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20010916 22;1017;23;20020108;15;20;

1 N = {1, 2, } Z + = {0, 1, 2, } Z = {0, ±1, ±2, } Q = { p p Z, q N} R = { lim a q n n a n Q, n N; sup a n < } R + = {x R x 0} n = {a + b 1 a, b R} u, v 1 R 2 2 R 3 2 u =(u 1,u 2,u 3 ), v =(v 1,v 2,v 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 ) R n u =(u 1,,u n ), v =(v 1,,v n ) u v = u 1 v 1 + +u n v n u = u u 1 R 2 2 R 3 P =(x, y) R 2 u : R 2 R 2 P u(p ) R 2 [a, b] R 2 ( ) R 2 [a, b] 1:1 1:1 d u (t) (0, 0), dt t [a, b], 1 n A (R R 2 ) 1 1.1 [1] u, v θ u v = u v cos θ

2 [2] (1) V(P )=(1, 0), P =(x, y) R 2. (2) V(P )=(x, y), P =(x, y) R 2. (3) V(P )=(y, x), P =(x, y) R 2. (4) V(P )=( y, x), P =(x, y) R 2. [3] (1) V(x, y) =(x, ( 0) ) x (2) V(x, y) = x 2 + y, y ( 2 x 2 + y 2 ) y (3) V(x, y) = x 2 + y, x ( 2 x 2 + y 2 ) y 2 x 2 +1 (4) V(x, y) = (x 2 + y 2 +1), 2xy 2 (x 2 + y 2 +1) 2 1.2 [4] = {(x, y) R 2 y 2 = x 2 (x +1)} ( ) [5] 1:1 (1) = {(x, y) R 2 cos x + y =0} (2) = {(x, y) R 2 3x 2 +4y 2 =1} (3) = {(x, y) R 2 ye x =1} [6] u 2 1 +u2 2 =1 [7] V (1) V(u) du (2) V(u) du V(u) ds(u).

3 (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 [8] V(x, y) =(2x, 3x + y) = {(x, y) R 2 1 4 x2 + y 2 =1} V(u) du [9] u V(x, y) =(x, y) : (, 0] R 2 t (e t cos t, e t sin t), t 0, V(u) du 1.3 grad [10] f, g grad fg= f grad g + g grad f [11] (1) V(x, y) =(y, x), (x, y) R 2, (2) V(x, y) =( y, x), (x, y) R 2,

4 1.4 rot Green [12] y = x 2 [13] } (1) = {(x, y) x2 a + y2 2 b =1 2 } (2) = {(x, y) x2 a y2 2 b =1 2 [14] 1:1 V P a>0 V (P ) a P V(u) n(u)du a [15] u (rot u) (1) u(x, y) =(x, y). (2) u(x, y) =(y, x). y (3) u(x, y) =( x2 + y, 2 y (4) u(x, y) =( x 2 + y, 2 x x2 + y 2 ). x x 2 + y 2 ). [16] f : (0, ) R R 2 \{(0, 0)} V(x, y) =( yf( x 2 + y 2 ),xf( x 2 + y 2 )) rot V(x, y) =1 R 2 \{(0, 0)} f [17] (1) V(x, y) = 1 ( y, x) rot V 2 (2) V ( 1 ) rot V =0 [18] (1) 1 (1, 0) 1 dxdy = 1 xdy)= D 2 D( ydx+ 1 xdy 1 ydx 2 D 2 D

5 (2) 1 (1, 0) ( 1, 0) (1, 0) 1:1 1 xdy 1 ydx=0 2 2 [19] U R 2 () F ( =(f 1,f 2 ): U R 2 \{0} 1 f1 U V F = f1 2 + f2 2 x f 2 f 2 x f f 1 1, y f 2 f ) 2 y f 1 (1) U rot V F =0 (2) U = {(x, y) R 2 1 2 <x2 + y 2 < 3 2 } F (x, y) = 1 x 2 +y 2 (x, y) V F (3) u(t) = (cos t, sin t) V F 2π 0 V F (u) du (4) F ˆF : R 2 R 2 \{0} U F [20] U = R 2 \{(x, 0) R 2 1 x 1}V(x, y) = ( y, x 1) (x 1) 2 + y2 1 (x +1) 2 + y ( y, x +1) 2 1 {(x, y) x 2 + y 2 =4,x 0} y {(0,y) 2 y 2} 2 {(x, y) (x 1) 2 + y 2 =16} 1 (1) V 2 (x, y) = (x +1) 2 + y ( y, x +1) 2 V 2 (u) du =0 1 1 (2) V 1 (x, y) = ( y, x 1) V(u) du = V (x 1) 2 + y2 1 (u) du 1 2 (3) f : U R U grad f = V (4) lim n (f(0, 1 n ) f(0, 1 n )) = 1 V(u) du [21], n, n N, u : [0, 1] R 2, u n : [0, 1] R 2, n N, u n u V R 2 (1) d u n d u lim V(u n ) du n = V(u) du dt dt n n (2) ɛ>0 Ω ɛ R 2 ɛ Ω ɛ 2ɛ ɛ A Ω ɛ 1 ɛ 1 [5, 1.9 (2)]

(3) D Ω ɛ D ɛ A rot V dx dy D A sup rot V(x, y) ɛ (x,y) D (4) Ω ɛ ɛ V du V du A sup rot V(x, y) ɛ ɛ (x,y) D (5) sup u n (t) u(t) <ɛ t [0,1] V du n V du A sup rot V(x, y) ɛ n ɛ (x,y) D (6) d u n d u dt dt lim V n n du = n 6 V du [22] Planimeter (Amsler ) planimeter 1.5 div Gauss [23] u (div) (1) u(x, y) =( y, x). (2) u(x, y) =(x, y). x (3) u(x, y) =( x2 + y, y )((x, y) (0, 0)). 2 x2 + y2 x (4) u(x, y) =( x 2 + y, y )((x, y) (0, 0)). 2 x 2 + y2 (5) u(x, y) =(xlog x 2 + y 2,ylog x 2 + y 2 )((x, y) (0, 0)). [24] f : (0, ) R R 2 \{(0, 0)} V(x, y) =(xf( x 2 + y 2 ),yf( x 2 + y 2 )) div V(x, y) =0 R 2 \{(0, 0)} f k f(r) =r k, r>0, k [25] f 1 V 1 (1) div(f V) = (grad f) V + f div V

7 (2) rot(f V) = (grad 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 [26] f 2 V 2 (1) rot grad f =0 (2) div grad f = 2 f x + 2 f 2 y 2 [27] [23] V (1) (2) (±1, ±1) 4 V nds [28] [23] V Gauss (0, 0) ɛ Gauss 1.6 [29] θ ( ) ( x x(x,y ) = y y(x,y ) ) ( = cos θ sin θ sin θ cos θ )( x y ) V rot V [30] V V(u) n ds(u) [31] f V grad f, rot V

8 2 2.1 grad, div, rot [32] u v R u v u, v, u v u v u =(1, 0, 0) v {(1, 0, 0), (0, 1, 0), (0, 0, 1)} u v [33] (1) u =(1, 1, 2), v =(2, 0, 1), w =(1, 0, 3) u (v w), v (w u), w (u v), u (v w), (u v) w (2) u (v w) [34] u (v w) u v v 0 [35] 3 u, v, w u (v w) =(u w)v (u v)w [36] u (v w) =(u v) w 0 u, v, w [37] u 0, v u v =0 v = u w w [38] u, v, A 3 3 t à A Ã(u v) =Au Av [39] f, g 1 U, V 1

9 (1) grad(f g) = (grad f) g + f grad g (2) div(f V) = (grad f) V + f div V (3) rot(f V) = (grad f) V + f rot V (4) div(u V) = (rot U) V U rot V (5) rot(u V) =(V grad)u + U (div V) (U grad)v V (div U) [40] f 2 V 2 (1) rot grad f =0 (2) div grad f = 2 f x + 2 f 2 y + 2 f 2 z 2 (3) div rot V = 0 (4) rot rot V = grad(div V) ( V x, V y, V z ) 2.2 [41] 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} [42] ϕ(s, t) =(s 2,t 2,st) ϕ Dϕ rank 2 ϕ [43] (0, 0, 1) x y = {(x, y, 0) x 2 + y 2 =1} z [44] S = {(x, y, z) x 2 + y 3 + z 4 =0} (0, 0, 0) (0, 0, 0) S

10 [45] S = {(x, y, z) R 3 ax + by + cz = d} (1) P 0 =(x 0,y 0,z 0 ) S 1 S (x, y, z) S a(x x 0 )+b(y y 0 )+c(z z 0 )=0 (2) S P P 0 P v (a, b, c) v λ R v = λ(a, b, c) 1 (3) 1 P 0 2 ( ) 2 2 [46] S = {(x, y, z) x 2 + 1 4 y2 + 1 9 z2 =1} [47] f : R 2 R 1 ϕ(s, t) =(s, t, f(s, t)) ϕ : R 2 R 3 ϕ S S f [48] S 2 R 3 ( ) ϕ : U R 3 U = {(s, t) 0 <s<2π, 0 < t<π} ϕ(s, t) = (cos s sin t, sin s sin t, cos t) ϕ : U R 3 U = {(s, t) s 2 + t 2 < 1} ϕ (s, t) =(s, t, 1 s 2 t 2 ) W U W U ψ : W W ϕ ψ(s, t) =ϕ(s, t), (s, t) W [49] [41] [50] E ds E V E V nds f E fds E [51] S = {(x, y, z) x 2 +y 2 +z 2 =1,z>0} V(x, y, z) = (x, y, z) V nds S

11 (1) ϕ : U R 3 ; U = {(s, t) π 2 <t<π 2, 0 <s<π}, ϕ(s, t) = (sin t, cos s cos t, sin s cos t) (2) ϕ : U R 3 ; U = {(s, t) s 2 + t 2 < 1}, ϕ (s, t) =(s, t, 1 s 2 t 2 ) [52] ( [41]) [53] (0, 0, 1) x y = {(x, y, 0) x 2 + y 2 =1} z S V(x, y, z) =(0, 0, 1), (x, y, z) S, V nds S 2.3 Gauss Green [54] Ω={(x, y, z) R 3 z < 1, x 2 + y 2 < 1} div V(x, y, z) dx dy dz = V nds. Ω S [4, ] [55] 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 [56] V : R 3 R 3 1 V(x, y, z) = (x, y, z) α>0 (x 2 + y 2 + z 2 ) α S V nds S [57] S = {(x, y, z) R 3 4x 2 +y 2 +z =1,z 3} V(x, y, z) =(3yz+7y+x, y, z+3) V nds S [58] S = {(x, y, z) x 2 + y 2 +4z 6 =4, z 0} V(x, y, z) =(e y,z,x 2 ) V nds S

12 2.4 Stokes [59] (1) {(x, y, z) R 3 1 <x 4 + y 2 + z 2 < 2} (2) {(x, y, z) R 3 0 <x 2 + z 4 < 2} (3) R 3 \{(x, y, 0) R 3 x 2 + y 2 =1} (4) R 3 \{(x, y, z) R 3 x 2 + y 2 + z 2 < 1} [60] S = {((2 + t cos s 2 ) sin s, (2 + t cos s 2 ) cos s, t sin s ) s R, t < 1} 2 S m V S rot V =0 u(s) = (2 sin s, 2 cos s, 0) S V dm =2 V du S [61] Ω = R 3 \{(0, 0,z) z R} ( ) ( y, x, 0) V(x, y, z) =,(x, y, z) Ω, Ω rot V =0 x 2 + y2 xy {(x, y, 0) x 2 + y 2 =1} V(u) du 0 [62] V(x, y, z) = 1 x2 + y 2 + z 23 (x, y, z) V R3 \{(0, 0, 0)} 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}, S i V n ds, i =1, 2, (1) V R 3 \{(0, 0, 0)} A V n ds = V n ds S 1 S 2 (2) V n ds, i =1, 2, V S i

13 Appendix. A A.1 [63] (1) a, b N 1 a + bn 2 = O(N 2 ), N, (2) k>0 e N = O(N k ), N, [64] f : R n R ( x R n )( ɛ >0) δ >0; ( y R n ) x y <δ f(x) f(y) <ɛ A R f 1 (A) [65] (1) f : R n R {x n } R n lim f(x n)=f( lim x n ) n n (2) f lim f(x n )=f( lim x n ) n n [66] R [a, b] a 1,a 2, [a, b] [67] (1) R (0, 1) (2) I =[a, b] f : I R [68] [a, b] f a x 0 b f(x) f(x 0 ), a x b, [69] (1) (a, b)

14 (2) [a, b] ( ) [70] I f n : I R, n =1, 2, 3,, [71] I f n : I R, n =1, 2, 3,, A.2 [72] f R 2 2 x = r cos θ, y = r sin θ g(r, θ) =f(r cos θ, r sin θ) (1) g r g f f (r, θ) (r, θ) (r cos θ, r sin θ) (r cos θ, r sin θ) θ x y (2) f x f y g r g θ A.3 [73] [a, b] n [a, b] n +1 a = t 0 <t 1 < <t n 1 <t n = b ={t 0,,t n } = (t i t i 1 ) sup i {1,,n} f [a, b] lim 0 n f(ξ i )(t i t i 1 ) i=1 ( ) {ξ i } {ξ i } t i 1 ξ i t i,1 i n, b f(t) dt ( ) a

15 [74] [a, b] f n, n N, b lim n a f n (x) dx = b a lim f n(x) dx n [75] (1) xe y dx dy, D = {(x, y) 0 x 1, 0 y 1} D (2) xy dx dy, D = {(x, y) y 0, 0 x 2 + y 2 x} D (3) cos(x + y) dx dy, D = {(x, y) x 0, y 0, x+ y π/2} D (4) e x+y dx dy, D = {(x, y) 0 y x 1} D [76] (1) (x 2 + y 2 ) dx dy, D = {(x, y) x 2 + y 2 1} D (2) x2 + y 2 dx dy, D = {(x, y) x 0, y 0, x 2 + y 2 1} D (3) y 2 dx dy, D = {(x, y) x 2 + y 2 2x} D A.4 R n R n, m 2 n m (rank) 2 2 2 2 ( ) n 2 m 2 1 0 [77] R n R ( ) ( ) 1 (2 ) R

16 [78] (1) R 3 1 1 0, 1 1 1 (2) [79] (1) R n (2) R 3 [80] (1) R 2 1 (2) R 2 [81] (1) R n u =(u 1,,u n ), v =(v 1,,v n ), u v = u 1 v 1 + + u n v n 2 (2) (1, 1, 1), (2, 1, 0) R 3 [82] n 2 n 2 A =(a ij ) rank 2 A 2 2 a 11. a 12. a n1 a n2 A.5 n N ( ) x R n i =1,,n x i x i x =(x 1,,x n ) R 2, R 3 x, y d : R n R n R (d(x, y) = 0 if and only if x = y) R n d 2 (x, y) = x y = n (x i y i ) 2 i=1 x R n r>0 Ball(x, r) ={y R n y x <r} ( )

17 R n A R n ( x A) r >0; Ball(x, r) A D r > 0; Ball(x, r) D x D D D R n D c D D o ()D D c D c R n E R n r >0 Ball(c, r) (E \{c}) c E E [83] (1) (2) d (x, y) = x y = max i=1,,n x i y i [84] A R n A {Ball(x λ r λ ) λ Λ} A = λ Λ Ball(x λ,r λ ) [85] [86] D R n (1) x R n x D x D \{x} (2) D =((D c ) o ) c (3) D = D D c (4) D o = D \ D (5) (D c ) o = D c = D c \ D (6) D D = D D D [87] (1) R n R n (2) R n

18 [1] [2] [3] 2001 [4] 2001 ver. 20010519 [5] 17