(x, y)dx 0 x S = f(x, y)dy 0 4 f 1:1 f(x, y)dx, f(x, y)dy u =(x, y) : [, b] R 2 ( 1:1 ) b f(x, y)dx = f(x(t),y(t)) dx (t) dt dt f(x, y)dy = b f(x(t),y(t)) dy (t) dt dt u() (u(b)) u(b) f =(f 1,f 2 ) f 1 (x, y) dx, f 1 (x, y) dy, f 2 (x, y) dx, f 2 (x, y) dy, ( ) u : [, b] R 2 V =(V 1,V 2 ): R 2 V(u) du = V 1 (x, y)dx + V 2 (x, y)dy ( ) 5 V 1:1 V(u) du u : [, b] R 2 ( 1:1 ) b V(u) du = V(u(t)) d u (t) dt dt

10 V(u) du V(u) ds(u).. u v u v 7 V (1) V(u) du (2) 6 (3) ( ) (1) V(x, y) =(x, 0) : (0, 0) (1, 0) (1, 0) (1, 1) (2) V(x, y) =(x, 0) : (0, 0) (0, 1) (0, 1) (1, 1) (3) V(x, y) =(y, 0) : (0, 0) (1, 0) (1, 0) (1, 1) (4) V(x, y) =(y, 0) : (0, 0) (0, 1) (0, 1) (1, 1) (5) V(x, y) =(y, x), : (1, 0) (0, 0) (6) V(x, y) =(y, x), : (x(t),y(t)) = (e t cos t, e t sin t), 0 t T T >0 (7) V(x, y) =(y, x), : (1, 0) (0, 0) (8) V(x, y) =(y, x), : (x(t),y(t)) = (e t cos t, e t sin t), 0 t T T >0 (9) V(x, y) =(x, y), : x 2 + y 2 =1 (10) V(x, y) =(y, x), : x 2 + y 2 =1 (11) V(x, y) =( y, x), : x 2 + y 2 =1 [2, 56] 12 7 V (1) M>0 V V(u) M L V(u) du ML (2) {V n } V V n (u) du = V(u) du. lim n (3) ={u i i =1,,n} Γ ɛ>0 V(u) du V(u) du <ɛ 13 Γ 12 13

11 grd grd ( ) 2 f : R 2 R 1 1 f x = f x, f y = f y 2 ( 34) f(x + t, y + tb) =f(x, y)+t(f x (x, y) + f y (x, y) b)+o(t). (10) o(t) o( ) lim =0 14 f f(x + t 0 t th, y + tk) f(x, y) t t t(f x (x, y) + f y (x, y) b) f : R 2 R 15 grd f : R 2 R 2 ( ) f f (grd f)(x, y) = (x, y), (x, y), (x, y) R, (11) x y V(x, y) = ( f x ) f (x, y), (x, y) y 2 f f V 8 ( rot V =0 ) f 2 (10) f(x + t, y + tb) =f(x, y)+t(, b) (grd f)(x, y)+o(t) (12) t (, b) =1( (, b) ) (t) 1 f(x + t, y + tb) f(x, y) t(, b) (grd f)(x, y) =t (grd f)(x, y) cos θ(, b) (13) θ(, b) (, b) (grd f)(x, y) 1 (13) (1 ) θ(, b) =0 (, b) = (grd f)(x, y) (grd f)(x, y) (grd f)(x, y) f 1 f 1 t (grd f)(x, y) 14 (t, tb) (0, 0) 15 1

12 grd f 8 D R 2 f : D R 1 1:1 u : [, b] D b (grd f)(u) du = f(u(b)) f(u()).. 5 ( 35) ( 42) b grd f(u) du = grd f(u(t)) d u b df(u(t)) (t) dt = dt = f(u(b)) f(u()). dt dt 9 D R 2 V : D R 2 D 1:1 V(u) du u(), u(b). V f V = grd f 8 9 (1) V(x, y) =(y, x), (x, y) R, (2) V(x, y) =( y, x), (x, y) R, 10 D R 2 V 2 (1) V =(V 1,V 2 ) f V = grd f (2) D 1:1 V(u) du u(), u(b). 9 P 0 D P 0 P D (P ) f(p )= V(u) du P (P ) (P ) f : D R ( ) (grd f)(p )=V(P ), P D, P =(x, y) R 2 { (P ) u : [0, 1] D (u(0) = P 0, u(1) = P ) ɛ >0 u(t), 0 t 1, u ɛ (t) = u ɛ P 0 P +(ɛ, 0) (x + t 1,y), 1 t 1+ɛ, f 1+ɛ f(x + ɛ, y) = V(u ɛ (t)) d u ɛ (t) dt, ɛ > 0. dt f(x + ɛ, y) f(x, y) = 1+ɛ 1 0 V(x + t 1,y) (1, 0) dt = 1+ɛ 1 V 1 (x + t 1,y) dt. ( 43) f x f x (P )=V 1(P ) f y (P )=V 2(P ) 10 ( ) cf. 15

13 rot Green 10 V (rot) :1 u ( 1:1 ) v P = u(t 0 ) (tngent vector) v = c d u dt (t 0) c 0 v 0 (norml vector) v d u dt (t 0)=0 1 1 t u u ( ) t = 1 u u (14) 2 ( ) u, v u(s) =v(t) s, t d u v (s) =cd (t) c>0 dt dt c<0 2 1:1 u, v u() =v() u(b) =v(b) u() =v(b) u(b) =v() (oriented) u (v 1,v 2 )= d u dt 90 (v 2, v 1 ) n u =(u 1,u 2 ) u n = 1 u (u 2, u 1 ) (15) 11 } (1) = {(x, y) x2 2 + y2 b 2 =1 (2) = { (x, y) y = x 2} } (3) = {(x, y) x2 2 y2 b 2 = V 1:1 V(u) du u : [, b] R 2 ( 1 :1 ) V(u) du = V(u) t(t) ds(u) t u

14 (14) 3 t(t) t u ( ) 1:1 u 12 1:1 1 : (Jordn ) R 2 Jordn ( ) c D R 2 ( ) D D ( ) ( ) 2 2 D D ( ) D R 2 D Jordn ( 1.2.1) ( ) D ( A.5) 17 ( 1.4.1) D D D D Jordn ( 52 ) D D Green ( A.3.2) ( ) D R 2 1:1 f : D R 1 f (x, y) dxdy = f(x, y) dx, y D D D f (x, y) dxdy = f(x, y) dy, x D ( R 2 Jordn ) R 2 Jordn

15 D b φ 1 (x) φ 2 (x), x b,, b R φ i : [, b] R, i =1, 2, D = {(x, y) R 2 φ 1 (x) y φ 2 (x), x b} (16) x y c d ψ 1 (y) ψ 2 (y), c y d, c, d R ψ i : [c, d] R, i =1, 2, D = {(y, x) R 2 ψ 1 (y) x ψ 2 (y), c y d} (17) D D D [2, 102]. D (16), (17) D u = (u 1,u 2 ): [0, 4] D +(b )t, 0 t<1, b, 1 t<2, u 1 (t) = b +( b)(t 2), 2 t<3,, 3 t 4, φ 1 ( +(b )t), 0 t<1, φ 1 (b)+(φ 2 (b) φ 1 (b))(t 1), 1 t<2, u 2 (t) = φ 2 (b +( b)(t 2)), 2 t<3, φ 2 ()+(φ 1 () φ 2 ())(t 3), 3 t 4, f : D R 1 47 D f (x, y) dxdy = y b f(x, φ 2 (x)) dx 4 f(x, y) dx = f(u(t)) du 1 (t) dt D 0 dt 1 =(b ) f( +(b )t, φ 1 ( +(b )t)) dt +( b) = b 0 f(x, φ 1 (x)) dx + b f(x, φ 2 (x)) dx. = b 3 2 b f(x, φ 2 (x)) dx + 1 f (x, y) dxdy = f(x, y) dx D y D D f (x, y) dxdy = f(x, y) dy x D f(x, φ 1 (x)) dx. f(b +( b)(t 2),φ 2 (b +( b)t)) dt b f(x, φ 1 (x)) dx. D 47 dy 2 D

16 rot plnimeter V =(V 1,V 2 ): R 2 R 2 D R 2 ( V2 V(u) du = x (x, y) V ) 1 (x, y) dxdy (18) y D D ( ) V =(V 1,V 2 ) (rottion) rot V : R 2 R rot V = V 2 x V 1 y (19) (18) 14 D R 2 1:1 V : D R 2 1 rot V(x, y) dxdy = V(u) du. D D (1) V(x, y) = 1 2 ( y, x) rot V (2) V ( 1 ) rot V =0 14 V 1 (x, y) = 1 2 y, V 2(x, y) = 1 2 x, V =(V 1,V 2 ) rot V(x, y) =1 dxdy = 1 xdy)= D 2 D( ydx+ 1 xdy 1 ydx (20) 2 D 2 D A.3.2 dxdy D S(D) D D D ( ) 14 ( A.3.2) (20) 1 xdy 1 ydx= 0 (21) (1) 1 (1, 0) 1 (20) (20) (2) 1 (1, 0) ( 1, 0) (1, 0) 1:1 (21) V 2 =

17 (20), (21) plnimeter Amsler [2, 41] 2 Jordn(1 : 1), 0 l>0 l A =(x, y), A 0 =(x 0,y 0 ), 0 Plnimeter OA 0 l A 0 A A 0 A 0 A A 0 k<l A 0 A K A 0 A x θ x = x 0 + l cos θ, y = y 0 + l sin θ. (22) A A 0 θ(s), x 0 (s), y 0 (s), s [0, 1], s 0 1 A 1 A 0 0 ( 1:1 ) 1 (20), (21) S S = (22) 0 (x(s)y (s) y(s)x (s)) ds, 1 0 (x 0 (s)y 0 (s) y 0(s)x 0 (s)) ds =0. S = l θ (s) ds + l (x 0 (s) cos θ(s)+y 0 (s) sin θ(s))θ (s) ds + l (cos θ(s) y 0 (s) sin θ(s) x 0 (s)) ds. l (x 0 (s) cos θ(s)+y 0 (s) sin θ(s))θ (s) ds 2 0 =(x 0 (s) sin θ(s) y 0 (s) cos θ(s)) 1 0 l 1 (x 0 2 (s) sin θ(s) y 0 (s) cos θ(s)) ds. A 0 A θ(0) = θ(1) S = l (cos θ(s) y 0 (s) sin θ(s) x 0 (s)) ds. A 0 A K K(s) =(x 0 (s)+kcos θ(s),y 0 (s)+ksin θ(s)) A 0 A A 0 A R 1 R = ( sin θ(s), cos θ(s)) dk(s) R = 1 0 (y 0 (s) cos θ(s) x 0 (s) sin θ(s)+kθ (s)) ds (y 0 (s) cos θ(s) x 0 (s) sin θ(s)) ds = S/l θ(0) = θ(1) plnimeter S ( ) ( 1.4.2) D R 2 1 : 1 V : D R 2 1 (1) V =(V 1,V 2 ) f V = grd f

18 (2) rot V = 0 D 17 ( ). rot V = 0 D 14 D 1:1 V(u) du =0. (23) 10 (23) V 15 D ( 18 V(x, y) = y ) x 2 + y 2, x x 2 + y 2 D = R 2 \{(0, 0)} V D rot V = π 0 ( sin θ, cos θ) ( sin θ, cos θ) dθ =2π 16 D = R 2 \{(0, 0)} 1 V : D R 2 D rot V =0 Π D (0, 0) V(u) du =Π V = grd f f D Π= , 2 (1 : 1 ) u 1, u 2 ɛ>0 D ɛ = {(x, y) R 2 x 2 + y 2 ɛ} 1, 2 1, 2 D 1, D 2 ɛ D ɛ D 1, D ɛ D 1 1, P ɛ, g ɛ : R 2 R 1 g ɛ (P )= 0, P < 1 2 ɛ, 1 2 ɛ P ɛ 0 1 g V (0, 0) 0 (0, 0) R D 1, D 2 (0, 0) P D ɛ rot(g V)(P ) = (rot V)(P )=0. 1, 2 g =1 10 V(u) du = i g(u)v(u) du = i rot(g V)(u)du = D i rot(g V)(u)du, D ɛ i =1, 2. Π=0 V(u) du 10 ( 10 ) V = grd f f R 2 \{(0, 0)} V = grd f f (0, 0) 2 P, Q 2 1 (P Q), 2 (Q P ) V(u) du = f(q) f(p )= V(u) du 1 2

19 V(u) du = div Guss ( ) 1 ( ) div grd, rot ( ) 3 V =(V 1,V 2 ): R 2 R 2 (div V)(x, y) = V 1 x (x, y)+ V 2 (x, y) y div div V : R 2 R V (divergence) 19 (1) V(x, y) =( y, x) (2) V(x, y) =(x, ( y) ) x (3) V(x, y) =, y ((x, y) (0, 0)) x 2 +y 2 x 2 +y 2 (4) V(x, y) =(x log x 2 + y 2,ylog x 2 + y 2 ) ((x, y) (0, 0)) 20 (1) f, g 1 V 1 () grd(f g) = (grd f) g + f grd g (b) div(f V) = (grd f) V + f div V (c) rot(f V) = (grd f) V + f rot V 2 U =(U 1,U 2 ), V =(V 1,V 2 ) U V = U 1 V 2 U 2 V 1 (2) f 2 V 2 () rot grd f =0 (b) div grd f = 2 f x f y 2 ( f (2 ) ) Guss V ( 1.4.1) u : [, b] R 2 1 :1 n(u) ( 90 ) V =(V 1,V 2 ): R 2 1:1 b V(u) n(t) ds(u) = V(u(t)) n(t) d u dt dt ( )

20 n :1 1 : V (1) (2) (±1, ±1) 4 V nds D R 2 1:1( 1:1 ) V : D R 2 1 div V(x, y) dxdy = V(u) n(t)ds(u). D n ( ). V ( V 2,V 1 ) 14 rot( V 2,V 1 ) = div V D div V(x, y) dxdy = D D ( V 2,V 1 )(u) du. 11 (14) (15) 1 = ( V 2,V 1 )(u) D u (u 1,u 2 )(u) ds(u) = V(u) n(t) ds(u) D Guss (0, 0) ɛ Guss 1.6 grd, rot, div V tds, V nds (V(P ) ) V = {V(P )} V R V =(V 1,V 2 ): R 2 R 2 V R 2 ( A.4.1) 2 V V : R 2 V

21 V (V R 2 ) R 2 V : R 2 V (x, y) V(x, y) (x, y) x 1 1 x 2 θ R θ 2 ( ) ( ) ( )( x x(x,y ) cos θ sin θ = y y(x,y = ) sin θ cos θ x y ) ( x y ) ( = x (x, y) y (x, y) ) ( = cos θ sin θ sin θ cos θ )( x y ) R 2 (x, y) (x,y ) R 2 20 ( ) R 2 R ( ) f : R 2 R f : R 2 R f(x,y )=f(x(x,y ),y(x,y )), (x,y ) R 2. (24) 36 1 f : R 2 R f x (x,y ) = cos θ f x (x(x,y ),y(x,y )) + sin θ f y (x(x,y ),y(x,y )), f y (x,y )= sin θ f x (x(x,y ),y(x,y )) + cos θ f y (x(x,y ),y(x,y )). (25) x = x(x,y ), y = y(x,y ) x,y x, y f x (x(x,y ),y(x,y )) = cos θ f x (x,y ) sin θ f y (x,y ), f y (x(x,y ),y(x,y )) = sin θ f x (x,y ) + cos θ f (26) y (x,y ) θ 90 (x, y) =(0, 1) (1, 0) = (x,y )

22 R 2 2 V V : R 2 V V e x, e y P =(x, y) R 2 V V(P )=V 1 (x, y)e x + V 2 (x, y)e y V 1 : R 2 R, V 2 : R 2 R V e x, e y e x e x = cos θ, e x e y = sin θ, e y e x = sin θ, e y e y = cos θ. Ṽ 1 (x,y )e x + Ṽ2(x,y )e y = Ṽ(x,y ) = V(x(x,y ),y(x,y )) = V 1 (x(x,y ),y(x,y ))e x + V 2 (x(x,y ),y(x,y ))e y, (x,y ) R 2. Ṽ 1 (x,y ) = cos θv 1 (x(x,y ),y(x,y )) + sin θv 2 (x(x,y ),y(x,y )), Ṽ 2 (x,y )= sin θv 1 (x(x,y ),y(x,y )) + cos θv 2 (x(x,y ),y(x,y )), (x,y ) R 2. (27) grd, rot, div f : R 2 R grd f grd f 1 (x,y )e x + grd f 2 (x,y )e y = grd f(x,y ) = grd f(x(x,y ),y(x,y )) = f x (x(x,y ),y(x,y ))e x + f y (x(x,y ),y(x,y ))e y, (x,y ) R 2, (27) grd f 1 (x,y ) = cos θ f x (x(x,y ),y(x,y )) + sin θ f y (x(x,y ),y(x,y )), grd f 2 (x,y )= sin θ f x (x(x,y ),y(x,y )) + cos θ f y (x(x,y ),y(x,y )), (x,y ) R 2. x(x,y ) y(x,y ) x = cos θ, x = sin θ, x(x,y ) y(x,y ) y = sin θ, y = cos θ, grd f 1 (x,y )= f(x(x,y ),y(x,y )) x, grd f 2 (x,y )= f(x(x,y ),y(x,y )) y. x = x(x,y ) x,y 21 grd f grd f(x(x,y ),y(x,y )) = grd f(x,y )= f(x(x,y ),y(x,y )) x e x + f(x(x,y ),y(x,y )) y e y. 21 g(x,y )=f(x(x,y ),y(x,y )) 2 g g 1 2

23 V : R 2 R 2 (27) div V(x(x,y ),y(x,y )) = V 1 x (x(x,y ),y(x,y )) + V 2 y (x(x,y ),y(x,y )) = x (x, y) V 1 (x(x,y ),y(x,y )) x x + y (x, y) V 1 (x(x,y ),y(x,y )) x y + x (x, y) V 2 (x(x,y ),y(x,y )) y x + y (x, y) V 2 (x(x,y ),y(x,y )) y y = cos θ V 1(x(x,y ),y(x,y )) x sin θ V 1(x(x,y ),y(x,y )) y + sin θ V 2(x(x,y ),y(x,y )) x + cos θ V 2(x(x,y ),y(x,y )) y = Ṽ1 x (x,y )+ Ṽ2 y (x,y ) div V (x,y ) 24 rot V u(t) =(x(t),y(t)) ũ(t) =(x (t),y (t)) = (cos θx(t) + sin θy(t), sin θx(t) + cos θy(t)) (x (t)) 2 +(y (t)) 2 = x(t) 2 + y(t) 2 u(t i ) u(t i 1 ) = ũ(t i ) ũ(t i 1 ). f (24) (8) S,ξ n n S,ξ = f(u(ξ i )) u(t i ) u(t i 1 ) = f(ũ(ξ i )) ũ(t i ) ũ(t i 1 ). i=1 i=1 0 f(u) ds(u) = f(ũ) ds(ũ). (9) S,ξ f V = V 1 e x + V 2 e y (27) n V 1 (x(ξ i ),y(ξ i ))(x(t i ) x(t i 1 )) i=1 = n (cos θṽ1(x (x(ξ i ),y(ξ i )),y (x(ξ i ),y(ξ i ))) sin θṽ2(x (x(ξ i ),y(ξ i )),y (x(ξ i ),y(ξ i )))) i=1 (cos θ(x (t i ) x (t i 1 )) sin θ(y (t i ) y (t i 1 ))), n V 2 (x(ξ i ),y(ξ i ))(y(t i ) y(t i 1 )) i=1 = n (sin θṽ1(x (x(ξ i ),y(ξ i )),y (x(ξ i ),y(ξ i ))) + cos θṽ2(x (x(ξ i ),y(ξ i )),y (x(ξ i ),y(ξ i )))) i=1 (sin θ(x (t i ) x (t i 1 )) + cos θ(y (t i ) y (t i 1 ))). V 1 (x, y)dx = (cos θṽ1(x,y ) sin θṽ2(x,y )) cos θdx (cos θṽ1(x,y ) sin θṽ2(x,y )) sin θdy.

24 V 2 (x, y)dy = (sin θṽ1(x,y ) + cos θṽ2(x,y )) sin θdx + V(u) du = (sin θṽ1(x,y ) + cos θṽ2(x,y )) cos θdy. V 1 (x, y)dx + V 2 (x, y)dy = Ṽ 1 (x,y )dx + Ṽ 2 (x,y )dy = Ṽ(ũ) dũ 25 V(u) n(t) ds(u) x = x(r, θ) =r cos θ, y = y(r, θ) =r cos θ, R 2 (x, y) (r, θ) R + R/(2πR) 36 1 f : R 2 R f(rcos θ, r sin θ) r f(rcos θ, r sin θ) θ = cos θ f x f (r cos θ, r sin θ) + sin θ (r cos θ, r sin θ), y = r sin θ f f (28) (r cos θ, r sin θ)+rcos θ x y (r cos θ, r sin θ). r, θ 22 x, y f x f f(rcos θ, r sin θ) (r cos θ, r sin θ) = sin θ + cos θ y r r (r cos θ, r sin θ) = cos θ f(r cos θ, r sin θ) r sin θ r f(rcos θ, r sin θ), θ f(rcos θ, r sin θ). θ (29) (28) x y e r e θ ( ) x e r (x, y) = x2 + y, y, ( 2 x2 + y 2 ) (30) y e θ (x, y) = x2 + y, x, 2 x2 + y 2 e r = e θ =1,e r e θ =0 V V(r cos θ, r sin θ) =V r (r, θ)e r (r cos θ, r sin θ)+v θ (r, θ)e θ (r cos θ, r sin θ) (29) V r (r, θ) =(V e r )(r cos θ, r sin θ), V θ (r, θ) =(V e θ )(r cos θ, r sin θ), V r x (r, θ) = cos θ V r sin θ V r (r, θ) (r, θ), r r θ V r y (r, θ) = sin θ V r θ V r (r, θ)+cos (r, θ). r r θ 22 f(r, θ) =f(r cos θ, r sin θ) 2 f f 1 2

25 x = r cos θ, y = r sin θ r(x, y), θ(x, y) r, θ V r r = r(x, y), θ = θ(x, y) x, y x, y x = r cos θ, y = r sin θ ( ) V θ (30) div e r (x, y) = 1 x2 + y 2, div e θ(x, y) =0, (x, y) (0, 0). (x, y) (0, 0) div V = grd V r e r + V r div e r + grd V θ e θ + V θ div e θ div V(r cos θ, r sin θ) = V r r (r, θ)+1 r V r(r, θ)+ 1 r V θ (r, θ)). θ r, θ, e r, e θ 26 grd f, rot V grd, div, rot (3 ) 3 (u 1,u 2,u 3 R) (u 1,u 2,u 3 ) R 3 ( ) R 3 (3 ) 3 u =(u 1,u 2,u 3 ) R 3, v =(v 1,v 2,v 3 ) R 3 u v R 3 u v =(u 2 v 3 u 3 v 2,u 3 v 1 u 1 v 3,u 1 v 2 u 2 v 1 ) u v = v u u u =0 19 u v R 3 u v ( u R 3 v R 3 3 ) ( u, v ) u, v, u v (1 2 3 ) u v. u, v 3 u, v 1 2 (u 1,u 2 ) (v 1,v 2 ) v 2 /v 1 >u 2 /u 1 3 u v 2 + u v 2 = u 2 v 2, u v θ u v = u v 1 cos 2 θ = u v sin θ.

26 u =(1, 0, 0) v {(1, 0, 0), (0, 1, 0), (0, 0, 1)} u v 19 R 3 R 3 R 3 u (v w) =v (w u) =w (u v) = det u 1 v 1 w 1 u 2 v 2 w 2 (31) u 3 v 3 w 3 20 (31) u, v, w 3. v w u θ S S = v w v w u π/2 θ S u sin θ = v w u cos( π 2 θ) = u (v w). 28 u, v, w 20 u (v w) =(u w)v (u v)w grd, div, rot V V : R 2 V V V R 2 ( A.4.1) R 2 V : R 2 R 2 (3 ) (3 ) 3 V V : R 3 V V R 3 ( A.4.1) R 3 V : R 3 R (R) grd, div, rot 1.3, 1.4.4, f : R 3 R V : R 3 R 3 ( f grd f = x, f y, f ), z div V = V 1 x + V 2 y + V 3 z, (32) ( V3 rot V = y V 2 z, V 1 z V 3 x, V 2 x V ) 1. y 3 grd 3 rot

27 (1) f, g 1 U, V 1 rot (x, 0, 0) ( y, x, 0) () grd(f g) = (grd f) g + f grd g (b) div(f V) = (grd f) V + f div V (c) rot(f V) = (grd f) V + f rot V (d) div(u V) = (rot U) V U rot V (e) rot(u V) =(V grd)u + U (div V) (U grd)v V (div U) (2) f 2 V 2 () rot grd f =0 (b) div grd f = 2 f x f y f z 2 ( f (3 ) ) (c) div rot V = 0 (d) rot rot V = grd(div V) V ( V =( V x, V y, V z ) 23 δ ij, i, j =1, 2, 3, i = j 1 0 (Kronecker ). E ijk, i, j, k =1, 2, 3, ijk ( 2 ) 0 (rot V) i = E ijk j V k (Einstein ) j = E ijk = E jik x j E ijk E ilm = δ jl δ km δ jm δ kl Einstein ( ) R 2 1:1 ( ) R 2 R 2 ( ) 23

28 S R 3 (surfce) P S ɛ = ɛ P > 0 2 U = U P R 2 ϕ = ϕ P : U R 3 25 (1) ϕ ϕ(u) {Q S d(q, P ) <ɛ} ϕ(u) S d R 3 (2) ϕ (3) ϕ (4) ϕ = ϕ 1 ϕ 2 Dϕ ( A.4.2) U 2 ϕ 3 ϕ 1 s (s, t) ϕ 1 (s, t) t (Dϕ)(s, t) = ϕ 2 s (s, t) ϕ 2 (s, t) t ϕ 3 s (s, t) ϕ 3 (s, t) t ϕ, (s, t) U, 2 S P U R 2 ϕ 1:1 (3) x, y, z ϕ(s, t) =(x(s, t),y(s, t),z(s, t)) = (s 2,t 2,st) ϕ : R 2 R 3 2s 0 (Dϕ)(s, t) = 0 2t t s rnk 2 (s, t) =(0, 0) x = X + Z 2, y = Z X 2, z = Y 2, (X(s, t),y(s, t),z(s, t)) = ( s 2 t 2 2 Z 2 = X 2 + Y 2, 2 st, s 2 + t 2 2 ) (0, 0, 0) Z ( ) 45 ±(s, t) 2:1 (s, t) =(0, 0) 1: TEX ϕ ϕ 26 (1) [5, 2.1 1] (0, 0) 1:1 Dϕ rnk (2) 2001 d X d Y X d = 2d +1 Y d =2d d =2d 1 R 3 (d =2,d =3) rnk 1 generic cse Arnold 1 Whittney umbrell ϕ(s, t) =(s 2,t,st) generic [9]

29 S ϕ P S P S ϕ i (U i ) i I {ϕ i : U i R 3 i I} 27 s>0 t>0 1:1 Dϕ rnk 2 31 S 2 = {(x, y, z) R 3 x 2 + y 2 + z 2 =1} ϕ(s, t) = (cos s sin t, sin s sin t, cos t) U = {(s, t) R 2 0 <s<2π, 0 <t<π} U 1:1 U U = S 2 \{(x, y, z) R 3 x 0, y =0} ϕ 2 (s, t) = (cos s sin t, cos t, sin s sin t) U 2 = {(s, t) R 2 π<s<π, 0 <t<π} 32 {(x, y, z) R 3 x 2 + y 2 + z 2 =1} {(x, y, z) R 3 x 2 + y 2 =1, z < 1} (Möbius) {((2 + t cos s 2 ) sin s, (2 + t cos s 2 ) cos s, t sin s 2 ) s R, t < 1} [5, 2.12] s R U 1 = {(s, t) R 2 π<s<π, t < 1} ϕ 1 U 2 = {(s, t) R π<s< 5 2 π, t < 1} ϕ 2 1:1 {ϕ 1,ϕ 2 } ϕ(0,t) ϕ(2π, 0) (ϕ(0,t) = ϕ(2π, 0)) ϕ 1 (0,t)= ϕ 2 (2π, t) B 28 {(x, y, z) R 3 x 2 + y 2 =1, z 1} ( z =1) S S P (P 1 :1 Dϕ rnk 2) ϕ(0, 0) = P ϕ U {(s, t) s 2 + t 2 < 1} {(s, t) s 2 + t 2 < 1, t 0} ([1, 328B]) P S S 27 R 3 (R 3 ) (s, t) [5, 2.1] (locl coordinte) (locl coordinte system) R 2 R 3 28 [1, 90A, 328B]

30 ( ) 1 0, b, c, d S = {(x, y, z)r 3 x + by + cz = d} S 34 S = {(x, y, z)r 3 x + by + cz = d} (1) P 0 =(x 0,y 0,z 0 ) S 1 S (x, y, z) S (x x 0 )+b(y y 0 )+c(z z 0 )=0 (2) S P P 0 P v (, b, c) v λ R v = λ(, b, c) 1 S (3) 1 P 0 2 ( ) 2 2 S P 0 p, q S = {P 0 + αp + βq (α, β) R 2 } p, q S = {P 0 + αp + βq (α, β) R 2 } P 0 p, q, b, c S = {(x, y, z)r 3 (x x 0 )+b(y y 0 )+c(z z 0 )=0} S R 3 P S ϕ P S P = ϕ(s 0,t 0 ) ( 2.2.1) Dϕ(s 0,t 0 ) rnk Dϕ(s 0,t 0 ) 2 ϕ s (s 0,t 0 ) ϕ t (s 0,t 0 ) ( 49) P ( 34) P + T P (S) S P 29 T P (S) P ( 34) 1 1 v v 2 21 S P ϕ : U R 3 P = ϕ(s 0,t 0 ) P S ϕ ± s ϕ t ϕ s ϕ (s 0,t 0 ) t ϕ. s ϕ t ϕ s ϕ t Normlizble 0 Dϕ rnk ϕ 1:1 1 ( 1.2.1) 29 P P [5, 2.1(c)] T P (S) 0 OP

31 S R 3 2 ϕ : U R 3, ϕ : U R 3 (U, U R 2 ) W = {(s, t) U ϕ(s, t) ϕ (U )}, W = {(s, t) U ϕ (s, t) ϕ(u)} 30 ψ : W W (1) ϕ ψ(s, t) =ϕ(s, t), (s, t) W (2) ψ ψ 1 : W W ψ, ψ 1 ψ (diffeomorphism). (s, t) W ϕ(s, t) =ϕ (u, v) (u, v) W ϕ (u, v) 1 ψ(s, t) =(u, v) ψ = ϕ 1 ϕ ψ : W W ϕ ψ = ϕ (u, v) W ϕ(s, t) =ϕ (u, v) (s, t) W 1 ψ 1 (u, v) =(s, t) ψ 1 ψ ψ ψ 1 (s, t) W (u, v) =ψ(s, t) W Dϕ (u, v) rnk W 2 ϕ (u, v) 2 R 2 V φ : V R 2 x, y 2 φ (ϕ 1,ϕ 2 )(x, y) =(x, y), (x, y) V ϕ ψ = ϕ (ϕ 1,ϕ 2) ψ =(ϕ 1,ϕ 2 ) φ (ϕ 1,ϕ 2 )=ψ ψ ( 2.2.2) ( 2.2.5) ( 1.4.1) ( ) S R 3 S R 3 n : S R 3 P S n(p ) n S (orientble) ϕ : U S S 21 S ϕ n(ϕ(s, t)) = ± s ϕ t ϕ s ϕ (s, t), (s, t) U, (33) t ϕ U ± (s, t) ϕ ( ) S (R 3 ) S ( 50) ( 32) S = ( ) [5, 2.1(d)] P 2.2 ϕ ϕ (ϕ(s, t)) (s, t) 2.2(b) (2.2) s s s ds [5, 1] [5, 2] R 2

32 S R 3 S c = R 3 \ S S 24 (R 3 ). S R 3 P S P n (2 1 ) ɛ>0 P ɛn P + ɛn S P 31 P ± ɛn S n(p ) P n : S R 3 R 3 Ω Ω Ω Ω Ω 2 ( Ω ) Ω ( 12 ) 1:1 36 ( 32) (33) t =0( ) n(ϕ(s, 0)) = ±( sin s sin s 2, cos s sin s 2, cos s 2 ). ϕ(0, 0) = ϕ(2π, 0) ( 32 ) n(ϕ(0, 0)) = n(ϕ(2π,0)) (0, 0, ±1) = (0, 0, 1) u(t i ) u(t i 1 ) ( 1.2.2, 1.2.3, 1.2.4) {ϕ(s i,t i )} ( ) [2, 97 ] 32 A.3.2 ϕ R 3 U R 2 U 1 [2, 97] ϕ : U S S R 3 (s 0,t 0 ) U S P 0 = ϕ(s 0,t 0 ) T S (P 0 )(P 0 S P 0 ) 31 1:1 S 32 Schwrz 1 1 2

33 Dϕ(s 0,t 0 ) ( ) 2 p 0 = ϕ s (s 0,t 0 ) q 0 = ϕ t (s 0,t 0 ) T S (P 0 ) (X, Y, Z) =αp 0 + βq 0, (α, β) R 2 R 3 P =(x, y, z) T S (P 0 ) Q =(X, Y, Z) λ R; Q = P + λn Q T S (P 0 ) 2 1 (X, Y, Z) = p 0 2 q 0 2 (p 0 q 0 ) 2 ) (( q 0 2 (x, y, z) p 0 p 0 q 0 (x, y, z) q 0 ) p 0 +( p 0 2 (x, y, z) q 0 p 0 q 0 (x, y, z) p 0 ) q 0 (34) P S (x, y, z) =ϕ(s, t) U 1 ɛ>0 1 P 0 = ϕ(s 0,t 0 ) ϕ 1 ( 34) 2 (s, t), (s + δs, t + δt) ϕ(s + δs, t + δt) ϕ(s, t) =p 0 δs + q 0 δt + O(ɛ 2 ). (35) O(ɛ 2 ) U (s 0,t 0,δs,δt M O(ɛ 2 ) Mɛ 2 ) 2 P Q δq ( ) (35), (34) δq = δs p 0 + δt q 0 + O(ɛ 2 ). 48 [2, 92] s δs t δt δs p 0 δt q 0 O(ɛ 2 ) p 0 q 0 + O(ɛ) R 2 R 2 ( 48) S U p q (s, t) ds dt = ϕ s ϕ t (s, t) ds dt U S R 3 n E S E 1 f : E R 1 V : E R 3 ds, fds, V nds E ϕ : U E (ϕ(u) =E) ds = ϕ E U s ϕ t (s, t) ds dt fds= f(ϕ(s, t)) ϕ E U s ϕ t (s, t) ds dt V n ds = V(ϕ(s, t)) ( ϕ s ϕ )(s, t) ds dt. t E V n ds = V(ϕ(s, t)) n(ϕ(s, t)) ϕ E U s ϕ t (s, t) ds dt ( ) ϕ ϕ = V(ϕ(s, t)) (s, t) (s, t) ds dt. s t (33) U U U E E E (36)

34 E = ϕ(u) U U i, i =1,,k, U i 1 ϕ i k ds = ϕ i s ϕ i t (s, t) ds dt 33 fds, E ds, fds, E E E E E i=1 U i V nds V nds E f V E S E = S 25(, b ) 3 ( ) 3 2 (, b) α β A = 3 c, d (c, d) =(, b) A γ δ. c d = (det A)( b). c d =(α + γb) (β + δb) =(αδ βγ)( b) = (det A)( b). =0,b = b 26. ϕ : U R 3 S φ = ϕ 1 ϕ : U U 22 well-defined U U ϕ = ϕ φ 48 g : R 3 R g(ϕ(s, t)) ds dt = U g(ϕ (u, v)) det Dφ(u, v) du dv. U (37) Dφ φ =(φ 1,φ 2 ) φ 1 Dφ = u ψ 2 u 35 ( ϕ u, ϕ v ) (u, v) = φ 1 v ψ 2 v. ( ) ϕ s, ϕ (φ(u, v)) Dφ(u, v). t 25 ϕ u ϕ (u, v) = det(dφ)(u, v) v ( ϕ s ϕ ) (φ(u, v)). t f : R 3 R (37) g(x) =f(x) ϕ s ϕ t (ϕ 1 (x)), x S, f(ϕ(s, t)) ϕ s ϕ t U (s, t) ds dt = f(ϕ (u, v)) ϕ u ϕ v (u, v) du dv U 33

35 f 1 V : R 3 R 3 (37) ( ϕ g(x) =V(x) s ϕ ) (ϕ 1 (x)), x S, t ϕ, ϕ det Dφ(u, v) > 0 ( ϕ V n ds = V(ϕ(s, t)) E U s ϕ ) (s, t) ds dt ( t ϕ = V(ϕ (u, v)) U s ϕ ) (φ(u, v))) det Dφ(u, v) du dv t ( ) ϕ = V(ϕ (u, v)) U u ϕ (u, v) du dv, v 37 E xy (s, t) =(x, y), ϕ(s, t) = (s, t, 0) ϕ(u) =E U = {(s, t) R 2 (s, t, 0) E} E E E ds = fds= V n ds = ϕ s ϕ =(0, 0, 1) t {(s,t) (s,t,0) E} {(s,t) (s,t,0) E} ds dt {(s,t) (s,t,0) E} f(s, t, 0) ds dt V(s, t, 0) n ds dt. E E f(x, y, 0), V(x, y, 0) n A.3.2 n =(0, 0, 1) ϕ(s, t) =(t, s, 0) n E yz x = V n ds = V(, s, t) n ds dt E {(s,t) (,s,t) E} n (±1, 0, 0) ( ) 38 ( 31) 2.3 Guss Green Guss R 2 Guss 18 R 3 Guss

36 (Guss ) Ω R 3 S = Ω V : Ω R 2 1 Ω div V(x, y, z) dx dy dz = S V nds. S n Ω ( A ) (18) 13 Ω x, y, z Ω 1 <b 1, 2 <b 2, 3 Ω={(x, y, z) R 3 1 <x<b 1, 2 <y<b 2, 3 <z<f(x, y)}. f {(x, y) R 2 1 <x<b 1, 2 <y<b 2 } ( 3, ). V =(V 1,V 2,V 3 ) 39 Ω ( 47 3 ) b1 b2 f(x,y) ( V1 div V(x, y, z) dx dy dz = dx dy Ω x + V 2 y + V ) 3 (x, y, z) dz. (38) z f(x,y) f(x,y) V 1 f V 1 dz = (x, y, z) dz + x 3 3 x x (x, y) V 1(x, y, f(x, y)). V 2 (38) 42 div V(x, y, z) dx dy dz Ω = + b2 2 b1 1 b1 1 dy dx dx f(1,y) 3 f(x,2) 3 b2 2 dz V 1 ( 1,y,z)+ dz V 2 (x, 2,z)+ ( dy b2 2 b1 dy f(b1,y) 3 f(x,b2) dz V 1 (b 1,y,z) b1 b2 dx dz V 2 (x, b 2,z) dx dy V 3 (x, y, 3 ) V 1 (x, y, f(x, y)) f x (x, y) V 2(x, y, f(x, y)) f ) y (x, y)+v 3(x, y, f(x, y)) Ω Ω 6 Ω = x, Ω x,+ Ω y, Ω y,+ Ω z, Ω z,+ Ω. n x, Ω={( 1,y,z) R 3 2 <y<b 2, 3 <z<f( 1,y)}, n =( 1, 0, 0), x,+ Ω={(b 1,y,z) R 3 2 <y<b 2, 3 <z<f(b 1,y)}, n =(1, 0, 0), y, Ω={(x, 2,z) R 3 1 <x<b 1, 3 <z<f(x, 2 )}, n =(0, 1, 0), y,+ Ω={(x, b 2,z) R 3 1 <x<b 1, 3 <z<f(x, b 2 )}, n =(0, 1, 0), z, Ω={(x, y, 3 ) R 3 1 <x<b 1, 2 <y<b 2 }, n =(0, 0 0, 1), z,+ Ω={(x, y, f(x, y)) R 3 x, f 1 y, 1 A 1 <x<b 1, 2 <y<b 2 }, n = ( f x, f ). y, 1 34 [5, 2.3(b)] (2.6) (2.7) f(x, y) (39)

37 ( z,+ Ω n z,+ Ω 21 s = x, t = y, ϕ(x, y) =(x, y, f(x, y)) ) 37 (39) V nds, V nds, x, Ω x,+ω V nds, V nds, V nds, (39) y, Ω y,+ω z, Ω (36) ϕ(s, t) =(s, t, f(s, t)) z,+ Ω ( ϕ s ϕ ) ( (s, t) = f t x, f ) y, 1 (s, t) z,+ω V n ds = {(s,t) R 2 1<s<b 1, 2<t<b 2} (39) (39) Ω div V(x, y, z) dx dy dz = ( V(s, t) f x, f ) y, 1 (s, t) ds dt V nds Ω V nds z,+ω 40 (1) Ω={(x, y, z) R 3 x 2 + y 2 + z 2 < 1} S = Ω ={(x, y, z) R 3 x 2 + y 2 + z 2 =1} V : R 3 R 3 V(x, y, z) =(x, y, z) div V(x, y, z) dx dy dz V nds Ω S 27 (2) V : R 3 R 3 1 V(x, y, z) = (x 2 + y 2 + z 2 (x, y, z) α>0 ) α S V nds S S 27 1 S ɛ>0 S ɛ S S ɛ Ω 27 S ɛ Green 28 (Green ) Ω R 3 S = Ω f : Ω R, g : Ω R, 2 (g f f g) dx dy dz = (g grd f f grd g) nds Ω 3 ( 29) = 2 f x f y f z 2. div(f V) = grd V + f div V, div grd f = f ( 29) ( 27) (g grd f f grd g) nds = div(g grd f f grd g)(x, y, z) dx dy dz = (g f f g) dx dy dz. S Ω S Ω

38 Stokes Guss Stokes ( 14) R 3 Guss Stokes R 3 ( 32) R 2 Guss Stokes R Stokes R 3 Stokes ( 1.2.1) R 2 R 3 [, b] R 3 ( )1 : 1 ( 1 ϕ (t) (0, 0, 0) ) ( 1.2.2) ( 1.2.3) ( ) ( 1.2.5) R 2 ( 1.4.1) v = c d u dt (t 0) c 0 v d u dt (t 0)=0 R S ( n 2 ( ) ) S P S ϕ : U R 3 P S P S U ϕ(p) =P p U p n p n 1 (P )=(Dϕ)(n p ) P S ( 2 ) t t = n 2 (P ) n 1 (P ) >0 n 2, n 1, t (n 2 n 1 t) 29 (Stokes ) S V : R 3 R 3 1 rot V n ds = V(u) du. S rot V n ds S S S S (S ) S = S rot V n ds =0 S 41 V R 3 S S S S V 29. ϕ : U S (ϕ(u) =S) Dϕ 1 = ϕ s, Dϕ 2 = ϕ Dϕ 1, 2 t Dϕ 1, Dϕ 2 (36) rot V n ds = (rot V(ϕ(s, t))) (Dϕ 1 Dϕ 2 )(s, t) ds dt. (40) S S 0 U U U m : [0, 1] U (m(1) = m(0)) l(t) =ϕ(m(t)), t [0, 1], S S m U l S 1 V(l) dl = V(l(t)) dl 1 (t) dt = (V(l(t))Dϕ(m(t))) dm (t) dt. dt dt S 0

39 V 1 3 Dϕ 3 2 m 2 1 R 2 ( 5) 2 Green 14 V(l) dl = (V(l(t))Dϕ(m(t))) dm = rot(v(ϕ(s, t))dϕ(s, t)) ds dt. (41) S U rot (19) 2 rot rot V = V 2 x V 1 y V = (V 1,V 2,V 3 ), Dϕ 1 =(Dϕ 11,Dϕ 12,Dϕ 13 ) ( 35 3 ) ( (s, t) ) rot((v ϕ) Dϕ) = s ((V ϕ) Dϕ 2) t ((V ϕ) Dϕ 1) 3 3 = (Dϕ 1 ) ((grd V j ) ϕ)(dϕ 2j ) (Dϕ 2 ) ((grd V j ) ϕ)(dϕ 1j ) j=1 3 3 ( Vj = x j=1 k k=1 ( Vk = x j (j,k); 1 j<k 3 (rot V) ϕ (Dϕ 1 Dϕ 2 ). U j=1 ) ϕ (Dϕ 1k Dϕ 2j Dϕ 2k Dϕ 1j ) V ) j ϕ (Dϕ 1j Dϕ 2k Dϕ 2j Dϕ 1k ) x k (40) (41) ( ) 3 Stokes ( 29) 2 Green ( 14) ( ) ( ) ( 1.3.2) 15 ( 1.4.5) R R 3 Ω R 3 Ω Ω S S = Ω R 3 1 V 3 (1) Ω V(u) u =0 (2) Ω V V(u) du (3) 2 f : Ω R V = grd f Ω V Ω b Ω V(u) du = f(b) f() (1) ( ) rot V =0 Ω 35 R 3 [5, 2.44]

40 (2) Ω rot V =0 Ω Ω V(u) u = ( 1.3.2) V = grd f 29 rot V = rot grd f =0 Ω rot V =0 Ω Ω S = S Ω Stokes ( 29) V(u) du = rot V n ds =0. S 2 ( 15) rot V =0 V 42 Ω=R 3 ( y, x, 0) \{(0, 0,z) z R} ( ) V(x, y, z) = x 2 + y 2, (x, y, z) Ω, Ω rot V =0 xy {(x, y, 0) x 2 + y 2 =1} V(u) u = 2π 0 ( sin θ, cos θ, 0) ( sin θ, cos θ, 0) dθ =2π Ω R 3 Ω ( ) Ω S S S ( S = S) S S S S S 0 Guss S S S S S S S S S S S S = S = S Ω V : Ω R 3 div V =0, on Ω, (42) Guss 27 S, S V nds =0 S S V nds = S V nds, S (43) V ( 2 ) A : Ω R 3 V = rot A, on Ω, (44) div rot A =0( 29) (42) (43) 29 (44) A V

41 (derhm ) Ω R 3 Ω Ω 1 V (42) V 2 A : Ω R 3 (44) 36 ( 1.4.5, 2.4.2) Ω Ω 1 43 V(x, y, z) = (x, y, z) V R 3 \{(0, 0, 0)} x 2 +y 2 +z 23 xy = {(x, y, 0) x 2 + y 2 =1} 2 S 1 = {(x, y, z) x 2 + y 2 + z 2 1, z 0}, S 2 = {(x, y, z) x 2 + y 2 + z 2 1, z 0}, V n ds, i =1, 2, S i V R 3 \{(0, 0, 0)} A V n ds = rot A n ds S i S i 29 i =1, 2 A(u) du xy z (x, y, z) S 1 n(x, y, z) = x2 + y 2 + z S (x, y, z) 2 2 n(x, y, z) = x2 + y 2 + z 2 z V n ds =2π, V n ds = 2π. S 1 S 2 V R 3 \{(0, 0, 0)} div V =0 rot grd f =0( 29) A V ϕ A + grd ϕ (44) 31 div A =0 A grd x =(x, y, z) R 3 t R ( ) R R 3 37 ( ) ( ) Poisson R 2 R

42 ( ) ( 39 x : R R 3 ( ) F ( ) x F R 3 F : R 3 R 3 F(x(t)) = m d2 x (t), t 0, (45) dt2 m>0 ( ) x(0), d x (0) dt F ( ) F 1 x 2 d x (0) = 0 ( ) dt F 40 rot F(x) =0, x R 3, (46) (45) d x (t) t [, b] dt b F(x(t)) d x dt (t) dt = m 2 b d dt d x dt (t) 2 dt = K(b) K() (47) K(t) = m d x 2 dt (t) 2, t 0, (48) 42 t 1:1 x(t), t [, b], ( 2.4.1, 1.2.5) 5 ( R 3 ) b F(x(t) d x (t) dt = F(u) du (49) dt x(t), t [, b], x(b) x() x ( ) F F (46) 30 V : R 3 R 41 (49) 30 b (47) F(x) = grd V (x), x R 3, (50) F(x(t)) d x (t) dt = V (x(b)) + V (x()). dt K(b)+V (x(b)) = K()+V (x()),,b 0, (51) x (45) F F 41

43 ( (45) ) x(t) E = K(t)+V (x(t)) = m d x 2 dt (t) 2 + V (x(t)) (52) ( 0 ) K(t) V (x(t)) E ( ) (45) F (46) ( ) (45) (46) (46) (52) 3.2 div ( ) ( ) (45) (t, x) R R 3 ( ) ρ(t, x) ( ) v(t, x) ( ) (45) ( ) 1 ρ ρ (t, x) + div(ρ v)(t, x) = 0 (53) t ρ v Guss 27 (53) Ω R 3 S = Ω ρ, v R R 3 1 t (53) x =(x, y, z) R 3 Ω t (1 44 ) div v 27 d ρ(t, x, y, z) dx dy dz = (ρ v n)(t, x, y, z)ds, (t, x, y, z) R R 3, (54) dt Ω n Ω S S (54) Ω ( ) S = Ω ( ) 42 ( ) 43

44 S ( ) ( ) 44 (54) Guss ( ) (53) (53) ( ) ( ) 3.3 rot ( ) E : R 4 R 3 B : R 4 R 3 (x, y, z) R 3 t R R ( div E = 1 ρ, ɛ 0 rot E = B t, div B =0, rot B = 1 E c 2 t + µ 0j. ɛ 0, µ 0 (MKSA) c c 2 =(ɛ 0 µ 0 ) 1 ( ) ρ : R 4 R j : R 4 R 3 ( ) (45) ( ) (45) F q F = F o + qe + q d x dt B (56) (F o E, B ) (56) ρ, j (55) ρ t = ɛ 0 div E t (55) = div j (57) div rot B =0( 29) ( 4 2 ) ( (53) ) div B =0 31 B = rot A (58) A (55) rot(e + A t )=0 44 (0 ) (0 )

45 A E = grd ϕ A t (59) ϕ (58), (59) A, ϕ f : R 4 R A = A + grd f ϕ = ϕ f t A ϕ E, B ( (58), (59) ) rot grd f =0( 29) f ( ) ϕ t + c2 div A = 0 (60) (60) (55) div grd ϕ = ϕ, rot rot A = grd(div A) A ( 29) ϕ = 1 ρ, ɛ 0 (61) A = µ 0 j. = 1 2 c 2 ((58), (59), (60) t2 ) (55) (61) (61) (55) 45 (61) (R 4 ρ =0,j = 0) v, k R 3 3 w = c k ϕ(t, x) =c k sin(k x wt), A(t, x) =(k + v k) sin(k x wt),, (t, x) R R 3, (61) (60) ( ) 4 46 (1) (2) (3) (4) Poisson Hermholtz (5) Poincré de Rhm 45 (56) E B

46 A.2 46 Appendix. A A.1 R d u z R d V f d, d ( ) D R d ( ) f : D R d D (D ) z f A ( ɛ >0) δ >0; 0 < z <δ, z D, f(z) A <ɛ, lim f(z) =A D f() f z lim f(z) =f() f D f D z δ D (ɛ ) f D 32 ([2, 11]) R d ( ( 50)) (1) (2) d =1 ( ) D R d {f n n N} f {f n } f D z D {f n (z)} f(z) {f n } f D lim f n f n D =0 E g g E = sup g(z) R d z z z E {f n } f D D f n f D 33 ( ) ( ɛ >0) n 0 N; n, m n 0 (1) D {f n } D f n f m D <ɛ, (2) A.2 d, d ( )

47 A ( [2, 25 29]) D R d d d f : D R d n (d > 1 n [2, 23,24]) x D, h R d ; x + h D, θ f(x + h) =f(x)+(h D)f(x)+ 1 2! (h D)2 f(x) n! (h D)n f(x + θh), 0 <θ<1, d =1(1 ) 1 (n 1)! (h D)n 1 f(x) f(x + h) =f(x)+hf (x)+ h2 2! f (x)+ + hn 1 (n 1)! f (n 1) (x)+ hn n! f (n) (x + θh), d =2,n =1 f(x + h, y + k) =f(x, y)+h f x f (x + θh, y + θk)+k (x + θh, y + θk). y 35 f : R 2 R x : R R, y : R 2 R df(x(t),y(t)) dt = dx dt (t) f x (x(t),y(t)) + dy dt (t) f y (x(t),y(t)) 35 2 ( ) 36 f : R 2 R x : R 2 R 2, y : R 2 R 2 f(x(t, u),y(t, u)) t = x t (t, u) f x (x(t, u),y(t, u)) + y t (t, u) f(x(t, u),y(t, u)) y y 37 ([2, 82 71]) D R 2 2 f : D R 1 P 0 =(x 0,y 0 ) D o f x (P 0 )= f x (P 0) 0 f y (P 0 )= f y (P 0) 0 f y (P 0 ) 0 f(x, y) =f(p 0 ) P 0 y y = φ(x) x 0 f(x, φ(x)) = f(p 0 ) y = φ(x) φ(x 0 )=y 0 φ φ (x) = f x (x, φ(x)) f y f r φ r 1 1 g : R R f(x, y) =y g(x) 37 1

48 A g : R R x 0 1 g (x 0 ) 0 y = g(x) x = g 1 (y) g(x 0 ) g 1 g g g 1 (y) = g 1 1 g r g 1 r 1 g (g 1 (y)) 2n n [2] 2 39 g : R 2 R 2 (x 0,y 0 ) 1 g 1 det x (x g 1 0,y 0 ) y (x 0,y 0 ) g 2 x (x g 2 0,y 0 ) y (x 0,y 0 ) g g(x 0,y 0 ) 1 g r r 0 A.3 A [, b] n [, b] n +1 = t 0 <t 1 < <t n 1 < t n = b ={t 0,,t n } = sup (t i t i 1 ) i {1,,n} 40 (Drboux [2, 30]) [, b] f : [, b] R sup inf n ( inf f(ξ i))(t i t i 1 )) = lim {ξ i} i=1 n i=1 (sup {ξ i} f(ξ i ))(t i t i 1 )) = n 0 i=1 n lim (sup 0 i=1 {ξ i} ( inf {ξ i} f(ξ i))(t i t i 1 ), (62) f(ξ i ))(t i t i 1 ), (63) [, b] {ξ i } t i 1 ξ i t i, 1 i n, ( ) 0 0. [2, 30] (63) (62) t i t i 1 t i t + t t i 1 sup sup 0 0 (lim 0 ) { n }, { n} { n } n 0 { n0 } (62) sup { n} sup { n} n 1/n 0 (f ) { n0 } { n } sup { n 0 } sup ( ) 0

49 A d(x, y) d(x, y) d(x, z)+d(z,y) lim ɛ 0 sup (x,y); x y ɛ d(x, y) =0 u d(x, y) = u(x) u(y) (62) t i t i 1 d(t i,t i 1 ) d(x, y) Riemnn-Stieltjes ( 32) (62) (63) 41 ([2, 30 31]) [, b] f : [, b] R lim 0 i=1 n f(ξ i )(t i t i 1 ) ( ) {ξ i } b f(t) dt ( ) f : R R F = f F : R R ( ) f u u du dt 42 ( [2, 32]) f F b f(t) dt = F (b) F () ([2, 32 35]) f(t) 1 t = x F (t) = F (x) =f(x) t f(t) dt t = x ( A.1) 44 ( [2, 48]) 2 f : [, b] [α 1,α 2 ] R α [α 1,α 2 ] F (α) = b f(x, α) dx f b α (x, α) F (α) = f α (x, α) dx A.3.2 D R 2 f : D R ( ) 1 [2, 90 93] D R 2 m, n D [, b] [c, d] = x 0 < x 1 < < x m 1 < x m = b c = y 0 < y 1 < < y n 1 < y n = d = ({x 0,,x m }, {y 0,,y n }) = mx{mx i {1,,m} (x i x i 1 ), mx j {1,,n} (y j y j 1 )} 45 (Drboux [2, 90]) D R 2 f : D R sup m,{ξ i,j} i=1 j=1 inf,{ξ i,j m i=1 j=1 n f(ξ i,j )(x i x i 1 )(y j y j 1 ) = n f(ξ i,j )(x i x i 1 )(y j y j 1 ) = lim m 0 i=1 j=1 lim m 0 i=1 j=1 n (sup f(ξ i,j )(x i x i 1 )(y j y j 1 ), {ξ i,j n ( inf f(ξ i,j )(x i x i 1 )(y j y j 1 ), {ξ i,j D ξ i,j R 2 [x i 1,x i ] [y j 1,y j ] D ( ) [x i 1,x i ] [y j 1,y j ] D = 0 ( f(ξ i,j )=0 )

50 A ([2, 92]) D R 2 D 0 D D f : D R 0 f(x, y) dxdy D f(x, y) dxdy = D lim m 0 i=1 j=1 n f(ξ i,j )(x i x i 1 )(y j y j 1 ). f(x, y) 1 D 1 ( ) 47 ([2, 93]) D R 2 b φ 1 (x) φ 2 (x), x b,, b R φ i : [, b] R, i =1, 2, D = {(x, y) R 2 φ 1 (x) y φ 2 (x), x b} f : D R x b D f(x, y) dxdy = b ( ) φ2(x) f(x, y) dy dx φ 1(x) φ2(x) φ 1(x) f(x, y) dy 48 ([2, 96]) Ω R 2 Ω = du dv x = x(u, v), x x y = y(u, v) Ω R 2 1 J (u, v) = u v y y (u, v) Ω 0 u v Ω={(x(u, v),y(u, v)) R 2 (u, v) Ω } Ω Ω= dx dy = J (u, v) du dv Ω Ω du dv = Ω Ω ρ (u, v) Ω ρ 0 Ω Ω lim ρ 0 Ω = J (u, v). f : Ω R f(x, y) dx dy = f(x(u, v),y(u, v)) J (u, v) du dv Ω Ω A.4 A.4.1 R 2 ( ) (u 1,u 2 )+(v 1,v 2 )= (u 1 + v 1,u 2 + v 2 ) R 2 r (u 1,u 2 )=(ru 1,ru 2 ) R 2 ( ) R 2 (0, 0) (u 1,u 2 )=( u 1, u 2 ) R 2 R 2 R 2 R 3 2 R n (u 1,,u n ) (v 1,,v n )=u 1 v u n v n R n R n R

51 A.5 51 u = u u ( ) u 0, ru = r u, u + v u + v, u =0 u = 0 V ( K ) ( ) K K K = R K = R v 1,,v k V 1,, n R 1 v k v k =0 1 = = n =0 R 2 (1, 0), (0, 1) n n n +1 ( ) (K ) 2 (, ) : V V R (u, u) 0 u =0 n e i e j = δ ij, i, j =1,,n, e i, i =1,,n, n V V R n 1:1 ( 1,, n ) R n 1:1 ( ) V ( ) A.4.2 n, m 2 n m (rnk) ( ) n 2 m2 1 0 rnk ( A.4.1) ( ) rnk n 2 n 2 A =( ij ) rnk 2 A n1 n2 A.5 [1, 14B,N,O] R n D D o (D c ) o D (D = D o ) (D c =(D c ) o ) c E r >0 (c, r) (E \{c}) c E \{c} D D D D D R n 50 (1) z R n z D z D \{z} (2) D =((D c ) o ) c (3) (4) D = D D c (5) D o = D \ D (6) (D c ) o = D c = D c \ D

52 B.0 52 (7) D D = D D D (8) D D R n S D (D ) (D ) R n O S = D O D \ S D S D D D A, B D = A B, A B = 51 D 2 D ( ) D R n D Jordn D D D Jordn D D B A.2 37 ([5, 1.31, 2.4, 2.7]) Appendix 1.4.2, 2.2.1, R 2 D R 2 1 f : D R c = {P R 2 f(p )=c} grd f(p ) 0, P, D R 3 (D = R 3, c =0) 3 1 f : R 3 R S = {P R 3 f(p )=0} grd f(p ) 0, P S, S P S S T P (S) ={P + v R 3 v (grd f)(p )=0} grd f(p ) ( ). R 2 P = (x, y) D grd f(p ) 0 f y (P ) 0 f x (P ) 0 37 δ 1,δ 2 > 0 φ : (x δ 1,x+ δ 2 ) R {Q Q P <ɛ} = {(t, φ(t)) t (x δ 1,x+ δ 2 )} f 1 grd f φ = f x 53 f y t (t, φ(t)) 1:1 R 3 f z (P ) 0 P =(x 0,y 0,z 0 ) (x 0,y 0 ) U R 2 2 g : U R f(x, y, g(x, y)) = 0, (x, y) U g ϕ(s, t) =(s, t, g(s, t)), (s, t) U, ϕ P f =0 ϕ ( 2.2.1) S P ϕ = ϕ 1 ϕ 2 ϕ 3 P = ϕ(s 0,t 0 ) S f(ϕ(s, t)) 0 35 (grd f) ϕdϕ=0 Dϕ ( 2.2.1) T P (S) Dϕ(s 0,t 0 ) 2 grd f(p ) ( 2.2.1) v (grd f)(p )=0 v

53 B R 2 2 (1) (2) P {Q Q P <ɛ} ɛ>0. R n ( 50) [1] 1985 [2] [3] [4] [5] 17 [6] [7] [8] [9] [5, 3] 11 [8, 4] 10 [5, 6, 7]