II 2 II
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- しげのぶ たけくま
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1 II 2 II 2005 yugami@cc.utsunomiya-u.ac.jp
2 i ii
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5 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 ) φ r r r r 2 ϕ 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 (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 ()
6 i = dq dt dv dt = v + (v )v t () 1 2 () 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
7 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 A 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 = = = 4 (2) A B = 1 ( 1) = = 2 9 a b cos θ 0 10
8 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 (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) a b sin θ(= a b ) 2 12
9 5 ( ) ( ) 1 (x, y, z) φ(x, y, z) (x, y, z) f(x, y, z) () () s ( ) = x, y, z (4) r P A(r) 19 a = (2, 1, 0) 20 A (B C) = (A C)B (A B)C ( A) = ( A) 2 A ( III) 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
10 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) (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 ( x z = 10 exp ( x2 + y 2 ) 4 28 z = 10 exp ( x2 + y 2 ) z 4 xy z z = 10 exp ( x2 + y 2 ) (x y 4 ) grad 15 16
11 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 A 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
12 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
13 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
14 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 = b a f(x)dx 24
15 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 = 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
16 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 = xdx (x 3y)dy = 1/x = 3 2 log x y 2 ( x 2x ) 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 z > 0 xy xy... dxdy (23 ) 28
17 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 x y (x + y + z)dxdydz = 1 8 (23 ) 29 30
18 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 =
19 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 dxdy = f (5) a (6) 2a A d A d 33 34
20 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
21 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
22 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
23 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)
24 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
25 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) 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
26 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
27 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
( : December 27, 2015) CONTENTS I. 1 II. 2 III. 2 IV. 3 V. 5 VI. 6 VII. 7 VIII. 9 I. 1 f(x) f (x) y = f(x) x ϕ(r) (gradient) ϕ(r) (gradϕ(r) ) ( ) ϕ(r)
( : December 27, 215 CONTENTS I. 1 II. 2 III. 2 IV. 3 V. 5 VI. 6 VII. 7 VIII. 9 I. 1 f(x f (x y f(x x ϕ(r (gradient ϕ(r (gradϕ(r ( ϕ(r r ϕ r xi + yj + zk ϕ(r ϕ(r x i + ϕ(r y j + ϕ(r z k (1.1 ϕ(r ϕ(r i
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
( 12 ( ( ( ( Levi-Civita grad div rot ( ( = 4 : 6 3 1 1.1 f(x n f (n (x, d n f(x (1.1 dxn f (2 (x f (x 1.1 f(x = e x f (n (x = e x d dx (fg = f g + fg (1.2 d dx d 2 dx (fg = f g + 2f g + fg 2... d n n
notekiso1_09.dvi
39 3 3.1 2 Ax 1,y 1 Bx 2,y 2 x y fx, y z fx, y x 1,y 1, 0 x 1,y 1,fx 1,y 1 x 2,y 2, 0 x 2,y 2,fx 2,y 2 A s I fx, yds lim fx i,y i Δs. 3.1.1 Δs 0 x i,y i N Δs 1 I lim Δx 2 +Δy 2 0 x 1 fx i,y i Δx i 2 +Δy
II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x (
II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, )
Chap11.dvi
. () x 3 + dx () (x )(x ) dx + sin x sin x( + cos x) dx () x 3 3 x + + 3 x + 3 x x + x 3 + dx 3 x + dx 6 x x x + dx + 3 log x + 6 log x x + + 3 rctn ( ) dx x + 3 4 ( x 3 ) + C x () t x t tn x dx x. t x
y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =
[ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =
120 9 I I 1 I 2 I 1 I 2 ( a) ( b) ( c ) I I 2 I 1 I ( d) ( e) ( f ) 9.1: Ampère (c) (d) (e) S I 1 I 2 B ds = µ 0 ( I 1 I 2 ) I 1 I 2 B ds =0. I 1 I 2
9 E B 9.1 9.1.1 Ampère Ampère Ampère s law B S µ 0 B ds = µ 0 j ds (9.1) S rot B = µ 0 j (9.2) S Ampère Biot-Savart oulomb Gauss Ampère rot B 0 Ampère µ 0 9.1 (a) (b) I B ds = µ 0 I. I 1 I 2 B ds = µ 0
i
009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3
1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,
[ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b
2.4 ( ) ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) Minoru TANAKA (Osaka Univ.) I(2011), Sec p. 1/30
2.4 ( ) 2.4.1 ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) I(2011), Sec. 2. 4 p. 1/30 (2) Γ f dr lim f i r i. r i 0 i f i i f r i i i+1 (1) n i r i (3) F dr = lim F i n i r i. Γ r i 0 i n i
II 1 3 2 5 3 7 4 8 5 11 6 13 7 16 8 18 2 1 1. x 2 + xy x y (1 lim (x,y (1,1 x 1 x 3 + y 3 (2 lim (x,y (, x 2 + y 2 x 2 (3 lim (x,y (, x 2 + y 2 xy (4 lim (x,y (, x 2 + y 2 x y (5 lim (x,y (, x + y x 3y
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06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
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![, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x](/thumbs/93/113428934.jpg ", 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x")
1 1.1 4n 2 x, x 1 2n f n (x) = 4n 2 ( 1 x), 1 x 1 n 2n n, 1 x n n 1 1 f n (x)dx = 1, n = 1, 2,.. 1 lim 1 lim 1 f n (x)dx = 1 lim f n(x) = ( lim f n (x))dx = f n (x)dx 1 ( lim f n (x))dx d dx ( lim f d
50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
i 18 2H 2 + O 2 2H 2 + ( ) 3K
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18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb
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r 1 r 2 r 1 r 2 2 Coulomb Gauss Coulomb 2.1 Coulomb 1 2 r 1 r 2 1 2 F 12 2 1 F 21 F 12 = F 21 = 1 4πε 0 1 2 r 1 r 2 2 r 1 r 2 r 1 r 2 (2.1) Coulomb ε 0 = 107 4πc 2 =8.854 187 817 10 12 C 2 N 1 m 2 (2.2)
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1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
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2.5 (Gauss) (flux) v(r)( ) S n S v n v n (1) v n S = v n S = v S, n S S. n n S v S v Minoru TANAKA (Osaka Univ.) I(2012), Sec p. 1/30
2.5 (Gauss) 2.5.1 (flux) v(r)( ) n v n v n (1) v n = v n = v, n. n n v v I(2012), ec. 2. 5 p. 1/30 i (2) lim v(r i ) i = v(r) d. i 0 i (flux) I(2012), ec. 2. 5 p. 2/30 2.5.2 ( ) ( ) q 1 r 2 E 2 q r 1 E
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[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2
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W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
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) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)
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II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
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II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
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13 13.1 O r F R = m d 2 r dt 2 m r m = F = m r M M d2 R dt 2 = m d 2 r dt 2 = F = F (13.1) F O L = r p = m r ṙ dl dt = m ṙ ṙ + m r r = r (m r ) = r F N. (13.2) N N = R F 13.2 O ˆn ω L O r u u = ω r 1 1:
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) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
![) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4](/thumbs/92/107866707.jpg ") ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4")
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