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3 ( ) A 89 A A A A A A A A A

4 4 A A

5 ,2 II, III

6 1 6 ( ) 3 1. ( ) 2. 3.

7 1 7 m g mg ( 9.8 [m/s 2 ]) 1 [kg] 1 [kg] 9.8 [m/s 2 ]

8 ( ) (,, 2006)

9 ( ) ( ) 2 3

10 2 10 ( ) ( ) ( ) ( ) 2.2

11 ( 3. ( ) ( ) ( ) 4. ( ) 5.

12 2 12

13 km 30km ( )30km 30 km/ ( ) 2[m] 1[m] 2 3.5[m] 1[m] (SI) SI 7

14 3 14 m kg s ( ) A K mol cd m 2 ( ) kg / m 3 ( ) m SI P T G M k h da d c m n p f a m 2km 0.05m 5cm 1 40Mm 40,000 km (0 2π, π ) ( 1)

3 15 1: 1 ( ) Hz ( ) ( ) 1 50 50 Hz /s [1/s] [Hz] SI

15 3 15 1: 1 ( ) Hz ( ) ( ) Hz /s [1/s] [Hz] SI 1. MKSA SI SI m kg s ( ) A MKSA 2. CGS cm g s ( ) CGS

16 3 16 [ ] [m/s] 100km (a) (b) (c) [ /s] 3.3 1[l]( ) 1[kg] l SI ( 1 [kg/l] ) SI [kg/m 3 ] 1[l] [m 3 ] 1[l] = 1000 [cm 3 ] 1[m 3 ] [cm 3 ] 1[m 3 ] 1 1[m] 1[m] = 100[cm] [c] 0.01 [cm] 0.01m 1[cm] = 1[0.01m] = 0.01[m] 100[cm] = 100[0.01m] = [m] = 1[m] [0.01m]

17 3 17 1[0.01m](0.01m 1 ) 0.01[m](1m 0.01 ) 1 100[cm] [cm 3 ] = [cm 3 ] [cm 3 ] = [(0.01m) 3 ] = [( )m 3 ] = [m 3 ] = 1[m 3 ] ((0.01m) 3 ) [cm 3 ] = 1[m 3 ] 1000[cm 3 ] = 1[l] [cm 3 ] = 1[m 3 ] [cm 3 ] = 1000[l] 1[m 3 ] = 1000 [l] 1[m 3 ] = 1000[l] [ ] 1 = m3 [l] [ m3] [ ] [ ] kg kg 1 = 1 1 l 1000 [ m3 = kg ] m 3 ] [ kg = 1000 m 3 1[l] 1[kg] 1000[l] 1000[kg] 1000[l] 1[m 3 ] 1000[kg/m 3 ] 1.2 [kg/m 3 ] [g/l]

18 (0) ( ) 1. A p A q = A p+q A p A q p + q 2. A p A q = A p q p q p + q {}}{ A A { A }} A { = A { }} A { A p A q p q p q p q {}}{ A A { A }} A { = { A }} A { 3. A 0 = 1 p = q Ap = A p p = A 0 1 A p A 0 = 1 4. A q = 1 A q A p A q = A p q p = 0 1 A q = A q 5. (AB) p = A p B p A B p A p B p

19 (A p ) q = A pq A p q A pq = p q 7. A = A 1 2 A ( A) 2 = A (A 1 2 ) 2 = A A = A A ( 3 A) 3 = A 3 1 A = A [cm 3 ] = 10 6 [(10 2 m) 3 ] = 10 6 [10 6 m 3 ] = [m 3 ] = 1[m 3 ]

20 x[m] y[m] z[m] x y z x y x, y z y x (x y ) 4

21 4 21 ( 2 ) x x x (x ) (x ) x x x x y, z x,y,z : 2: ( ) x y y x x, y, z x y ( ) ( 2) ( ) x y z (x, y, z)

22 4 22 x y z 3: 4 4: ( ) p = x y z

23 4 23 p = x y z (,,, ) p = (x, y, z), q = (u, v, w) 1. a 5: a p = a = x y z ax ay az a a a 1:a

24 4 24 6: p + q = = x y z + x + u y + v z + w u v w x[m] y[m] z[m] u[m] v[m] w[m] x + u[m] y + v[m] z + w[m] p q p q p + q 3. 7: r p p -1 r

25 4 25 r r = p + q r p = q r p r p r p p r r p ( r) ( q) 5[m] 3[m] 5[m]-3[m] = 2 [m] ( ) as ( ) ( ) 1 ( ) 6 y x x ( ) y 7 3

26 : 1 1 ( 9 ) ( 9 ) 9: ( ) ( )

27 4 27 ( 10) 10: 1 1 2

28 4 28

29 ( 11) 11: ( ) ( ) ( ) ( 12) 100 [m] [m] 11 t x x t

30 : (A) (B) A B ( 13 ) B A 13: A B A B A ( 13 ) B A ( B A 0 ) f(t) t ( f(t) t )

31 4 31 x 0 lim x 0 t 0 f t f lim x 0 t f(t) t df(t) d f(t) dt dt t f t (t) df dt df f dt t t df dt f = lim x 0 t f(t) = t 3 df = dt 3t2 f(t) = t n df = dt ntn 1 1. x t (1) t p(t) t

32 4 32 x x x x t t t t v v v v t t t t a a a a t t t t 14: t + t p (t + t) t p(t) p (t + t) p(t) t 15: 15 t t p (t + t) p(t) t p t velocity v

33 4 33 p(t + t) p(t) v = lim t 0 t = d dt p(t) p(t) 3 p x (t), p y (t), p z (t) v x (t), v y (t), v z (t) v x (t) v y (t) v z (t) = d dt = p x (t) p y (t) p z (t) dp x (t) dt dp y(t) dt dp z (t) dt acceleration a α v(t + t) v(t) a = lim t 0 t = d dt v(t) a x (t) a y (t) a z (t) = d dt = v x (t) v y (t) v z (t) dv x (t) dt dv y (t) dt dv z(t) dt 2 ( 2 2 ) a = d dt v(t) = d dt d dt x(t)

34 4 34 = p dt x(t) p dt y(t) p dt z(t) d d dt d d dt d d dt d d d2 dt dt dt 2 a = d2 dt x(t) 2 d 2 p dt 2 x (t) d 2 = dt 2 p y (t) d 2 dt 2 p z (t) ( ) ( ) ( ) 8

35 4 35 v F v F t v t m v 1 m 3 v F t m m v t F F t v m lim t 0 t = mdv dt F

36 4 36 A B B A ( ) ( ) ( ) ( ) ( ) A B B A ( ) 4.6

37 4 37 5[m] + 3[kg] (+) (-) ( ) ( ) 100[m] 9.68[s] 100[m] 9.68[s] = 10.33[m/s] [m/s] [m/s] 2 2[m/s] 20[m/s] 20[m/s] 2[m/s] 2[s] [m/s 2 ] 20[m/s] 2[m/s] 2[s] = 9[m/s 2 ] m dv dt m dv dt C F = CF 100 1

38 4 38 ( 1 =100 ) C 1 1 1[kg] 1[m/s 2 ] 1 2[kg] 1[m/s 2 ] 1[kg] 2[m/s 2 ] 2 [ kg m s N ( ) [N] = 2 ] [ kg m s 2 ]

39 4 39 F = m a SI 4.7 t f(t) f(t) t ( 15) 16: f(t) 16 t ( t ) t = 0 ( 17) ( )

40 : f(t) t = 0 ( )

41 :

42 4 42 ( ) a ( 18) ( ) ( ) a

43 ( ) ( ) = ( ) ( ) ( ) 9.8 [m/s 2 ]

44 4 44 ( ) = ( ) ( ) ( ) ( ) = ( ) ( ) ( ) = ( ) ( ) ( ) ( 19) a t 19: ( ) ( 20) (

45 4 45 v t 20: ) ( 21) x t 21:

46 4 46 ( SF ) ( ) ( ) ( )

47 : 22 ( 22 ) ( ) ( ) ( )

48 4 48 ( ) 2 ( ) ( ) = ( ) ( )2 ( ) ( ) ( )2 ( )

49 4 49 : F F = G m 1m 2 r 2 12 G :, [m 3 /kg s] m 1, m 2 : 1,2 ( [kg]) r 12 : 1,2 ( [m]) M, m ( ) r v ( ) = ( ) ( ) G Mm v 2 = m r 2 r 1. m G M r 2 = v2 r m m 2. v r G, M v r

50 4 50 ( r v ) r v = G M r (2)

51 4 51 ( ) ( ) 3 ( ) 2 ( ) (2) 2π ( ) ( ) ( ) 3 ( ) 2 = = = ( ) ( ) 2 ( ) ( ) 2 1 (2π ) 2 ( ) (2π) 2 ( ) 2 1 (2π) 2 ( )2 ( ) ( ) 2 ( ) (2) 2 r v 2 = G M r v 2 r = GM = (2) ( ) [ ] [ ] ( ) 3 ( )

52 4 52 R (2) V = V G M R 8 [km/s] 8 [km/s] ( ) 2 2GM R ( )

53 F AB F BA 23: A, B F AB = m A a A F BA = m B a B F AB = F BA 0 = m A a A + m B a B A,B t v A, v B v A 0 = m A t + m v B B t 0 = m A v A + m B v B (3)

54 4 54 A, B v A0, v B0 v A0, v B0 m A v A0 + m B v B0 = m A v A + m B v B m A ( v A v A0 ) + m B ( v B v B0 ) = m A v A + m B v B (3) ( 1001 ) m/s

55 m/s SF 30 [kg] 9.8 [m/s] 30 [kg] 9.8 [m/s] 30 [kg] 30 [m 3 ] 1/10 3 [m 3 ] 100 [m/s]

56 ( ) ( ) ( ) ( ) ( )

57 4 57 [kg (m/s) 2 ] = [kgm/s 2 m] = [Nm] [ J ]( ) [J] = [Nm] = [kg(m/s) 2 ] ( )

58 4 58 ( ) ( ) (Coriolis Force) ( )

59 (20 ) 4.14 ( ) 13 ( ) ( ) ( ) ( 24) 24: ( 25) 13

60 : ( ) ( ) ( ) ( 26) 26: : ( ) ( ) ( )

61 4 61 ( ) ( 27) 27:

62 4 62

63 ( ) ( ) 14

64 5 64 ( ) 5.2 ( ) ( ) ( ): F F = 1 4πε 0 q 1 q 2 r 2 12 r 12 : 1,2 ( [m]) q 1, q 2 : 1,2 ( [C]) ε 0 : ( ) [C 2 /N m 2 ] r 12

65 5 65 [C]( ) ε 0 ( ) ( ) ( ) 5.3 : F F = 1 4πµ 0 m 1 m 2 r 2 12 r 12 : 1,2 m 1, m 2 : 1,2 (N, [Wb]) µ 0 : ( ), 4π 10 7 [N/A 2 ] [Wb]( ) 4π 10 7

66 5 66 ( ) N S ( ) : N ( 28)

67 5 67 a : 28 F F = m 4π qv r R 2 r q : ( [C]) m : (N, [Wb]) v : r : R : a a a 29:

68 5 68 a : F F = m 4π qv r R 2 r q : ( [C]) m : (N, [Wb]) v : r : R : a a a (4.14 ) ( 30) 2. ( 31)

69 : 31: 7.3

70 ( q 1 ) q 2 q 2 (1[C]) 1[C] ( ) ( ) ( ) ( Q) ( 1[C] : ) E = Q 4πε 0 1 (x 2 + y 2 + z 2 ) 1 ( ) x2 + y 2 + z (x,y,z) x y z

71 x : ( ) x :

72 ( ) ( ) 3. ( ) ( ) (2 ) ( ) x :

73 5 73 (1[C]) (φ) A B ( 2 ) A B (φ) 1. φ 2. φ q q 3. [J/C] [V] ( 15 ) A B ( 2 ) A B ( ) ( ) ( ) ( )2 15

74 5 74 ( ) ( ) ( ) ( ) ( ) [V] 100V ( 200V) 100[V] 1 [C/s] [A]( ) 5.7.2

75 : ( ) ( ) ( ) ( )

76 5 76 ( ) ( ) ( ) ( ) E I R E = R I 5.7.3

77 5 77 ( ) 36: 37:

78 : (B) (I) (F) ( ) 16

79 F = m 1, m 2 [Wb] = [J/A] : [Wb 2 /N m 2 ] 1 4πµ 0 m 1 m 2 r 2 12 : [N/Wb] H = µ 0 4π m r 2 12 : [A/m] = [N/Wb] H = I 2πr : [Wb] = [N m/a] = [kg m 2 /A s 2 ] : [Wb 2 /N m 2 ] = [Wb/A m] = [kg m / A 2 s 2 ] = [N/A 2 ] F = qvb : [N/A m] = [Wb/A m][n/wb] B = µ 0 H

80 ( ) ( ) km : x ( ) ( ) ( )

81 6 81 kx = m d2 x dt 2 x x x ( ) k m x d2 x dt 2 x x 2 x ( ) x = A sin ωt ω d 2 dt A sin ωt = d (A ddt ) 2 dt sin ωt = d (Aω cos ωt) dt = Aω d cos ωt dt = Aω 2 sin ωt x = A sin ωt k(a sin ωt) = m( Aω 2 sin ωt) k = mω 2 m ω = k x = A sin m k t 1. ( ) 2. (sin) A sin ωt ( ) n n n

82 6 82 f(x) f(x) + C n dn dt n n A, φ 0 2 x = A sin(ωt + φ 0 ) v = dx dt = Aω cos(ωt + φ 0 ) (t = 0) A, φ x y ( ) F = mω 2 A F F F F x x A x = (A cos ωt, A sin ωt) x (A cos ωt) 2 + (A sin ωt) 2 = A 2 (cos 2 ωt + sin 2 ωt) = A 1 x 1 A F F F (F x, F y ) F x = F x A F y = F y A

83 6 83 x, y F x A = md2 x dt 2 F y A = md2 y dt 2 x y 6.3 ( ) 2 T l y l T x l = m d2 x dt 2 mg = m d2 y dt 2 θ sin θ = x l y, cos θ = l l T T T θ x y y 17 y y = 0 T mg = 0 17

84 6 84 x T = mg g l x = d2 x dt 2 x = A sin g l t x y 6.4 :, P ( ) : S z = A sin(ωt + P (x)) φ(x)

85 6 85 ( ) z = A sin(kx + Q(t))

86 ( ) 7.2

87 7 87 ( ) ( ) ( ) ( ) 7.3 ( )

88 7 88 ( ) ( ) 7.4 ( ) ( ) 20

89 A 89 A A.1 α A A β B B γ Γ Γ δ ɛ(ε) E E ζ Z Z η H H θ(ϑ) θ Θ ι I I κ K K λ Λ Λ µ M M ν N N ξ Ξ Ξ o O O π(ϖ) Π Π ρ(ϱ) P P σ(ς) Σ Σ τ T T υ Υ Υ φ(ϕ) Φ Φ χ X X ψ Ψ Ψ ω Ω Ω A.2 ( ) 3 (, ) 3

90 A 90 A A p A q = A p+q 2. A p A q = A p q 3. A 0 = 1 4. A q = 1 A q 5. (AB) p = A p B p 6. (A p ) q = A pq 7. A = A 1 2, 3 A = A 1 3, A.4 x y y x θ sin θ, cos θ, tan θ θ 1 ( ) 90 π/2, 180 π, 360 2π 6 1. co 2.

91 A (a) sin 2 θ + cos 2 θ = 1 (b) tan 2 θ = sec 2 θ (c) cot 2 θ = cosec 2 θ sin(α + β) = sin α cos β + cos α sin β cos(α + β) = cos α cos β sin α sin β e iθ = cos θ + i sin θ( i i 2 = 1 ) e i(α+β) = cos(α + β) + i sin(α + β) e iα e iβ = (cos α + i sin α) (cos β + i sin β) = cos α cos β sin α sin β + i ( cos α sin β + sin α cos β )

92 A 92 A.5 f(x) x f(x) x x f(x) f(x) x x x f(x) f f x f x x f ( f/ x, ) x dx n dx d sin x dx de x dx = nx n 1 = cos x = e x d (af(x)) = adf(x) dx dx d df(x) (f(x) + g(x)) = dx dx d df(x) (f(x) g(x)) = dx dx a df(g(x)) dx = df(g) dg + dg(x) dx dg(x) g(x) + f(x) dx dg(x) dx A t n 2 a x = f(t) = at 2

93 A 93 40: t (t, f(t)) t t = t o A (t o, at 2 o ) B A t = t o + t t 18 B (x) a(t o + t) 2 AB A B t a(t o + t) 2 at 2 o a(t o + t) 2 at 2 o t B A t 2 a(t o + t) 2 at 2 o t = a(t2 o + 2t 0 t + t 2 ) at 2 o t = at2 o + 2at 0 t + a t 2 at 2 o t = 2at 0 t + a t 2 t = 2at 0 + a t t t t t 2at 0 + a t t t 2at o 18 t difference d

94 A 94 lim t 0 a(t o + t) 2 at 2 o lim t 0 t = lim t 0 {2at 0 + a t} = 2at o t o t x = f(t), f(t) = at 2 t 2at A.5.2 f(t) f (t) x = at 2 f(t) = at 2 f (t) = 2at x = at 2 dx = 2at dt d dx x ( ) dt t ( ) dx x f(t) dt f(t) = at 2 df(t) dt = 2at (f(t)) d f(t) = 2at dt A dt n dt = nt n 1

95 A 95 t h (t + h) n t n lim h 0 h (t n + nht n 1 + n(n 1) h 2 t n 2 ) t n 2 = lim h 0 h nht n 1 + n(n 1) h 2 t n 2 2 = lim h 0 { h } = lim nt n 1 n(n 1) + ht n 2 h 0 2 = nt n f(t) = at f (t) = a y = ax + b 1 a 2 2. f(t) = at 2 f (t) = 2at d sin t dt = cos t sin(t + h) sin t lim h 0 h = lim h 0 (sin t cos h + cos t sin h) sin t h = lim h 0 sin t cos h sin t + cos t sin h h = lim sin t cos h 1 + lim cos t sin h h 0 h h 0 h cos h 1 sin h = sin t lim + cos t lim h 0 h h 0 h lim cos h 1 h 0 h sin h, lim h 0 h lim h 0 cos h 1 h =

96 A 96 cos(0+h) cos 0 lim cos t h 0 h sin h sin(0+h) sin 0 lim = lim h 0 h h 0 h 1 d sin t dt 3. = cos t f(t) = a t t t t a a f(t) a f(t) f(t) f(t) f (t) = f(t) a e e = f(t) = e t f (t) = e t A.5.4 f f(t + t) f(t) (t) = lim t 0 t t t A B f (t) f(t + t) f(t) t f (t) t f(t + t) f(t) f(t) + f (t) t f(t + t) f(t), f (t) t ( t) (f(t + t))

97 A 97 A d (af(t)) = adf(t) dt dt af(t 0 + t) af(t o ) lim t 0 t = lim af(t 0 + t) f(t o ) t 0 t f(t 0 + t) f(t o ) = a lim t 0 t d df(t) (f(t) + g(t)) = + dg(t) dt dt dt (f(t 0 + t) + g(t 0 + t)) (f(t o ) + g(t o )) lim t 0 t (f(t 0 + t) f(t o )) + (g(t 0 + t) g(t o )) = lim t 0 t f(t 0 + t) f(t o ) g(t 0 + t) g(t o ) = lim + lim t 0 t t 0 t 3. d df(t) (f(t) g(t)) = g(t) + f(t) dg(t) dt dt dt (f(t 0 + t) g(t 0 + t)) (f(t o ) g(t o )) lim t 0 t f(t 0 + t) f(t 0 ) + f (t o ) t ((f(t 0 ) + f (t o ) t) (g(t 0 ) + g (t o ) t)) (f(t o ) g(t o )) t = f(t 0)g(t 0 ) + f(t o )g (t o ) t + f (t o )g(t o ) t + f (t o )g (t o ) t 2 f(t o )g(t o ) t = f(t o)g (t o ) t + f (t o )g(t o ) t + f (t o )g (t o ) t 2 t = f(t o )g (t o ) + f (t o )g(t o ) + f (t o )g (t o ) t

98 A 98 f(t o )g (t o ) + f (t o )g(t o ) 4. df(g(t)) dt = df(g) dg dg(t) dt f g df(g) g dg g f f df(g) dg 2 f 2 g g t dg(t) dt dg(t) 3 g 3 t dt f 2 g 2(3 t) 6 t f df(g) dg dg(t) t dt 19 A cos t cos t = sin ( ) t + π f(g) = sin g, 2 g(t) = t + π 2 d dt cos t = d (t dt sin + π ) 2 = d dt f(g(t)) ( ) ( ) df dg = dg dt 19

99 A 99 ( ) ( d d = dg sin g dt = (cos g) (1) = cos g ( = cos t + π ) 2 = sin t ( t + π ) ) sin t t f(g) = 1 = g g 1, g(t) = sin t d 1 dt sin t = d dt f(g(t)) = df(g) dg dg dt ( ) ( ) d d = dg g 1 dt sin t = ( ( 1)g 2) (cos t) = 1 sin 2 t cos t 3 d dt f(at + b) = af (at + b) d dt d dt 1 = f (t) f(t) f 2 (t) g(t) = g (t)f(t) f (t)g(t) f(t) f 2 (t)

100 100 N, 38, 35, 20, 88, 14, 86, 14, 74, 77, 56, 21, 75, 65, 65, 46, 56, 73, 34, 7, 36, 39, 53, 54, 54, 54, 57 SI, 13 SI, 14 x, 21 x, 21, 56, 57, 57 MKSA, 15, 58, 5, 46, 75, 59, 59, 87, 33, 29, 87, 20, 20, 84, 87, 34, 57, 34, 57, 46, 55, 59, 14, 14, 13, 9, 30, 89, 76, 14, 64, 64, 13, 29, 51, 14, 63, 20, 86, 77, 88, 14, 75, 13, 11, 20

101 101, 14, 58, 21, 20, 36, 36 3, 21, 73 CGS, 15, 65, 65, 34, 11, 18, 90, 96, 90, 96, 79, 7, 14, 35, 46, 22, 89, 11, 70, 79, 51, 22, 89, 75, 15, 44, 46, 7, 43, 7, 43, 70, 52, 73, 57, 40, 44, 11, 65, 49, 80, 22, 70, 11, 22, 39, 65, 14, 88, 11, 88, 52, 31, 32, 32, 52, 92, 94, 52, 84, 22, 13, 66, 81, 83, 47, 83, 75, 63, 75, 13, 13, 89, 14, 86, 14, 14, 76, 73, 11, 63, 71, 58, 63, 11

102 102, 63, 79, 75, 64, 70, 11, 74, 11, 84, 47, 29, 75, 72, 72, 9, 74, 14, 80 2, 33 2, 92, 20, 38, 56, 11, 70, 88, 80, 33, 88, 36, 48, 63, 70, 67, 58, 14, 21, 30, 30, 14, 88, 80, 80, 14, 13, 52, 52, 14, 22, 89, 70, 14, 15, 80, 86, 54, 76, 73, 14, 63, 58, 21, 14, 14, 14, 65, 65, 75, 84, 21, 63, 74, 14, 42, 42, 11, 20, 56

103 103, 11, 68, 78

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. 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)