曲面のパラメタ表示と接線ベクトル

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1 L11( Wed) :Time-stamp: Wed 13:08 JST hig 1,, () (L11) Wed 1 / 18

2 ( ) 1 V = (xy2 ) x + (2y) y = y V = 4y., D V ds = 2 2 ( ) 4 x 2 4y dy dx = 0 = [ 2y 2 ] 4 x 2 0 dx 2(4 x 2 ) dx = () (L11) Wed 2 / 18

3 3 C 1 r(t) = (t, 0) ( 2 t 2). n R π dr 2 dt (t) = (0, 1). I 1 = 2 2 (0, ) (0, 1) dt = 12. C 2 r(t) = (2 cos t, 2 sin t) (0 t π). n R π dr 2 dt (t) = (2 cos t, 2 sin t). I 2 = = π 0 π 0 (0, 8 sin 2 t 3) (2 cos t, 2 sin t) dt (5 8 cos 2 t)(2 sin t) dt =[ 10 cos t cos3 t] π 0 = 28 3., r(t) = (t, 4 t 2 ) ( 2 t +2)., () (L11) Wed 3 / 18

4 ( 2) C. r(t) = (0, 2) + (1, 1)t (0 t 2), N(t) = (1, 1).,, 2 VdS = V n ds = t(2 t)(2 3, e 22 ) (1, 1) dt = (8+e 4 )(4 8 3 ). D C ( ) 0 1 V = 5. C 3 C 4 D,, V n ds = V ds = 5 (D ) = 5(π 2). D D () (L11) Wed 4 / 18

5 2 C 3 r(t) = (2 cos t, 2 sin t) ( 1 2π t 0), N(t) = (2 cos t, 2 sin t). C 3 V n ds = π V(r(t)) N(t) dt = 5π + 2. C 4 r(t) = (0, 2) + (1, 1)t (0 t 2), N(t) = ( 1, 1). V C 4 V n ds = 1 0 V(r(t)) N(t) dt = 12. -,, (2003) p.130, 6.8 () (L11) Wed 5 / 18

6 : (=divergence-free) C C V n ds = 0, ()V , C n V n ds = C 1 V n ds,, C 2 V n ds C 1 V n ds = 0 C 2., () (L11) Wed 6 / 18

7 [a,b] F (x) dx = : x [a,b] ±F (x) = F (b) F (a) 1 = 0 C ( f(r)) dr =f(r(t )) f(r(t )) 1 = (0 ) ( V) z ds = V dr D D 2 = 1 VdS = V n ds D D 2 = 2 () (L11) Wed 7 / 18

8 3 3 3 r = (x, y, z) 3 A = (A 1, A 2, A 3 ) A B =. 2 A =. 2 3 r(t) = (x(t), y(t), z(t)) p.61 1 r 1 (t) = (5 cos t, 5 sin t, t). 2 r 2 (t) = (1 + 3t, 2 + 4t, 3 + 5t). 3 r(t) = A + Bt. 2 z x y 5 () (L11) Wed 8 / 18

9 3 3 dr dt (t 0). 2 3 r (t) = r(t 0 ) + dr dt (t 0)(t t 0 ). 3 (3 ) 2 r(t) = (5 cos t, 5 sin t, t), r(t 0 ) = ( 5, 0, π). () (L11) Wed 9 / 18

10 3 f(r) = f(x, y, z). f(r) = x + 2y + 3z + 4, f(r) = r 2 9. f(r) = 0. f(r) = x 2 + y 2 + z = 0 f(r) = z 2 = 0, f(r) = x + 2y + 3z + 4 = 0 f(r), x, y, z 1, f(r) = 0. () (L11) Wed 10 / 18

11 r 1 (t) = (t 2, 2t + 1, 2t), r 2 (t) = (t 2, 2t + 2, 2t), r 3 (t) = (t 2, 2t + 3, 2t),. r s (t) r(s, t) = (t 2, 2t + s, 2t). s, t r(s, t) = (x(s, t), y(s, t), z(s, t)) r(s, 1), r(s, 2), r(s, 3),.... () (L11) Wed 11 / 18

12 r(s, t) = A + Bs + Ct (s, t 1 ), (A ). 1 s r(s 0, t) = (A + Bs 0 ) + Ct ( t) t r(s, t 0 ) = (A + Ct 0 ) + Bs ( s) 2 A = 0. () (L11) Wed 12 / 18

13 ( ) r(s, t) = (1, 0, 3) + (1, 0, 1)s + (0, 2, 1)t. () (L11) Wed 13 / 18

14 ( ) 3x + 2y + z + 6 = 0. () (L11) Wed 14 / 18

15 r s0 (t) = r(s 0, t) t. t = t 0 r t0 (s) = r(s, t 0 ) s. s = s 0 r(s 0, t 0 ),. r t (s 0, t 0 )t + r s (s 0, t 0 )s () (L11) Wed 15 / 18

16 r(s, t) r(s 0, t 0 ), r (s, t) = r(s 0, t 0 ) + r s (s 0, t 0 )(s s 0 ) + r t (s 0, t 0 )(t t 0 ). 2 f(x, y) (x, y) = (a, b) 1 f(x, y) = f(a, b) + f (a, b) (x a) + t f (a, b) (y b). x y () (L11) Wed 16 / 18

17 ( ) r(s, t) = (s + 2t, t + t 3, s 3 + st), r(1, 2) = (5, 10, 3). ( ) r(s, t) = (2 sin t cos s, 2 sin t sin s, 4 cos t) (0 t π, 0 s < 2π), r( 2 3 π, 1 6 π) = ( 1 2, 3 2, 2 3). () (L11) Wed 17 / 18

18 : 1. 26:00= 02:00. quiz (p.62) 3 1 [2.9](p.65) (p.63), [2.9](p.65) () (L11) Wed 18 / 18

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