II 2 II

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................................. 6 3 9 4 11 16 35 17 37 18 1 39 19 2 41 20 43 21 45 22 47 23 49 5 13 6 15 7 17 8 19 9 21 10 23 11 25 12 27 13 29 14 31 15 33 i ii

1 (3 ) 1 1.1 1.2 1.3 1.4 1.5 1.6 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3 3.1 3.2 3.3 1 3.4 3.5 3.6 3.7 3.8 4 4.1 4.2 4.3 1 4.4 4.5 1 4.6 5 5.1 5.2 5.3 5.4 5.5 ( ) 5.6 5.7 1 2

6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 7 7.1 8.4 8.5 8.6 7.2 7.3 1 7.4 7.5 7.6 7.7 8 8.1 8.2 8.3 3 4

2 2.1 (x, y, z) gradφ = φ = φ x i + φ y j + φ z k diva = A = A x x + A y y + A z ( z Az rota = A = y A ) ( y Ax i + z z ( ) (r, ϕ, z) ˆr ˆϕ ẑ gradφ = φ r ˆr + 1 φ φ ˆϕ + r ϕ z ẑ diva = A = 1 r r (ra r) + 1 A ϕ r ϕ + A z ( z 1 A z rota = r ϕ A ) ( ϕ Ar ˆr + z z A ) z ˆϕ + 1 r r 2 φ = 1 ( r φ ) + 1 2 φ r r r r 2 ϕ 2 + 2 φ z 2 (r, θ, ϕ) ˆr ˆθ ˆϕ gradφ = φ = φ r ˆr + 1 φ r θ ˆθ + 1 r sin θ diva = A = 1 r 2 r (r2 A r ) + 1 rota = A = 1 r sin θ 2 φ = 1 ( r 2 φ ) + r 2 r r A ) ( z Ay j + x x A ) x k y [ r (ra ϕ) A ] r ẑ ϕ φ ϕ ˆϕ r sin θ θ (A θ sin θ) + 1 A ϕ [ r sin θ ϕ θ (A ϕ sin θ) A ] θ ˆr + 1 [ 1 A r ϕ r sin θ ϕ ] r ra ϕ ˆθ + 1 [ r r (ra θ) A ] r ˆϕ ( θ 1 sin θ φ ) 1 2 φ + r 2 sin θ θ θ r 2 sin 2 θ ϕ 2 5 2.2 (1) (φϕ) = φ ϕ = ϕ φ (2) (φa) = φ A + A φ (3) (φa) = φ A + φ A (4) ( φ) = 0 (5) ( A) = 0 (II) (6) (A B) = B A A B (7) A (B C) = (A C)B (A B)C (8) (A B) = A B B A + (B )A (A )B (9) (A B) = A ( B) + B ( A) + (B )A + (A )B (10) ( φ) = φ = 2 φ (I) (11) ( A) = ( A) 2 A (:III) 2.3 E = ρ ε B = 0 E = B t B = µj + µε E t 2.4 F = q(e + v B) db = µi 4πr dl r 3 j + ρ t = 0 B = A E = φ E = A t φ 6 ()

i = dq dt dv dt = v + (v )v t () 1 2 () 3 4 5 6 a = (a x, a y, a z ) b = (b x, b y, b z ) (a )b ( ) x y z i j k x y z e x e y e z ˆ (1) (2) (3) 7 8

3 a = (a x, a y, a z ) b = (b x, b y, b z ) a b (a, b) a b = (a x, a y, a z ) (b x, b y, b z ) = a x b x + a y b y + a z b z = a b cos θ (1) θ a b 1 a b b ( ) θ = 0 (a b ) a b = ab θ = π 2 (a b ) a b = 0 7 A B A B (1) A = (2, 4, 6), B = (1, 3, 2) (2) A = (1, 1, 0), B = (0, 1, 3) (3) A = (1, 0, 2), B = (1, 2, 3) (4) A = (1, 0, 0), B = (0, 1, 0) (5) A = (0, 2, 1), B = (1, 2, 1) (6) A = (3, 1, 2), B = (1, 0, 1) (7) A = (2, 2, 1), B = (1, 2, 0) (8) A = (1, 1, 0), B = (1, 2, 3) 8 (A 1 B 1 + A 2 B 2 + A 3 B 3 ) 2 (A 2 1 + A 2 2 + A 2 3)(B1 2 + B2 2 + B3) 2 9 A = (2, 4, 6) B = (1, 3, 2) θ 10 A = (2, 3, 1) B = (1, 0, 2) a b θ a cos θ b 1: A B A B (1) A = (2, 1, 1), B = (0, 1, 3) (2) A = (1, 1, 1), B = ( 1, 2, 1) (1) (1) A B = 2 1 + 1 1 + 1 3 = 0 + 1 + 3 = 4 (2) A B = 1 ( 1) + 1 2 + 1 1 = 1 + 2 + 1 = 2 9 a b cos θ 0 10

4 a = (a x, a y, a z ) b = (b x, b y, b z ) a b a b = (a x, a y, a z ) (b x, b y, b z ) = (a y b z a z b y, a z b x a x b z, a x b y a y b x ) (2) (2) i j k a b = a x a y a z b x b y b z a b 4 a b a b (ab sin θ) θ = 0 (a b ) a b = 0 θ = π (a b ) a b = ab 2 a b a b (3) 11 A B A B (1) A = (2, 4, 6), B = (1, 3, 2) (2) A = (1, 1, 0), B = (0, 1, 3) (3) A = (1, 0, 2), B = (1, 2, 3) (4) A = (1, 0, 0), B = (0, 1, 0) (5) A = (0, 2, 1), B = (1, 2, 1) (6) A = (3, 1, 2), B = (1, 0, 1) 12 a b a b 13 A = (2, 1, 1) B = (1, 1, 2) 14 A (B C) = (A C)B (A B)C 15 A = (1, 2, 3), B = (2, 1, 1) C = ( 1, 1, 1) (a) A (B C) 14 (b) (A B) C a b θ b a (A B) C = (A C)B (B C)A 16 14 (A B) C = (A C)B (B C)A 1 A B A B (1) A = (2, 1, 1), B = (0, 1, 3) (2) A = (1, 1, 1), B = ( 1, 2, 1) (2) (1) A B = (2, 1, 1) (0, 1, 3) = (2, 6, 2). (2) A B = (1, 1, 1) ( 1, 2, 1) = ( 1, 2, 3). 11 0 a b sin θ(= a b ) 2 12

5 ( ) ( ) 1 (x, y, z) φ(x, y, z) (x, y, z) f(x, y, z) () () s ( ) = x, y, z (4) 17 18 r P A(r) 19 a = (2, 1, 0) 20 A (B C) = (A C)B (A B)C ( A) = ( A) 2 A ( III) 88 88 u = (u x, u y, u z ) [ ( = (u x, u y, u z ) x, y, ( = u x x + u y y + u z z = (u )u )] (u x, u y, u z ) z ) (u x, u y, u z ) ( u x u x x + u u x y y + u u x z z, u u y x x + u u y y y + u u y z z, u u z x x + u u z y y + u z u u ( ) ) u z z 13 14

6 φ(x, y, z) φ grad φ φ = grad φ = φ x i + φ y j + φ z k (5) (cφ) = c φ (6) (φ + ψ) = φ + ψ (7) (φψ) = φ ψ + ψ φ (8) φ ψ c φ(x, y, z) = x 2 + y 2 + z 2 P(1, 1, 0) Q(2, 1, 1) φ = (x2 + y 2 + z 2 ) i + (x2 + y 2 + z 2 ) j + (x2 + y 2 + z 2 ) k x y z = 2xi + 2yj + 2zk (1, 1, 0) (2, 1, 1) P (2, 2, 0) Q (4, 2, 2) 43 21 (8) 22 φ φ = 3x 2 yz (1, 1, 1) φ 23 φ = r = r = (x 2 + y 2 + z 2 ) 1/2 φ 24 φ = φ(x, y, z) φ dr = dφ 25 r = r = (x 2 +y 2 +z 2 ) 1/2 n (r n ) (1/r) ) 26 y = 10 exp ( x2 4 27 z = 10 exp ( x2 + y 2 ) 4 28 z = 10 exp ( x2 + y 2 ) z 4 xy z 2 4 6 8 29 z = 10 exp ( x2 + y 2 ) (x y 4 ) grad 15 16

7 (divgergence : ) div A = A = lim V 0 1 A d (9) V 2 A = A x x + A y y + A z z A = A x x + A y y (grad) (rot) (10) (11) 30 f = (x y)i+(y z)j +(z x)k g = (x 2 +yz)i+(y 2 +zx)j +(z 2 +xy)k f, g 31 f = 3xyz 2 i + 2xy 3 j x 2 yzk (1, 1, 1) f 32 x-y (0 x 5, 0 y 5) A = 0 i + exp ) ( x2 j 10 i j x y 33 32 A 34 32 A( ) 33 (1) A = xyzi + x 2 y 2 zj + yz 3 k (2) A = 2x 2 zi xy 2 zj + 3yz 3 k (3) A = 3x 2 yi 2y 3 z 2 j + xy 2 zk (10) (1) A = (xyz) x (2) A = (2x2 z) x (3) A = (3x2 y) x + (x2 y 2 z) y + ( xy2 z) y + ( 2y3 z 2 ) y + (yz3 ) z + (3yz3 ) z + (xy2 z) z = yz + 2x 2 yz + 3yz 2 = 4xz 2xyz + 6yz = 6xy 6y 2 z 2 + xy 2 2: 0 div 17 18

8 (rotation) 1 rot A = A = n max. lim 0 n max. 1 A dr C ( Az A = y A ) ( y Ax i + z z A ) z j + x (grad) C A dr (12) ( Ay x A x y ) k (13) 35 r = (x, y, z) r 36 f = (xyz, x 2 y 2 z, yz 3 ) f 37 f = (x y)i+(y z)j +(z x)k g = (x 2 +yz)i+(y 2 +zx)j +(z 2 +xy)k f, f, g, g 38 φ = 3x 2 yz f = 3xyz 2 i + 2xy 3 j x 2 yzk (1, 1, 1) (a) φ (b) f (c) f (d) f φ (e) (φf) (f) (φf) (g) 2 φ 39 φ ( φ) = 0 40 f ( f) = 0 41 f = yi + xj + zk g = 3xyz 2 i + 2xy 3 j x 2 yzk φ = xyz (1, 1, 1) (f )φ (f )g (1) A = xyzi + x 2 y 2 zj + yz 3 k (2) A = (x y)i + (y z)j + (z x)k (1) A = (z 3 x 2 y 2 )i + xyj + (2xy 2 z xz)k (2) A = i + j + k 42 f = f(x, y, z, t) dr = dxi + dyj + dzk df = (dr )f + f t dt 43 (φf) = φ f + f φ 44 (φf) = φ f + ( φ) f = φ f f φ 45 f = (2z, x 2, x) φ = 2x 2 y 2 z 2 (1, 1, 1) (f )φ 46 c cf = φ f f = 0 rot 19 20

9 t r(t) = x(t)i + y(t)j + z(t)k (14) s t r(s, t) = x(s, t)i + y(s, t)j + z(s, t)k (15) s t (1)z = 4 + x 2 + y 2 (2) a(z 0) 47 r(x, y) = xi + yj + z(x, y)k (1) r x r y (2) r x r y (3) (4) r x r y 48 d n (1) x = s, y = t, z = 4 + x 2 + y 2 = 4 + s 2 + t 2 (1) (2) xy xy xy z xy r(s, t) = si + tj + (4 + s 2 + t 2 )k (2) x 2 + y 2 + z 2 = a 2 z = a 2 x 2 y 2 x = s y = t z = a 2 s 2 t 2 r(s, t) = si + tj + a 2 s 2 t 2 k 21 22

10 y = f(x) x = a x = b x 1. :x = a x = b n x 2. x = a b n x n i x x i = a + a b n i ( f(x i ) = f a + a b ) n i x ( f(x i ) x = f a + a b ) a b n i n 3 ( f a + a b ) ( a b n 1 n + f a + a b ) a b n 2 n + 23 y 0 a b y = f(x) x 3: ( + f a + a b ) ( a b n i n + + f a + a b ) n n n ( f a + a b ) a b n n i n = f (a + x i) x i=1 4 n lim n i=1 1. i=1 n f (a + x i) x = 2. 3. 4. b a f(x)dx 24

11 x(x ) f C f C (23 ) C (n ) L P 1 P n i L lim n n i f(p i ) L = f(p )dl x C (0, 0) (1, 1) (x 2 + y 2 )ds C C s C x = t, y = t 0 t 1 s = 2t ds = 2dt C (x 2 + y 2 )ds = 1 0 (t 2 + t 2 ) 2dt = 2 2 3 25 C 49 f(x, y) = z = x 2 + y 2 (1) (2) C( (0, 0) (1, 1) ) (3) s t (4) g(x) x = a x = b L( ) 50 z = f(x, y) = x(y + 1) 2 x(y + 1) 2 ds C C (x 1) 2 + y 2 = 1 (0, 0) (1, 1) (1) C (2) C (x, y) θ (3) 51 f(x, y) = z = x 2 + y 2 C 1 :x 2 + y 2 = 1 (x 2 + y 2 )ds C (23 ) 26

12 f f (23 ) (n ) P 1 P n i lim n n i f(p i ) = f(p )d I = (x 2 3xy)d : 1 x 2, I = 2 1 2 xdx (x 3y)dy = 1/x = 3 2 log 2 35 2. 27 2 1 1 x y 2 ( x 2x 7 + 3 ) 2x 2 dx 52 R ρ r ( ρ(r) = ρ 0 1 r ) R M M (1) ρ(r) (2) r r + dr d (3) r r + dr dm (4) M 53 f : x 2 + y 2 + z = 4, f(x, y, z) = 2y 2 + z 4x2 + 4y 2 + 1 z > 0 xy xy... dxdy 22 48 (23 ) 28

13 x V φ(x, y, z) R φ(x, y, z)dv R dv = dxdydz 3 φ = 1 R V (x + y + z)dxdydz, V : x > 0, y > 0, z > 0, x + y + z < 1 V 54 R ρ(x, y, z) ρ(x, y, z)dv R 55 n(x, y, z) M 56 R ρ r M (1) ( ρ(r) = ρ 0 1 r ) R (2) r r + dr dv (3) r r + dr dm (4) M I = 1 dx 1 x dy 0 0 0 1 x y (x + y + z)dxdydz = 1 8 (23 ) 29 30

14 A dl (16) C C ( ) dl A dl C dl = dl dl A θ A dl = A dl cos θ C f = xi + 2yj + zk f dr C (0, 0, 0) (1, 1, 1) C r = ti + tj + tk C dr = dti + dtj + dtk 57 (1) dl C dl = dl (2) dl A 58 C xy a (1) C (2) (x, y) θ (3) (x, y) dl x y θ (4) A C 60 A = A A dl dl dl (5) A dl C 59 ( y A = x 2 + y, x ) 2 x 2 + y, 0 2 (1) xy (2) C xy a A dl C f f = ti + 2tj + tk f dr = 1 0 (t + 2t + t)dt = 1 0 4tdt = 2 31 32

15 A d (17) d A d d = d 60 f = xi + yj + 2zk x 2 + y 2 = 1 z z 0 f d 61 r A A = k 1 r 2 r r f = 0i + 0j + 2k z = 2 0 x 2 0 y 2 f d d z d = dk r r r = r k k > 0 (1) r r (2) (3) a d nd = d (4) d A A d f d = 2d = 2dxdy 2 2 2 dxdy = 8 0 0 f (5) a (6) 2a A d A d 33 34

16 V A V AdV = A d (18) 3 2 E = ρ ε 0, B = 0 ρ EdV = dv V V ε 0 = E d = 1 ρdv ε 0 V 62 (1) V (2) d 63 r V r d = 3V 64 V r r r = r V 1 r dv = r d 2 r 2 65 A ( A) d = 0 Q E d = Q ε 0 35 36

17 C A C 1 A d = A dl (19) C E = B t, B = µ 0J + µ 0 ε 0 E t B E d = t d Φ E dl = B d = Φ C t t 66 (1) C (2) dl 67 r = xi + yj + zk r dr = 0 68 C C C φ dr = 0 69 Φ B Φ = B d A B = A Φ A 37 38

18 1 1 f(x, y) x y y = dy = f(x, y) (20) dx f(x, y) f(x) dx dy = f(x)dx ydy = f(x)dx y = F (x) + C C F = m dv dt 70 θ m t = 0 t g 71 v 0 t = 0 g (1) t (2) t 1 (3) t 2 (4) v 72 h m v 0 (1) y = 2x + 1 (x = 0 y = 1) y = dy dx = 2x + 1 dy = (2x + 1)dx dy = (2x + 1)dx + C = x 2 + x + C C = 1 y = x 2 + x + 1 39 40

19 2 2 a b, c ay + by + cy = 0 y = dy dx y = d2 y dx 2 λ aλ 2 + bλ + c = 0 (21) λ = b ± b 2 4ac 2a i) b 2 4ac 0 2 λ 1 λ 2 (λ 1 λ 2 ) i)-1. b 2 4ac > 0 λ 1 λ 2 C 1, C 2 y = C 1 e λ1x + C 2 e λ2x (22) i)-2. b 2 4ac < 0 λ 1 λ 2 cos z = eiz + e iz, sin z = eiz e iz 2 2i A B (A = C 1 + C 2, B = i(c 1 C 2 )) y = e µx (A cos ωx + B sin ωx) (24) ii) b 2 4ac = 0 λ y = C 1 e λx + C 2 xe λx = (C 1 + C 2 x)e λx (25) 73 (y = dy dx, y = d2 y dx ) 2 (1) y + 9y = 0 (2) y 5y + 6y = 0 (3) y 6y + 9y = 0 (4) y + 4y + 5y = 0 (5) y 3y + 2y = 0 (6) y 3y = 0 (7) y + 4y + 4y = 0 (8) y + 2y + 2y = 0 74 k (1) x = 0 y = h dy dx = 0 d 2 y dx 2 + k2 y = 0 (2) x = 0 y = 0 dy dx = v 0 y = C 1 e λ1x + C 2 e λ2x (23) λ λ 1,2 = µ ± iω µ, ω i i 2 = 1 y = C 1 e µx+iωx + C 2 e µx iωx = e ( µx C 1 e +iωx + C 2 e iωx) 41 3 42

20 f(x, y, ) x x f(x + h, y, ) f(x, y, ) lim h 0 h x f x () f x, f x 75 u u = u xx + u yy = 0 u xx x 2 x (1) u = x 2 + y 2 (2) u = log x 2 + y 2 76 z = f(x, y) = 4 x 2 y 2 (1) 2 x 2 2 y 2 (2) x y (3) (2, 0) (4) (1) x f ( ) 2 f(x, y) df(x, y) = f(x, y) dx + x f(x, y) dy (26) y (1) f = x 2 + y 2 (y ) (2)f = a 2 + x 2 (a ) (1) (x2 + y 2 ) y = 2y (2) (a2 + x 2 ) a = 2a 43 44

21 f(x) x = a () f(x) = f(a) + f (a) 1! (x a) + f (a) 2! (x a) 2 + f (n) (a) (x a) n + (27) n! x = a f(a) x = a + x f(x) = f(a + x) f(x) = f(a + x) x = a f(a) 77 y = f(x) x = a y f(a) x = a+ x y f(a + x) f(a + x) f(a) + df(x) dx x 78 f(x) = cos x g(x) = (1 + x) α x = a 79 a b (1) a 2 + b 2 (a 2 + b 2 ) 1/2 80 x 0 (1) 1 1 + x (2) (3) sin x(x ) 1 + x f(x) = sin x x = 0 (27) x = a (x a) (x a)2 (x a)3 f(x) = sin a + cos a sin a cos a + 1! 2! 3! a = 0 f(x) = x x3 6 + x 1 x 2 0 (1 ± x) α 1 ± αx, x 0 e x 1 + x, x 0 sin θ θ, θ 0 cos θ 1, θ 0 tan θ θ, θ 0 f(x) = sin x x 45 46

22 1 i a b a + ib e ±iθ = cos θ ± i sin θ (28) 81 sin θ cos θ 82 e iθ cos θ sin θ 83 i 84 e x x = iθ cos θ sin θ ( ) 85 sinh x cosh x sinh x = ex e x, cosh x = ex + e x 2 2 sinh iθ cosh iθ E 0 ω E E 0 e iωt E 0 (cos ωt + i sin ωt) E 0 cos ωt E 0 cos ωt 47 48

23 v 2 φ = 1 v 2 2 φ t 2 (29) 2 φ x 2 = 1 2 φ v 2 t 2 (30) φ t = K 2 φ (31) 86 x Ak ω 87 E(r, t) k 88 E(r, t) = E exp i(k r ωt) B = ε 0 µ 0 E t E = B t ( ) K φ t = K 2 φ x 2 (32) (30) φ(x, t) = A sin(kx ωt) = Ak 2 sin(kx ωt), = 1 v 2 ω2 A sin(kx ωt) v = ω k 49 4 50