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i 1 1 1.1................................... 2 1.2............................... 3 1.2.1.................................... 3 1.2.2............................... 4 1.2.3 SI.............................. 4 1.3............................. 5 1.4................................... 7 1.5........................................ 7 1.5.1................................ 7 1.5.2........................... 7 1.5.3.................................. 8 2 9 2.1...................................... 9 2.1.1............................... 9 2.1.2.......................... 10 2.1.3................................ 11 2.1.4................................ 12 2.2.................................... 15 2.2.1.................................. 15 2.2.2............................... 16 2.3.................................... 16 2.3.1............................. 17 2.3.2........................... 18 2.3.3..................... 18 2.4............................. 19 2.4.1.................................. 19 iii

2.4.2................................ 20 2.5........................................ 21 2.5.1................................... 21 2.5.2............................... 21 2.5.3................................ 21 2.5.4................................ 21 3 23 3.1.................................... 23 3.1.1................................ 23 3.1.2..................................... 24 3.1.3.................................. 24 3.2 3................................... 25 3.3............................. 26 3.4.................................. 27 3.4.1............................ 27 3.4.2........................... 27 3.4.3........................ 28 3.4.4.................................... 30 3.4.5 ( )............................ 32 3.4.6........................... 34 3.5............................. 34 3.5.1........................... 36 3.6........................ 37 3.6.1.................................. 38 3.7........................................ 39 4 43 4.1................................. 43 4.1.1.................................. 44 4.1.2............................. 45 4.1.3............................... 50 4.1.4......................... 51 4.2................................. 52 4.3............................... 53 4.4........................................ 57 4.4.1 1....................................... 57 iv

5 71 5.1................................... 71 5.2........................................ 71 5.2.1...................... 72 5.3............................. 73 5.3.1........................... 75 5.4............................. 78 5.4.1................................ 79 5.5............................. 80 5.5.1.................................. 80 5.5.2....................... 82 5.6...................................... 83 5.7................................... 84 5.7.1 85 5.7.2..................... 86 5.8........................................ 86 5.8.1............................ 86 5.8.2........................... 87 5.8.3........................ 89 5.8.4........................... 89 5.8.5................................ 90 5.8.6................................... 90 5.8.7.................................... 90 5.8.8.................................. 91 6 93 6.1.................................. 93 6.1.1.................................. 94 6.2................................... 95 6.3........................................ 97 6.3.1............................. 97 6.3.2............................... 97 6.3.3.............................. 97 6.3.4............................... 97 7 103 7.1.................................. 103 7.2.................................. 103 v

7.3.................................... 104 7.3.1.................................. 105 7.3.2................................ 106 7.3.3................................ 107 7.4.................................. 107 7.5............................. 109 7.5.1.................................. 109 7.5.2.................................. 110 7.6........................................ 113 7.6.1.......................... 113 7.6.2................................ 113 7.6.3................................... 113 7.6.4................................... 113 8 115 8.1...................................... 116 8.1.1............................... 116 8.2............................... 117 8.2.1............................. 118 8.2.2........................... 118 8.3.................................. 119 8.3.1.............................. 119 8.4........................................ 120 8.4.1..................... 122 8.4.2............................... 123 8.5.................................. 125 8.5.1........................... 125 8.5.2............................. 126 8.5.3............................. 126 8.5.4......................... 127 8.6........................................ 127 8.6.1.................................. 127 8.6.2 1................................... 127 8.6.3 2................................... 128 8.6.4................................ 128 vi

9 129 9.1.................................... 129 9.2....................................... 130 9.3.................................. 131 9.4...................................... 133 9.5...................................... 134 9.5.1.................................. 134 9.6........................................ 135 9.6.1................................... 135 9.6.2............................... 135 9.6.3.................................. 136 9.6.4.................................. 136 9.6.5............................... 136 10 137 10.1........................................ 137 10.2............................... 138 10.3................................. 138 10.3.1.................................... 139 10.4................................... 139 10.4.1............................. 139 10.4.2.................................... 140 10.4.3.................................. 141 10.4.4.................................. 141 10.5........................................ 143 10.5.1................................... 143 10.5.2.................................... 143 10.5.3.............................. 143 10.5.4.................................... 144 10.5.5................................... 144 10.5.6................................... 144 11 153 11.1................................... 153 11.1.1............................... 153 11.1.2............................... 154 11.1.3............................... 154 11.2....................................... 155 vii

11.2.1................................ 155 11.2.2............................... 156 viii

1 [ ] 1

1.1 19 2

1.2 1.2.1 3

A = BC, (1.1) d F = DE, dt (1.2) A B C F D E 1.2.2 MKS (SI) 1.2.3 SI Kg 4

1 299792458 1 Kg Kg 90 10 mm 1 91 9263 1770 m s m s s m s 2 x = m 2 1.3 5

a b c = L a M b T c 1m 1m 2m m m m (1.3) 1m 1m 2 1m 1m 2 1m 2m 2 1m + 2m 2 (1.4) l 1 l 2 l 1 + l 2 (1.5) A + A 2 (1.6) exp A = (1 + A + A2 2! + A3 +...) : A (1.7) 3! A A [A] [A] = L a 1 M b 1 T c 1, [B] = L a 2 M b 2 T c 2 (1.8) [AB] = L a 1+a 2 M b 1+b 2 T c 1+c 2 (1.9) A B = La 1 a 2 M b 1 b 2 T c 1 c 2 (1.10) 6

1.4 1.5 1.5.1 1.5.2 7

1.5.3 8

2 2.1 2.1.1 9

0 1, 2, 3 x = x(t) (2.1) t t x(t) 2.1.2 x(t) t x v = x t = x(t + t) x(t) t (2.2) t t 1 x(t) t 2 v t t + t t x(t) t x(t + t) x(t) v = lim t 0 t = dx dt t x(t) t (2.3) v(t + t) v(t) a = lim t 0 t = dv dt (2.4) 10

t t a = (2.5) dv dt = a = v = at + v 0 (2.6) (2.7) dx dt = v(t) = v = at + v 0 x = 1 2 at2 + v 0 t + x 0 (2.8) 2.1.3 t n d dt t = 1 (2.9) d dt t2 = 2t, d dt t3 = 3t 2 (2.10) d dt tn = nt n 1 (2.11) x (x + h) n = x n + nx n 1 h + n(n 1) x n 2 h 2 + (2.12) 2 d dt t = lim (t + t) (t) = 1 t 0 t (2.13) d (t + t) 2 (t) 2 dt t2 = lim = 2t t 0 t (2.14) d (t + t) 3 (t) 3 dt t3 = lim = 3t 2 t 0 t (2.15) d dt tn = nt n 1 (2.16) 11

n z d dt tz = zt z 1 (2.17) sin(α + β) = sin(α) cos(β) + cos(α) sin(β) (2.18) cos(α + β) = cos(α) cos(β) sin(α) sin(β) (2.19) sin( θ) = θ + O(( θ) 2 ) (2.20) cos( θ) = 1 + O(( θ) 2 ) (2.21) O(( θ) 2 ) θ d dt d cos at = lim dt t 0 sin a(t + t) sin a(t) sin at = lim t 0 t cos a(t + t) cos a(t) t = a cos at (2.22) = a sin at (2.23) (2.21) 2.1.4 12

1 2 x(t) = a 2 t2 + bt + c (2.24) v(t) = d x(t) = at + b (2.25) dt a(t) = d v(t) = a (2.26) dt 2 R ω t x(t) = R cos ωt (2.27) v(t) = d x(t) = Rω sin ωt (2.28) dt a(t) = d dt v(t) = Rω2 cos ωt (2.29) T = 2π ω T 13

3 x(t) = e at (2.30) v(t) = d dt x(t) = aeat = ax(t) (2.31) a(t) = d dt v(t) = a2 e at = a 2 x(t) (2.32) x(t) = v 0 (t 1 a e at ) (2.33) v(t) = d dt x(t) = v 0(1 e at ) (2.34) z(t) = d dt v(t) = v 0ae at (2.35) t 14

t x(t) = x 0 log(t) (2.36) t = e x(t) x 0 (2.37) 1 = e x(t) x 0 d x(t) (2.38) dt x 0 d dt x(t) = x 0e x(t) x 0 = x 0 t (2.39) 2.2 2.2.1 (x, y, z) (x i ; i = 1, 3) x(t) = (x(t), y(t), z(t))(= (x 1 (t), x 2 (t), x 3 (t))) (2.40) v(t) = d dt x(t) = ( d dt x(t), d dt y(t), d dt z(t)) = ( d dt x 1(t), d dt x 2(t), d dt x 3(t)) (2.41) 15

x i (t), v i (t), a i (t), i = 1, 3 (2.42) d dt x i(t) = v i (t), (2.43) d dt v i(t) = a i (t) (2.44) a(t) = d dt d2 d2 d2 d2 v(t) = ( x(t), y(t), z(t))(= ( dt2 dt2 dt2 dt x 1(t), d2 2 dt x 2(t), d2 2 dt x 3(t))) (2.45) 2 2.2.2 t x 1 (t) = (2t, 3, 4.9t 2 ), x 2 (t) = (0, 3t, 4.9t 2 + 5t) (2.46) v 1 (t) = (2, 0, 9.8t), v 2 (t) = (0, 3, 9.8t + 5) (2.47) a 1 (t) = (0, 0, 9.8), a 2 (t) = (0, 0, 9.8) (2.48) 2.3 R m 16

θ t θ = ωt (2.49) ω 2.3.1 δt x(t + δt) x(t) (2.50) δt δt vδt = Rωδt (2.51) v = Rω (2.52) Rω θ t θ = ωt (2.53) π 2 17

2.3.2 t a = Rω 2 (2.54) θ = ωt (2.55) 2.3.3 R ω 18

2.4 2.4.1 x(t) g x 0 v 0 x(t) = 1 2 gt2 + v 0 t + x 0 (2.56) d dt x(t) = gt + v 0 (2.57) x 0 = v 0 = 0 (2.58) g = 9.8 m 2 /sec t = 1sec, v = 9.8m/sec, x = 4.9m (2.59) t = 10sec, v = 98m/sec, x = 490m (2.60) t = 100sec, v = 980m/sec, x = 49000m (2.61) 1 3600(60 60) 3600 v = 9.8m/sec = 9.8 3600m/ = 36Km/ (2.62) v = 98m/sec = 360Km/ (2.63) v = 980m/sec = 3600Km/ (2.64) 340m/sec 490m 360Km/ 100Km/ 19

2.4.2 F = b v (2.65) a = g b m v (2.66) v = mg b (2.67) 20

2.5 2.5.1 a, b, c, d n sin ax, cos bx, exp(cx + d), log x, x n 2.5.2 x 1 (t) = (1, 2t + 3t 2, 4t), x 2 (t) = (5 sin 2t, 5 cos 2t, 0), x 3 (t) = (6 sin 2t, 6 cos 2t, 4t) 2.5.3 ω 2.5.4 z(t) = (2.68) 21

d sin ax = a cos ax, dx d cos bx = b sin bx, dx d exp(cx + d) = c exp(cx + d), dx d dx log x = 1 x, d dx xn = nx n 1 2 x 1 (t) = (1, 2t + 3t 2, 4t), (2.69) d dt x1 (t) = (0, 2 + 6t, 4), d2 dt 2 x1 (t) = (0, 6, 0) (2.70) x 2 (t) = (5 sin 2t, 5 cos 2t, 0), (2.71) d dt x2 (t) = (10 cos 2t, 10 sin 2t, 0), d2 dt 2 x2 (t) = ( 20 sin 2t, 20 cos 2t, 0) (2.72) x 3 (t) = (6 sin 2t, 6 cos 2t, 4t) (2.73) d dt x3 (t) = (12 cos 2t, 12 sin 2t, 4) (2.74) d 2 dt 2 x3 (t) = ( 24 sin 2t, 24 cos 2t, 0) (2.75) 22

3 3.1 a m F F a m a = F m (3.1) (3.2) 3.1.1 23

9.8m/s 2 1/7 3.1.2 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 3.1.3 24

F = kx, (3.3) F : k : x : (3.4) F = kx (3.5) F = mg (3.6) 3.2 3 F = ma (3.7) 25

3.3 26

3.4 3.4.1 A = A 1, A 2, A 3 (3.8) 3.4.2 a + b = c (3.9) 27

a λ λ a = c (3.10) c λ a λ a a a = b + c (3.11) a + b = c (3.12) 3.4.3 e 1 x x e 1 = x (3.13) 28

e 2 y y e 2 = y (3.14) r = x + y, (3.15) x = x e 1, y = y e 2 (3.16) e 1 e 2 e 1 e 2 29

3.4.4 θ ( a, b) = ab cos θ (3.17) a : a, b : b θ (3.18) a cos θ a b ( a + b, c) = ( a, c) + ( b, c) (3.19) ( a, b + c) = ( a, b) + ( a, c) (3.20) ( a, b) = ab cos θ = 0 (3.21) 30

e 1 = (1, 0, 0), e 2 = (0, 1, 0), e 3 = (0, 0, 1) (3.22) ( e 1, e 1 ) = 1, ( e 1, e 2 ) = 0, ( e 1, e 3 ) = 0 (3.23) ( e 2, e 1 ) = 0, ( e 2, e 2 ) = 1, ( e 2, e 3 ) = 0 ( e 3, e 1 ) = 0, ( e 3, e 2 ) = 0, ( e 3, e 3 ) = 1 a = a 1 e 1 + a 2 e 2 + a 3 e 3 (3.24) b = b1 e 1 + b 2 e 2 + b 3 e 3 (3.25) ( a, b) (3.26) = (a 1 e 1 + a 2 e 2 + a 3 e 3, b 1 e 1 + b 2 e 2 + b 3 e 3 ) = a 1 b 1 ( e 1, e 1 ) + a 1 b 2 ( e 1, e 2 ) + a 1 b 3 ( e 1, e 3 ) +a 2 b 1 ( e 2, e 1 ) + a 2 b 2 ( e 2, e 2 ) + a 2 b 3 ( e 2, e 3 ) +a 3 b 1 ( e 3, e 1 ) + a 3 b 2 ( e 3, e 2 ) + a 3 b 3 ( e 3, e 3 ) = a 1 b 1 + a 2 b 2 + a 3 b 3 ( a, b) = ab cos θ = a 1 b 1 + a 2 b 2 + a 3 b 3 (3.27) cos θ = ( a b) a b (3.28) x(t) ( e i e j ) = δ ij (3.29) 31

d x(t + h) x(t) x(t) = lim dt h 0 h (3.30) x(t) v(t) a(t) d x(t) = v(t) dt (3.31) d v(t) = a(t) dt (3.32) ( a(t), b(t)) (3.33) d dt ( a(t), b(t)) = ( d dt a(t), b(t)) + ( a(t), d dt b(t)) (3.34) 3.4.5 ( ) a b (a 2 b 3 a 3 b 2, a 3 b 1 a 1 b 3, a 1 b 2 a 2 b 1 ) (3.35) a b (3.36) θ a b = a b sin θ, (3.37) a b a b 32

a a = (a 2 a 3 a 3 a 2, a 3 a 1 a 1 a 3, a 1 a 2 a 2 a 1 ) = (0, 0, 0) (3.38) a 0 = 0 (3.39) a = b (3.40) a c = b c (3.41) c (λ a + µ b) = λ c a + µ c b (3.42) e 1 e 1 = 0, e 1 e 2 = e 3, e 1 e 3 = e 2, (3.43) e 2 e 1 = e 3, e 2 e 2 = 0, e 2 e 3 = e 1, e 3 e 1 = e 2, e 3 e 2 = e 1, e 3 e 3 = 0 e i e j = ϵ ijk e k (3.44) ϵ ijk ϵ ijk ijk ϵ ijk = 0, i = j, i = k, j = k (3.45) ϵ 123 = ϵ 231 = ϵ 312 = 1, ϵ 123 = ϵ 123 = ϵ 123 = 1, 33 (3.46)

ϵ ijk = ϵ jik = ϵ ikj (3.47) (a 1 e 1 + a 2 e 2 + a 3 e 3 ) (b 1 e 1 + b 2 e 2 + b 3 e 3 ) = a i b j e i e j (3.48) = ϵ kij e k a i b j i ( a b) i = ϵ ijk a j b k (3.49) 3.4.6 (3.23) (3.44) 3.5 34

F a m m a = F (3.50) a = d2 x(t) (3.51) dt2 35

3.5.1 m mg mg = g m m R ω R ω 2 F F = mrω 2 (3.52) Rω 36

x 0 x F = k(x x 0 ) (3.53) x = x 0 x(t) = x 0 (3.54) x(t) = x 0 (3.55) t = 0 3.6 A B A B F B B A F A F B = F A (3.56) 37

3.6.1 A B A F a A f B B f A M a a = F f (3.57) M b a = f (3.58) (M a + M b )a = F (3.59) M a + M b 38

3.7 ( a, b) a b ( a, b) 2 + ( a b) 2 = a 2 b 2 (3.60) F a 39

A, B a 1 5 a, b c a = b (3.61) a c = b c (3.62) a c = b c c a c = b c (3.63) a = b (3.64) a c = b c (3.65) ( a, b) 2 = (a 1 b 1 + a 2 b 2 + a 3 b 3 ) 2 (3.66) ( a b) 2 = (a 2 b 3 a 3 b 2 ) 2 + (a 3 b 1 a 1 b 3 ) 2 + (a 1 b 2 a 2 b 1 ) 2 (3.67) ( a, b) 2 + ( a b) 2 (3.68) = (a 1 b 1 + a 2 b 2 + a 3 b 3 ) 2 + (a 2 b 3 a 3 b 2 ) 2 + (a 3 b 1 a 1 b 3 ) 2 + (a 1 b 2 a 2 b 1 ) 2 40

a 1 a 2 1 b 2 1 + b 2 3 + b 2 2 (3.69) a 1 2b 1 (a 2 b 2 + a 3 b 3 ) 2b 3 a 3 b 1 2b 2 a 2 b 1 = 2b 1 (a 2 b 2 + a 3 b 3 a 2 b 2 a 3 b 3 ) = 0 (3.70) a 2 a 3 a 2 2(b 2 1 + b 2 3 + b 2 2) (3.71) a 2 3(b 2 1 + b 2 3 + b 2 2) (3.72) (a 2 1 + a 2 2 + a 2 3)(b 2 1 + b 2 3 + b 2 2) = a 2 b 2 (3.73) a b = a b cos θ (3.74) a b = a b sin θ (3.75) 41

4 4.1 f (x) y = f(x) y = f(x) x S(x) S (x) = lim h 0 S(x + h) S(x) h (4.1) S(x + h) = S(x) + hf(x) + O(h 2 ) (4.2) 43

S S(x + h) S(x) (x) = lim h 0 h = f(x). (4.3) f(x) S(x) S(x) f(x) S(x) = x x 0 dxf(x) (4.4) x 0 x x 0 x 4.1.1 F d 2 dt 2 x(t) = F (4.5) t 1 t 0 t1 t 0 dt d2 dt 2 x(t) = t1 t 0 dt F (4.6) d dt x(t) t 1 t0 = F t t 1 t0 (4.7) d dt x(t) t=t 1 d dt x(t) t=t 0 = F (t 1 t 0 ) (4.8) 44

t 1 t t 0 v 0 d dt x(t) = v 0 + F (t t 0 ) (4.9) v 0 = d dt x(t) t=t 0 (4.10) t 0 x 0 x(t) = v 0 (t t 0 ) + F 1 2 (t t 0) 2 + x 0 (4.11) x 0 = x t=t0 (4.12) F v 0 x 0 t = t 0 4.1.2 m m d2 x(t) = kx(t) (4.13) dt2 x, y F x = kr cos θ = kx (4.14) F y = kr sin θ = ky (4.15) 45

a d dt cos ωt = ω sin ωt, d sin ωt = ω cos ωt (4.16) dt d 2 dt cos ωt = 2 ω2 cos ωt, d2 dt sin ωt = 2 ω2 sin ωt (4.17) d 2 dt 2 eat = a 2 e at (4.18) e iat = A(t) + ib(t) (4.19) iae iat = A (t) + ib (t) (4.20) ia(a(t) + ib(t)) = ab(t) + iaa(t) (4.21) A (t) = ab(t), B (t) = aa(t) (4.22) t f(t) t = 0 f(t) = f(0) + f (0)t + 1 2! f (2) (0)t 2 + 1 3! f (3) (0)t 3 + 1 4! f (4) (0)t 4 + + 1 n! f (n) (0)t n + (4.23) t = 0 f(0), f (0), f (2) (0), f (3) (0), f (4) (0), f (n) (0) (4.24) t = 0 t t = 0 t t 46

n n t f(t) ϵ f(t) f(0) = tf (ϵt), ϵ 1 (4.25) t 0 ϵt f(t) f(0) t = f (ϵt), ϵ 1 (4.26) f(t) t = 0 t = t t = ϵt f(t) (f(0) + tf (0) + + tn 1 (n 1)! f (n 1) (0)) = ϵt n f (n) (ϵt), ϵ 1 (4.27) 47

f(t) = e iat f (n) (t) = (ia) n e iat f (n) (0) = (ia) n f(t) = 1 + iat + 1 2! (iat)2 + 1 3! (iat)3 + 1 4! (iat)4 + 1 n! (iat)n + (4.28) = (1 1 2! (at)2 + 1 4! (at)4 + 1 (2n)! ( 1)n (at) 2n + ) +i(at 1 3! (at)3 + g(t) = cos at 1 (2n + 1)! ( 1)n (at) 2n+1 + ) (4.29) g(t) = 1 1 2! (at)2 + 1 4! (at)4 + 1 (2n)! ( 1)n (at) 2n + (4.30) h(t) = sin at h(t) = at 1 3! (at)3 + f(t), g(t), h(t) 1 (2n + 1)! ( 1)n (at) 2n+1 + (4.31) e iat = cos at + i sin at (4.32) e iat = cos at i sin at (4.33) 48

(4.13) cos at = 1 2 (eiat + e iat ) (4.34) sin at = 1 2i (eiat e iat ) (4.35) d 2 k dt x(t) = 2 ω2 x(t), ω = m (4.36) cos ωt, sin ωt (4.37) x(t) = A cos ωt + B sin ωt (4.38) A, B t A, B C, α x(t) = C cos(ωt + α) (4.39) A = C sin α, B = C cos α (4.40) t = 0 x(0), ẋ(0) A, B ẋ(t) = Aω sin ωt + Bω cos ωt (4.41) x(0) = A, ẋ(0) = B (4.42) 49

(4.13) x(t) = l a l t l (4.43) a l d dt x(t) = l a l lt (l 1) (4.44) ( d dt )2 x(t) = l a l l(l 1)t (l 2) (4.45) a l l(l 1)t (l 2) + ω 2 l l t a l+2 (l + 2)(l + 1)t l + ω 2 l l a l t l = 0 (4.46) a l t l = 0 (4.47) (a l+2 (l + 2)(l + 1) + ω 2 a l )t l = 0 (4.48) l t (a l+2 (l + 2)(l + 1) + ω 2 a l ) = 0 (4.49) ω 2 a l+2 = (l + 2)(l + 1) a l (4.50) l cos ωt sin ωt 4.1.3 z x d 2 x(t) = 0, dt2 (4.51) d 2 z(t) = g dt2 (4.52) 50

x(t) = x 0 + v x (0)t, (4.53) z(t) = g 2 t2 + v z (0)t + z(0) (4.54) ẋ(t) = v x (0), (4.55) ż(t) = gt + v z (0) (4.56) t = 0 x(0) = x 0, (4.57) z(0) = z(0) (4.58) ẋ(0) = v x (0), (4.59) ż(t) = v z (0) (4.60) 4.1.4 d 2 dt z(t) = g b d z(t) (4.61) 2 dt d v(t) = g bv(t) (4.62) dt d = bṽ(t) (4.63) dtṽ(t) ṽ(t) = v(t) g b (4.64) ṽ(t) = e bt ṽ(0) (4.65) 51

4.2 x y(x) y (x) x,y(x) y F (x, y, y ) = 0 (4.66) y = f(x, y) (4.67) x y = sin x (4.68) y = y dx (4.69) = dx sin x = cos x + c c x y 1 = cos x 1 + c (4.70) c = y 1 + cos x 1 2 52

y = exp ax (4.71) y = y dx (4.72) = dx exp ax = 1 exp ax + c a 3 y = X(x)Y (y) (4.73) dy Y (y) = X(x)dx (4.74) dy Y (y) = X(x)dx y = ay (4.75) dy = adx (4.76) y dy = adx y log y = ax + c (4.77) y = y 0 e ax, y 0 = e c (4.78) 4.3 1 53

y (x) y(x) x a y y = ay (4.79) y y = a log(y) = ax + C y = C exp( ax) y x y dx dy = 1 1 a y (4.80) y x = 1 a log y + C (4.81) y y(x) a a y + (a + b)y + aby = 0 (4.82) 54

y + (a + b)y + aby (4.83) = ( d dx + a)( d dx + b)y = 0 D = d dx D D D m D m = D m+n (4.84) Da = ad a b (D + a)(d + b) = D 2 + (a + b)d + ab (4.85) ( d d + a)y = 0, ( + b)y = 0 (4.86) dx dx y = C exp( ax) + C exp( bx) (4.87) x y y 1 = C exp( ax 1 ) + C exp( bx 1 ) y 2 = C exp( ax 2 ) + C exp( bx 2 ) (4.88) x y y y 1 = C exp( ax 1 ) + C exp( bx 1 ) d 1 = ac exp( ax 2 ) bc exp( bx 2 ) (4.89) a b ( d dx + a)2 y = 0 (4.90) 55

( d + a)y = 0, (4.91) dx y = C 1 exp( ax) ( d dx + a)y = C 1e ax (4.92) (4.92) y = g(x)e ax (4.93) g(x) y (4.92) y = g e ax age ax (4.94) g e ax age ax + age ax = C 1 e ax (4.95) g = C 1 g = C 1 x + C 2 y = (C 1 x + C 2 )e ax (4.96) y + (a + b)y + aby = c (4.97) c = 0 y = c ab (4.98) y = C exp( ax) + C exp( bx) (4.99) 56

C, C y = c ab + C exp( ax) + C exp( bx) (4.100) y + ω 2 y = 0 (4.101) ( d dx + iω)( d iω)y = 0 (4.102) dx y = C exp(iωx) + C exp( iωx) (4.103) y = A cos(ωx) + B sin(ωx) (4.104) A = C + C, B = ic ic (4.105) e iωx = cos ωx + i sin ωx (4.106) 4.4 4.4.1 1. L T M 2. ( x x 57

(1 + x) n, log(1 + x), e ax, cos ωx, sin ωx log(1 + x) 3. 4. (4-1) M F x(t) (4-2) (4-1) (4-3) t = 0 x(0) ẋ(0) 5. ( -1) M F x(t) ( -2) ( -1) ( -3) t = 0 x(0) ẋ(0). (6-1) M F = kx x(t) (6-2) (6-1) (6-3) t = 0 x(0) = A ẋ(0) = 58

1. L T M = LT 1 (4.107) = LT 2 (4.108) = MLT 2 (4.109) = (4.110) = T 1 (4.111) 2. ( x x f(x) d dx (1 + x)n = n(1 + x) n 1 (4.112) d log(1 + x) = 1 1 + x (4.113) dx d dx eax = ae ax (4.114) d cos ωx = ω sin ωx dx (4.115) d sin ωx = ω cos ωx dx (4.116) f(x) = f(0) + f (0)x + f 2 (0) x2 2 + + f n (0) xn n! + (4.117) f(x) = log(1 + x) n f (1) (x) = 1 1 + x 59 (4.118)

f (2) 1 (x) = (1 + x) 2 (4.119) f (n) (x) = ( ) n 1 1 (n 1)! (1 + x) n (4.120) n f (1) (0) = 1 (4.121) f (2) (x) = (4.122) f (n) (x) = ( 1) n 1 (n 1)! (4.123) log(1 + x) = x x2 2 + + ( )n 1 xn n + (4.124) 3. 1 a = 1 M F (4.125) A B B A 4. (4-1) M F x(t) (4-2) M d2 x(t) = F (4.126) dt2 60

(4-1) F M d 2 dt x(t) = F 2 M (4.127) d dt x(t) = F M t + C (4.128) x(t) = F t 2 + Ct + D M 2 (4.129) C, D (4-3) t = 0 x(0) ẋ(0) C, D t = 0 d dt x(t) = F M t + C (4.130) x(t) = F t 2 + Ct + D M 2 (4.131) d x(0) = C = ẋ(0) dt (4.132) x(0) = D = x(0) (4.133) x(t) = F t 2 M 2 + ẋ(0)t + x(0) (4.134) 61

6 (6-1) M F = kx x(t) (6-2) M M d2 x(t) = kx(t) (4.135) dt2 d 2 k dt x(t) = 2 ω2 x(t), ω = M (4.136) ( d dt + iω)( d iω)x(t) = 0 (4.137) dt ( d iω)x(t) = 0 (4.138) dt ( d dt + iω)x(t) = 0 ( d dt iω)x(t) = 0 x(t) = eiωt (4.139) ( d + iω)x(t) = 0 x(t) = e iωt dt C D x(t) = Ce iωt + De icω = C(cos ωt + i sin ωt) + D(cos ωt i sin ωt) (4.140) = (C + D) cos ωt + i(c D) sin ωt (6-3) t = 0, x(0) = (C + D), ẋ(0) = ωi(c D) (4.141) x(0) = A ẋ(0) = 0 C = D, 2C = A (4.142) x(t) = A cos ωt. (4.143) 62

2009 6 24 10:30-12:00 1. L T M 2. ( x (1 + x) n, log(1 + x), e ax, cos ωx, sin ωx, cos 1 x n a, ω t x(t) = A cos ωt M d2 x(t) = kx(t) (4.144) dt2 A ω M k 3. 4. (4-1) M F x(t) (4-2) (4-1) (4-3) t = 0 x(0) ẋ(0) 1. L T M LT 1 LT 2 MLT 2 M 0 L 0 T 0 T 1 L 2 L 3 63

2. ( x d dx (1 + x)n = n(1 + x) n 1 (4.145) d log(1 + x) = 1 1 + x (4.146) dx d dx eax = ae ax (4.147) d cos ωx = ω sin ωx dx (4.148) d sin ωx = ω cos ωx dx (4.149) d 1 dx cos 1 x = 1 x 2 (4.150) y = cos 1 x (4.151) cos y = x (4.152) d y( 1) sin y = 1 (4.153) dx d dx y = ± 1 1 x 2 t x(t) = A cos ωt (4.154) M d2 x(t) = kx(t) (4.155) dt2 M d2 dt 2 x(t) = Mω2 A cos ωt (4.156) 64

ω ω = k M (4.157) A 3. 4. (4-1) M F x(t) (4-2) (4-1) d 2 x(t) = F (4.158) dt2 d dt x(t) = F t + v 0 (4.159) x(t) = 1 2 F t2 + v 0 t + x 0 (4.160) v 0, x 0 (4-3) t = 0 x(0) ẋ(0) x(t) = 1 2 F t2 + ẋ(0)t + x(0). (4.161) 65

y =X(x)Y(y) dy dx = X(x)Y (y) dy Y (y) = dxx(x) (4.162) dy dx = ay dy dx = axy dy dx = axy2 y -ay=f(x) dy y = dy y = dy y = 2 dxa log y = ax + c y = e c e ax = Ce ax (4.163) dxax log y = a/2x 2 + c y = e c e a/2x2 = Ce a/2x2 (4.164) dxax 1/y = a/2x 2 + c y = 1 ax 2 /2 + c (4.165) ( d dx a)y = ebx (4.166) e ax y = e ax u(x) y = (e ax ) u(x) + e ax u = ae ax u(x) + e ax u (4.167) (au(x) + u au(x))e ax = e bx (4.168) u = e (b a)x 1 u = (b a) e(b a)x b a = 0 y = 1 (b a) ebx (4.169) (au(x) + u au(x))e ax = e ax (4.170) u = 1 u = x y = xe ax (4.171) 66

() (1-1) M F = kx x(t) x(t) d x(t) dt (1-2) (1-1) (1-3) t = 0 x(0) = 0 ẋ(0) = B 2. v(t) ( d + c)v(t) = g, c > 0 dt 3. dx(t) dt = ax(t)t, a > 0 X(t) dx(t) dt M d2 x(t) = F = kx(t) (4.172) dt2 M d2 x(t) dt 2 dx(t) dt = kx(t) dx(t) dt (4.173) d dt [M 2 (dx(t) ) 2 ] = d dt dt [ k 2 (x(t))2 ] (4.174) 67

d dt [M 2 (dx(t) ) 2 + k dt 2 (x(t))2 ] = 0 (4.175) [ M 2 (dx(t) ) 2 + k dt 2 (x(t))2 ] = E (4.176) (1-2) x(t) = C cos ωt + D sin ωt, ω = k M C, D x(0) = C (4.177) ẋ(0) = Dω x(0) = C = 0 (4.178) ẋ(0) = Dω = B C = 0 (4.179) D = B ω x(t) = B ω sin ωt (4.180) 2. ( d + c)v(t) = g, c > 0 dt v(t) g = 0 e ct v(t) v(t) = e ct u(t) (4.181) u(t) d dt v(t) = d dt e ct u(t) = ce ct u(t) + e ct d u(t) (4.182) dt 68

u(t) v(t) ( d dt + c)v(t) = d e ct u(t) = g (4.183) dt d dt u(t) = gect (4.184) u(t) = g c ect + u 0 (4.185) v(t) = e ct u(t) = e ct ( g c ect + u 0 ) = g c + u 0e ct (4.186) lim t v(t) = g c (4.187) 3. dx(t) dt = ax(t)t, a > 0 X(t) X(t) t dx(t) X(t) = atdt (4.188) log X(t) = at2 2 + C 0 (4.189) X(t) = exp at2 2 X 0 = e C 0 + C 0 = X 0 exp at2 2 (4.190) 69

d dx 70

5 5.1 p p = m v (5.1) d dt p = F (5.2) t2 t 1 dt d dt p = t2 p(t 1 ) p(t 2 ) = t 1 dt F (5.3) t2 t 1 dt F (5.4) 5.2 71

F x W = F x (5.5) 5.2.1 m v F m d dt v = F (5.6) v = F m t + v 0 (5.7) x = F 2m t2 + v 0 t + x 0 (5.8) v 0, x 0 t 1 t 2 v(t 2 ) v(t 1 ) = F m (t 2 t 1 ) (5.9) x2 W = F dx = x 1 F dx dt dt = F (x 2 x 1 ) (5.10) 72

t x2 W = F dx = F vdt (5.11) x 1 = dt(mv d v) (5.12) dt = dt 1 2 m d dt v2 (5.13) = 1 2 m(v2 2 v 2 1) (5.14) 1 2 mv2 2 = 1 2 mv2 1 + W (5.15) v 1 = 0 1 2 mv2 2 = W (5.16) 5.3 W = dr F (5.17) W = dr F (5.18) = dr d m dt v 73

= = dt d dt r m d dt v dt 1 2 m d ( v v) dt = 1 2 m( v v) t 2 t1 = 1 2 m( v 2) 2 1 2 m( v 1) 2 dr dr = dt = vdt dt (5.19) d dt r m d d v = v dt dt v = d v v dt (5.20) (5.17) (5.17) F = m n z g = mg(0, 0, 1) (5.21) 1 x h y h, z h 2 (1,1,1) h 74

5.3.1 f(x) x = x F (x) = x a dx f(x ) (5.22) f(x) = d F (x) (5.23) dx x, y f(x, y) x f(x + h, y) f(x, y) f(x, y) = lim = f x (x, y) (5.24) x h 0 h y x y f(x, y + h) f(x, y) f(x, y) = lim y h 0 h = f y (x, y) (5.25) x y f(x, y) f x (x, y), f y (x, y) f xx (x, y) = x f x(x, y) (5.26) f yx (x, y) = y f x(x, y) (5.27) f xy (x, y) = x f y(x, y) (5.28) f yy (x, y) = y f y(x, y) (5.29) 75

f(x+h 2,y+h 1 ) f(x+h 2,y) f xy (x, y) = lim lim h 1 f(x,y+h 1) f(x,y) h 1 h1 0 h 2 0 = lim h1 0 lim h 2 0 f yx (x, y) = lim h2 0 lim h 1 0 = lim h2 0 lim h 1 0 h 2 f(x + h 2, y + h 1 ) f(x + h 2, y) f(x, y + h 1 ) + f(x, y) h 1 h 2 (5.30) f(x+h 2,y+h 1 ) f(x,y+h 1 ) h 2 h 1 f(x+h 2,y) f(x,y) h 2 f(x + h 2, y + h 1 ) f(x + h 2, y) f(x, y + h 1 ) + f(x, y) h 1 h 2 (5.31) f xy (x, y) = f yx (x, y) (5.32) x y b dx f(x, y) = f(b, y) f(a, y) (5.33) a x d c dx f(x, y) = f(x, d) f(x, c) (5.34) y b dxdy a x f(x, y) = d dydx y f(x, y) = c dy(f(b, y) f(a, y)) (5.35) dx(f(x, d) f(x, c)) (5.36) x2 y2 I 1 = dx dy( x 1 y 1 x f(x, y)) (5.37) y I 2 = x2 x 1 y2 dx dy( y 1 x V y(x, y) y V x(x, y)) (5.38) I 1 I 1 = f(x 2, y 2 ) f(x 2, y 1 ) f(x 1, y 2 ) + f(x 1, y 1 ) (5.39) 76

I 2 I 2 = y2 y 1 dy(v y (x 2, y) V y (x 1, y)) I 2 = = x2 x2 x 1 dxv x (x, y 1 ) + x 1 dxv x (x, y 1 ) + y2 y 1 y2 x2 dyv y (x 2, y) x 1 dx(v x (x, y 2 ) V x (x, y 1 )) (5.40) x2 x 1 x1 dxv x (x, y 2 ) y1 y2 y 1 dyv y (x 1, y) (5.41) dyv y (x 2, y) + dxv x (x, y 2 ) + dyv y (x 1, y) y 1 x 2 y 2 d l V (5.42) I 2 (5.42) V (x, y) ( x V y(x, y) y V x(x, y)) = 0 (5.43) d l V = 0 (5.44) C 1, C 2 77

d l V = C 1 d l V C 2 (5.45) V F C i (5.43) U(x, y) F = ( U(x, y), x U(x, y) ) (5.46) y 5.4 U(x, y, z) F = ( x U(x, y, z), y U(x, y, z)), U(x, y, z)) (5.47) z C W = P2 P 1 d l F = = (U(P 1 ) U(P 2 )) P2 P 1 (dx x U(x, y, z) + dy y U(x, y, z) + dz U(x, y, z))(5.48) z U(x, y, z) U(x, y, z) x(t) P2 W = = P 1 dt( dx dt x U(x, y, z) + dy dt dz U(x, y, z) + y dt U(x, y, z)) z dt d dt U(x(t), y(t), z(t)) = (U(P 1) U(P 2 )) (5.49) U m d2 dt 2 x(t) = F (5.50) m d2 dt x(t) d 2 dt x(t) = F d x(t) (5.51) dt 78

U( x) t2 t 1 dtm d2 dt x(t) d 2 dt x(t) = t2 dt( ) d U( x(t)) (5.52) t 1 dt t2 t 1 dt d dt (m 2 ( d dt x(t))2 + U( x(t))) = 0 (5.53) m 2 ( d dt x(t))2 + U( x(t)) t=t1 m 2 ( d dt x(t))2 + U( x(t)) t=t2 = 0 (5.54) E E = m 2 ( d dt x(t))2 + U( x(t)) (5.55) 5.4.1 () z z U = mgz (5.56) F = (0, 0, mg) (5.57) 79

U = k 2 (x2 + y 2 + z 2 ) = k 2 r2 (5.58) F = k(x, y, z) (5.59) U = 1 r (5.60) F = 1 (x, y, z) (5.61) r3 5.5 5.5.1 m d2 d kx 2 x(t) = kx(t) = dt2 dx 2 (5.62) d x(t) dt m d d2 x(t) dt dt x(t) = d 2 dt x(t) d kx 2 dx 2 (5.63) 80

d m dt 2 ( d dt x(t))2 = d dt (k 2 x2 ) (5.64) E E = m 2 ( dx(t) dt ) 2 + k 2 (x(t))2 (5.65) (5.65) dx dt = ±( 2 m )1/2 (E k 2 x2 ) 1/2 (5.66) ( ) 2 dx(t) 0 (5.67) E k 2 (x(t))2 = m 2 dt x E k 2 x2 = 0 (5.68) x (0) ± = ±x 0 (5.69) x 0 = ( 2E k )1/2 (5.70) x 0 x x 0 + (5.71) dt dx = (m 1 k )1/2 ( 2E (5.72) k x2 ) 1/2 x t t t 0 = ( m k )1/2 sin 1 ( x x 0 ) (5.73) 81

t 0 x 0 x = x 0 sin (ω(t t 0 )) (5.74) ω = ( k m )1/2 (5.75) ω T = 2π ω f = ω 2π x 0 5.5.2 E = m/2(ṙ 2 + r 2 ϕ2 ) + U(r) (5.76) M z = mr 2 ϕ (5.77) E M z ϕ r r ṙ dr dt E = m/2(ṙ 2 + r 2 M 2 z ) + U(r) (5.78) mr 2 dr dt = 2 m (E U(r)) M 2 z m 2 r 2 (5.79) 82

t r t r dt dr = 1 (5.80) 2 (E U(r)) M z 2 m m 2 r 2 1 t = dr (5.81) 2 (E U(r)) M z 2 m m 2 r 2 dr dt dϕ dt d dr ϕ = M z 1 r 2 (5.82) 2 (E U(r)) M z 2 m m 2 r 2 ϕ = dr M z 1 r 2 (5.83) 2 (E U(r)) M z 2 m m 2 r 2 U(r) = α r (5.84) ρ = 1 r M mα M ϕ = cos 1 r 2mE + m 2 α 2 (5.85) M 2 z 5.6 dx log x = x log x x (5.86) 83

2 x ξ 1 dx a2 x = 2 sin 1 ( x a ) (5.87) x = a cos ξ (5.88) dx = a sin ξdξ a2 x 2 = a sin ξ (5.89) 1 dx a2 x = 2 dξ = ξ = cos 1 ( x a ) (5.90) 3 1 I F = dx ax2 + bx + c = 1 log 2ax + b + 2 a(ax 2 + bx + x), a > 0 a = 1 log 2ax + b 2 a(ax 2 + bx + x), a > 0 a (5.91) = 1 sin 1 2ax + b a b2 4ac, a < 0 dx ax 2 + bx + c = 2ax + b 4a ax2 + bx + c b2 4ac I F (5.92) 8a 5.7 84

5.7.1 2 F = µn (5.93) µ (5.94) F = µ N (5.95) µ (5.96) 85

5.7.2 5.8 5.8.1 z M m d2 z(t) = mg (5.97) dt2 d dt z(t) = gt + v 0 (5.98) 86

z 1 z 2 W = z(t) = g 2 t2 + v 0 t + z 0 (5.99) z2 z 1 F dz = mg(z 2 z 1 ) (5.100) z 1 z 2 t 1 t 2 W = mg(z 2 z 1 ) m 2 g2 (t 2 2 t 2 1) mgv 0 (t 2 t 1 ) (5.101) T = m 2 (( gt 2 v 0 ) 2 ( gt 1 + v 0 ) 2 ) (5.102) = m 2 g2 (t 2 2 t 2 1) mgv 0 (t 2 t 1 ) (5.103) (5.101) (5.103) 5.8.2 d 2 dt z(t) = g c d z(t) (5.104) 2 dt d dtṽ = cṽ, ṽ = v + g c (5.105) v(t) = v 0 + ˆv 0 e ct, v 0 = g c (5.106) 87

z = g c t ṽ0 c e ct + z 0 (5.107) ṽ 0 (5.104) d z(t) dt d d2 z(t) dt dt z(t) + g d 2 dt z(t) = c( d dt z(t))2 (5.108) d dt (1 2 ( d dt z(t))2 + gz(t)) = c( d dt z(t))2 (5.109) c c = 0 c > 0 c 1 2 v2 + gz = g2 2c 2 + g(z 0 g c t) 2g c ṽ0e ct + 1 2ṽ2 0e 2ct (5.110) cv 2 = ( g2 c 2ˆv 0ge ct + cˆv 2 0e 2ct ) (5.111) d dt (1 2 ( d dt z(t))2 + gz(t)) = g2 c + 2gṽ 0e ct cṽ 2 0e 2ct (5.112) c( d dt z(t))2 = g2 c + 2gṽ 0e ct cṽ 2 0e 2ct (5.113) (5.109) t d dt E = g2 c (5.114) 88

(5.109) t2 t 1 d dt E t2 = dtcv(t) 2 (5.115) t 1 E (t 2 ) E (t 1 ) = t2 t 1 dtcv(t) 2 (5.116) E (t 1 ) E (t 2 ) = t2 v(t) t2 t 1 t 1 dtcv(t) 2 (5.117) dtcv(t) 2 = g2 c (t 2 t 1 ) + 2v 0g c (e ct 2 e ct 1 ) v2 0 2 (e 2ct 2 e 2ct 1 ) (5.118) z(t 1 ) z(t 2 ) 5.8.3 V (x) M E x(t) V (x) = 1 2 kx2 + cx 4 5.8.4 M z 89

5.8.5 M C i O(0, 0, 0) P (10, 10, 10) C 1 :OP C 2 (0, 0, 0) (10, 0, 0) (10, 10, 0) (10, 10, 10) C 3 (0, 0, 0) (20, 0, 0) (20, 10, 0) (20, 10, 10) (10, 10, 10) C 4 (0, 0, 0) (0, 10, 0) (10, 10, 0) (10, 10, 10) 5.8.6 F F = (0, 0, mg) 2. F = 1 1 4πϵ 0 r r 3 3. F = k x F = U (5.119) 5.8.7 F = U( r) (5.120) F ( ) 2 U( r) (5.121) F ( ) 2 U( r) (5.122) 90

5.8.8 F = kv (5.123) F = U (5.124) 91

6 6.1 F m d2 dt 2 x(t) = F (6.1) r r F = rf(r), r = x 2 + y 2 (6.2) L = r p (6.3) 93

d L dt = d r p (6.4) dt = d dt r p + r d p (6.5) dt d dt r = 1 m p, d dt p = F (6.6) F = f r (6.7) d L dt = 1 p p + f r r = 0 (6.8) m L = x p (6.9) L x = yp z zp y, L y = zp x xp z, Lz = xp y yp x (6.10) d dt L = 0 (6.11) z xy xy 6.1.1 θ L = m r v = mrv sin θ (6.12) 94

L r = 0, L p = 0 (6.13) 6.2 r θ x(t) = r(t) cos θ(t), y(t) = r(t) sin θ(t) (6.14) d dt x(t) = ṙ(t) cos θ(t) r(t) sin θ(t) θ(t) (6.15) d dt y(t) = ṙ(t) sin θ(t) + r(t) cos θ(t) θ(t) (6.16) 95

f(t) f(t) = d f(t) (6.17) dt L = m(y d dt x x d dt y) = mr2 θ (6.18) E = m 2 v2 (6.19) = m 2 ((ṙ(t) cos θ(t) r(t) sin θ(t) θ(t)) 2 + (ṙ(t) sin θ(t) + r(t) cos θ(t) θ(t)) 2 ) = m 2 ((ṙ(t))2 + r 2 θ2 ) 1 U(r) F E E = E + U(r) (6.20) = m 2 (ṙ2 + r 2 θ2 ) + U(r) r(t) ṙ(t) θ(t) (6.8) (6.18) θ(t) d dt θ = L mr 2 (6.21) E = E + U(r) (6.22) = m 2 (ṙ(t)2 + L2 m 2 r 2 ) + U(r) = m 2 ṙ(t)2 + L2 2mr 2 + U(r) r(t) 96

6.3 6.3.1 F = k r (6.23) M 6.3.2 (r, θ) E M z r(t) 6.3.3 r = (A, 0), v = (0, 0) (6.24) r = (0, 0), v = (B, 0) (6.25) A B r = r 1, v = v 1 (6.26) 6.3.4 U = c 1 r (6.27) 97

x(t) y(t) m d2 x(t) = kx(t) (6.28) dt2 m d2 y(t) = ky(t) (6.29) dt2 x = x 0 cos(ωt + α) (6.30) y = y 0 cos(ωt + β) (6.31) x 0, y 0, α, β ẋ = ωx 0 sin(ωt + α) (6.32) ẏ = ωy 0 sin(ωt + β) (6.33) m(xẏ yẋ) = mωx 0 y 0 sin(β α) (6.34) m 2 v2 + k 2 x2 = k 2 (x2 0 + y 2 0) (6.35) E L z E = m 2 v2 + k 2 x2 = m 2 (ṙ2 + r 2 θ2 ) + k 2 r2 (6.36) L z = mr 2 θ (r(t), θ(t)) v = d (r(t) cos θ(t), r(t) sin θ(t)) (6.37) dt = (ṙ(t) cos θ(t) + r(t) θ(t)( ) sin θ(t), ṙ(t) sin θ(t) + r(t) θ(t) cos θ(t)) 98

θ(t) L z r E = m 2 (ṙ2 + dt dr v 2 = (ṙ(t)) 2 + r 2 ( θ) 2 (6.38) θ = L z mr 2 (6.39) L2 z m 2 r 2 ) + k 2 r2 (6.40) dt dr = 1 (6.41) 2E/m L 2 z/m 2 r 2 km r2 1 t + t 0 = dr (6.42) 2E/m L 2 z/m 2 r 2 km r2 s r 2 = s, 2rdr = ds (6.43) ds 1 2 1 s 2E/m L 2 z/m 2 s km s (6.44) = ds 1 1 2 2Es/m L 2 z/m 2 k m s2 ds 1 1 E2, A = 2 A k (s E/k)2 mk L2 z (6.45) m 2 m ds 1 1 2 A k (s = 1 m k m E/k)2 2 k sin 1 ( ma (s E )) (6.46) k 99

s = r 2 t + t 0 = 1 m k 2 k sin 1 ( ma (r2 E )) (6.47) k k sin 2ω(t + t 0 ) = ( ma (r2 E )) (6.48) k, r(t) 2 E k = ma k sin 2ω(t + t 0) (6.49) x(0) = x 0 cos α, ẋ(0) = ωx 0 sin α (6.50) y(0) = y 0 cos β, ẏ(0) = ωy 0 sin β (6.51) r = (A, 0), v = (0, 0) (6.52) A = x 0 cos α, 0 = ωx 0 sin α (6.53) 0 = y 0 cos β, 0 = ωy 0 sin β (6.54) α = 0, x 0 = A (6.55) β = 0, y 0 = 0 (6.56) x(t) = A cos ωt, (6.57) y(t) = o 100

r = (0, 0), v = (B, 0) (6.58) 0 = x 0 cos α, B = ωx 0 sin α (6.59) 0 = y 0 cos β, 0 = ωy 0 sin β (6.60) α = π 2, x 0 = B ω (6.61) β = 0, y 0 = 0 (6.62) x(t) = B ω cos(ωt + π 2 ) = B ω y(t) = o sin(ωt), (6.63) (6.57) (6.63) A = B ω (6.64) (6.20)(6.22) U(r) = c 1 r 101

7 7.1 3 7.2 n F = C 1 r n (7.1) m R ω F = mrω 2 (7.2) T ω T = 2π ω (7.3) 103

C 1 R n = mr(2π T )2 (7.4) T 2 = m(2π)2 C Rn+1 (7.5) n + 1 n C G F = C 1 r 2 (7.6) C = GMm (7.7) 7.3 F = C 1 r2 n (7.8) n = r r (7.9) U(r) = C 1 r (7.10) (5.85) cos ϕ = M z r mc M z 2mE + m 2 C 2 M 2 z (7.11) 104

E M z (7.11) r = l 1 + e cos θ (7.12) e l e = l = M 2 z mc 1 + 2EM 2 z mc 2 (7.13) (7.14) e < 1, ; E < 0 (7.15) e = 1, ; E = 0 (7.16) e > 1, ; E > 0 (7.17) E M z x(0) v(0) xy z E = v(0)2 2m C x(0) (7.18) Mz = m x(0) v(0) (7.19) 7.3.1 e < 1 E < 0 θ r θ = π θ = 0 r = l 1 e, θ = π (7.20) r = 1 1 + e, θ = 0 (7.21) 105

r (7.13) r = l (7.22) 1 + 2EM 2 z mc 2 = 0 (7.23) E = mc2 2M 2 z (7.24) 7.3.2 e = 1 E = 0 θ = 0 r r 106

7.3.3 e > 1 E > 0 θ r > 0 cos θ > 1 e (7.25) θ θ c, θ c = cos 1 ( 1 e ) (7.26) 7.4 1.521 10 8 Km 1.471 10 8 Km 0.0167, 0.2056, 0.0068, 0.0167, 0.0934, 0.0485, 0.0555 107

, 0.0463, 0.0090, 0.2490 384400Km 0.0548799 θ θ l i, i = r i, i = θ = R l (7.27) θ = R l (7.28) θ = 16 11.4 (7.29) θ = 15 50.7 (7.30) θ = 15 46.8 (7.31) θ = 16 00.4 (7.32) 6 108

7.5 m d2 dt 2 x(t) = F (7.33) 7.5.1 x 0 (t) (7.34) x 0 (t) x x(t) = x(t) x 0 (t) (7.35) m d2 dt 2 ( x(t) x 0(t)) = F m d2 dt 2 x 0(t) (7.36) m d2 dt 2 x(t) = F (7.37) F = F m d2 dt x 0(t) 2 (7.38) F m d2 dt 2 x 0 (t) F = 0 (7.39) 109

7.5.2 (x, y, z = z) (x, y, z) θ x = x cos θ + y sin θ (7.40) y = x sin θ + y cos θ (7.41) θ = ωt (7.42) v x, v y d dt x = d dt x cos θ x ω sin θ + d dt y sin θ + y ω cos θ (7.43) v x = ṽ x cos θ + ṽ y sin θ (7.44) ṽ x = (v x + y ω), ṽ y = (v y x ω) (7.45) d dt y = d dt x sin θ x ω cos θ + d dt y cos θ y ω sin θ (7.46) v y = ṽ x sin θ + ṽ y cos θ (7.47) d dt θ = ω (7.48) v x, v y v x = d dt x (7.49) v y = d dt y (7.50) d dt v x = dtṽ d x cos θ ṽ xω sin θ + dtṽ d y sin θ + ṽ yω cos θ (7.51) a x = ã x cos θ + ã y sin θ (7.52) ã x = ( d dtṽ x + ṽ yω), ã y = ( d dtṽ y ṽ xω) (7.53) d dt v y = dtṽ d x sin θ ṽ xω cos θ + dtṽ d y cos θ ṽ yω sin θ (7.54) a y = ã x sin θ + ã y cos θ (7.55) ma x = F x (7.56) ma y = F y (7.57) 110

a a m(ã x cos θ + ã y sin θ) = F x (7.58) m(ã y cos θ ã x sin θ) = F y (7.59) ã i(i = x, y) ṽ i, (i = x, y) (7.53) mã x = F x cos θ F y sin θ (7.60) mã y = F y sin θ + F y cos θ (7.61) m( d dtṽ x + ( d dt y x ω) = F x cos θ F y sin θ (7.62) m( d dtṽ y ṽ xω) = F y sin θ + F y cos θ (7.63) m( d dt ( d dt x + y ω) + ( d dt y x ω)ω) = F x cos θ F y sin θ (7.64) m( d dt ( d dt y x ω) ( d dt x + y ω)ω) = F y sin θ + F y cos θ (7.65) m d dt v x = 2mv yω + mx ω 2 + F x cos θ F y sin θ (7.66) m d dt v y = +2mv xω + my ω 2 + F y sin θ + F y cos θ (7.67) 1 2 f 111

t = 0 112

7.6 7.6.1 F = k r (7.68) r 3 M E M z r(t) 7.6.2 E M z 7.6.3 7.6.4 113

8 N N x 1, x 2, x N F i (i = 1, N) j F i = F (0) i + j F (i,j) (8.1) F (i,j) = F (j,i) (8.2) 115

8.1 m 1 x 1 (t) m 2 x 2 (t) m 1 d 2 dt 2 x 1(t) = F (0) 1 + F (1,2) (8.3) m 2 d 2 dt 2 x 2(t) = F (0) 2 + F (2,1) (8.4) F (1,2) = F (2,1) = F ( x 1 x 2 ) (8.5) d 2 m 1 dt x 1(t) = F ( x 2 1 x 2 ) (8.6) d 2 m 2 dt x 2(t) = F ( x 2 1 x 2 ) (8.7) 8.1.1 d 2 dt 2 (m 1 x 1 (t) + m 2 x 2 (t)) = 0 (8.8) p 1 = m 1 x1, p 2 = m 2 x2 d dt ( p 1 + p 2 ) = 0 (8.9) X G = m 1 x 1 + m 2 x 2 m 1 + m 2 (8.10) 116

d 2 dt 2 X G = 0 (8.11) r x 1 = X G + m 2 m 1 + m 2 r (8.12) x 2 = X G m 1 m 1 + m 2 r (8.13) r = x 1 x 2 (8.14) µ d2 dt 2 r(t) = F ( r) (8.15) µ = m 1m 2 m 1 + m 2 (8.16) µ m 1 m 2 8.2 i d 2 m i dt x i(t) = F (0) 2 i + j F (i,j) (8.17) 117

i i j F (0) i = F (0) i ( x i ) (8.18) F (i,j) = F (0) i ( x i x j ) (8.19) 8.2.1 N X G = 1 N (m i x i ) M total i=1 (8.20) N M total = m i (8.21) i=1 F (ij) = F (ji) d 2 dt 2 X G (t) = 0 (8.22) M total P total = M total d dt X G (t) (8.23) P total =, d dt P total = 0 (8.24) 8.2.2 U (0) ( x) F (0) i = U (0) ( x) x= xi (8.25) 118

U int ( x) F i,j = U int ( x) x= xi x j (8.26) E = m i 2 v2 + i U (0) ( x i ) + i j U int ( x i x j ) (8.27) d dt E = m i v d dt v + v ( i U (0) ( x i ) + i j U int ( x i x j )) = 0 (8.28) 8.3 10 10 m 10 23 6 10 23 8.3.1 i F i d dt p i = F i (8.29) z x y 119

z d dt pz i = F z i (8.30) δt δp z δp z i = F z i δt (8.31) δp z = 2p z (8.32) n F z n z 2l z 2np z i = F z (8.33) n = v z 2l z, v z = pz m (8.34) F z = pz v z l z = pz2 ml z (8.35) S P P = F z S = pz2 = pz2 (8.36) msl z mv V = l z S (8.37) 8.4 120

i x i m i j F (0) i F (i,j) m i d 2 dt 2 x i = F (0) i + F (i,j) (8.38) M total X G X G = 1 x l, M total = M total i l M total d 2 dt 2 X G = l m l (8.39) F (0) l (8.40) L = l m l x l x l (8.41) d dt L = N, N = i x i F (0) i (8.42) F (i,j) = 0 (8.43) (i,j) ( x i x j ) F (i,j) = 0, F (i,j) = ( x i x j )F (i,j) (8.44) (i,j) 121

N x i = Ω x i, v i = ϵ ijk Ω j x k (8.45) E = 1 l 2 m l v l 2 (8.46) = 1 l 2 I ijω i Ω j (8.47) (8.48) Ω i I ij I ij = l ( x 2 l δ ij x i lx j l ) (8.49) 8.4.1 122

8.4.2 123

M x(t) M d2 dt 2 x(t) = F + F (8.50) x b x i, i = 1, 8 x = 1 8 M (m b x b + m i x i ), M = m b + i=1 i m i (8.51) r i r i = x i x b (8.52) v c 1, c 2, c 3, 124

n F = (c 1 v + c 2 v 2 + c 3 v 3 + ) n v (8.53) F i F f i O mega f i L L l n LF i (L l) f i = L d Ω n (8.54) dt 8.5 8.5.1 M L F 1, F 2 125

F 1 + F 2 Mg = 0 (8.55) x 1 F 1 Mg L 2 + x 2F 2 = 0 F 1 = (8.56) F 2 = 8.5.2 8.5.3 B 126

8.5.4 V M ρ 1 ρ 2 8.6 8.6.1 8.6.2 1 127

8.6.3 2 8.6.4 128

9 9.1 R ω 2π ω t x(t) = R cos ωt (9.1) y(t) = R sin ωt v x (t) = Rω sin ωt (9.2) v y (t) = Rω cos ωt a x (t) = Rω 2 cos ωt (9.3) a y (t) = Rω 2 sin ωt F x (t) = mrω 2 cos ωt (9.4) F y (t) = mrω 2 sin ωt 129

(9.2) m d2 dt 2 x(t) = mω2 x(t) (9.5) m d2 dt 2 y(t) = mω2 y(t) (9.6) 9.2 x(t) m m d2 x(t) = kx(t) (9.7) dt2 x(t) = A cos(ωt + α), ω = k m (9.8) k A α t = 0 T x(t + T ) = x(t) (9.9) T = 2π ω (9.10) E = m 2 v(t)2 + k 2 x(t)2 (9.11) 130

= k 2 A2 (sin 2 (ωt + α) + cos 2 (ωt + α)) = k 2 A2 (9.12) γ γ γ ± x(t) = Ae γt (9.13) (mγ 2 + k)ae γt = 0 (9.14) γ = ±iω (9.15) e iωt, e iωt (9.16) (9.8) A + e iωt + A e iωt (9.17) A + = A e iα, A = A e iα (9.18) 9.3 131

c m d2 dt x(t) = kx(t) c d x(t) (9.19) 2 dt (9.19) γ x(t) = Ae γt (9.20) (9.19) γ γ mγ 2 + k + cγ = 0 (9.21) γ = c ± c 2 4mk 2m c c (9.22) γ = c 2m ± iω, ω = c2 4mk 2m (9.23) x(t) = Ae c 2m t e ±iωt (9.24) 2m c c c γ = c ± c 2 4mk 2m = c 1 ± 1 4mk c 2 2m (9.25) γ 1 = c m, γ 2 = k m (9.26) 132

9.4 m d2 x(t) + kx(t) = f cos γt (9.27) dt2 γ f γ 0 = k m x(t) = A cos γt (9.28) ( mγ 2 + k)a cos γt = f cos γt (9.29) A A = f mγ 2 + k (9.30) x(t) = f cos γt = f/m mγ 2 + k cos γt (9.31) γ0 2 γ2 γ γ γ 0 1/0 133

γ γ 0 f x(t) = lim γ γ0 cos γt (9.32) mγ 2 + k f = lim γ γ0 cos γt γ ( mγ2 + k) γ = f t sin γt 2mγ 0 t sin γt t t 9.5 x i (t) m d2 dt 2 x i(t) = k i x i (t), i = 1, N (9.33) A i, α i x i (t) = A i cos(ω i t + α i ) (9.34) 9.5.1 x i (t) i i m d2 dt 2 x i(t) = k(x i (t) x i 1 (t)) k(x i (t) x i+1 (t)), i = 1, N (9.35) ω x i (t) = A i cos(ωt), i = 1, N (9.36) 134

N ma i ω 2 cos(ωt) = k(a i cos(ωt) A i 1 cos(ωt)) (9.37) k(a i cos(ωt) A i+1 cos(ωt)), i = 1, N ma i ω 2 = k(a i A i 1 ) k(a i A i+1 ), i = 1, N A i ω 9.6 9.6.1 9.6.2 135

9.6.3 9.6.4 9.6.5 136

10 10.1 x t x h(x, t) = A cos(ωt kx) (10.1) 2 t h(x, t) = 2 ω2 A cos(ωt kx) (10.2) = ω 2 h(x, t) (10.3) 2 x h(x, t) = 2 k2 A cos(ωt kx) (10.4) = k 2 h(x, t) (10.5) ωt kx = α (10.6) x = ω k t + α v = ω k v 137

[ 2 x 2 1 v 2 2 t 2 ]h(x, t) = k2 (1 ( ω vk )2 )h(x, t) = 0 (10.7) 10.2 h(x, t) = A cos(ωt kx) (10.8) A ω 2π 2π v = ω ω k λ = 2π T λ k h(x + λ, t) = h(x, t) = h(x, t + T ) (10.9) 10.3 h 1 (x, t) h 2 (x, t) h(x, t) = h 1 (x, t) + h 2 (x, t) (10.10) h 1 = A 1 cos(ω 1 t k 1 x), h 2 = A 2 cos(ω 2 t k 2 x) 138

10.3.1 θ h 1 (t, x) = A cos(ωt 1 k 1 x) (10.11) h 2 (t, x) = A cos(ω 1 t k 1 x + θ) (10.12) h(t, x) = h 1 (t, x) + h 2 (t, x) (10.13) θ = 0; h(t, x) = 2h 1 (t, x) (10.14) θ = π; h(t, x) = 0 θ = 2π; h(t, x) = 2h 1 (t, x) θ 10.4 10.4.1 v s λ = (c v s)t f s t = (c v s) f s (10.15) λ = c = v s = f s = 139

f l = (c + v L) λ = (c + v L) c v s f s (10.16) c = v L = f s = 10.4.2 c n 140

10.4.3 δω x(t) = A cos(ωt) + A cos((ω + δω)t) (10.17) = A(cos(ωt) + cos(ωt) cos(δωt) sin(ωt) sin(δωt)) = A(1 + cos(δωt)) cos(ωt) A sin(δωt) sin(ωt) = B(t) cos(ωt + α(t)) B(t) α(t) B(t) = A (1 + cos δωt) 2 + sin(δωt) 2 (10.18) sin δωt tan α(t) = 1 + cos δωt B(t) = A 4 cos δωt 2 tan α(t) = tan δωt 2 2 = 2A cos δωt 2 ω/2 (10.19) 10.4.4 141

n c n 142

10.5 10.5.1 10.5.2 10.5.3 143

10.5.4 10.5.5 10.5.6 144

1. L T M :[= d2 x(t) ] = LT 2 dt 2 := m d2 x[= LT 2 M] dt 2 : = d l F [L 2 T 2 M] : L 2 T 2 M : LT 1 M : [F/x = T 2 M] : T : T 1 :,L : LT 2 : [F ]L 2 M 2 = L 3 T 2 M 1 2. M 1 M 2 M 1 + M 2 M 1, M 2 F f d 2 M 1 dt x(t) = F f 2 (10.20) d 2 M 2 dt x(t) = + f 2 (10.21) (M 1 + M 2 ) d2 dt 2 x(t) = F (10.22) 145

(3) 3. v 0 M v 1 4. ( ) M x(t) ( ) ( ). M v v. ( ) M k ( ) ( ) t = 0 x(0) = A, ẋ(0) = 0. v 0 M v 1.. 146

M F v v. ( ) M k ( ) ( ) t = 0 x(0) = A, ẋ(0) = 0 M 2 v2 k 2 x2. 2M F v v 12. 13. i ii iii 3 147

r, θ xy- E = m 2 (ṙ2 + r 2 θ2 ) M z = mr 2 θ ṙ = d dt r(t), θ = d dt θ(t) 148

1. L T M 2. M 1 M 2 M 1 + M 2 1. 4 149

1. M 1 M 2 M 1 + M 2 x(0) = x 0 v(0) = v 0 t x(t) x 0, v 0.. i ii iii 3 r, θ xy- E = m 2 (ṙ2 + r 2 θ2 ) M z = mr 2 θ ṙ = d dt r(t), θ = d dt θ(t) 150

1. M 1 a 1 M 2 a 2. F U(x, y, z) F = ( x U(x, y, z), y U(x, y, z) U(x, y, z)) (10.23) z. i ii iii 3 r, θ xy- E = m 2 (ṙ2 + r 2 θ2 ) M z = mr 2 θ ṙ = d dt r(t), θ = d dt θ(t) 151

11 11.1 11.1.1 f(x) a < b x = a x = b x = c f(a) = f(b) = 0 (11.1) f (c) = 0, a < c < b (11.2) ( ) x = c f f(c + h) f(c) (c) = lim h +0 h f (c) = lim h 0 f(c + h) f(c) h 0, (11.3) 0 (11.4) f (c) = 0 (11.5) 153

11.1.2 f(x) g(x) g(x) = f(x) f(a) k(x a), k = f(b) f(a) b a (11.6) g(a) = 0, g(b) = 0 (11.7) g (c) = 0, a < c < b (11.8) c a b g (c) = f (c) k = 0 (11.9) c f (c) = k = f(b) f(a) b a (11.10) 11.1.3 1 n! x a f (n+1) (t)(x t) n dt (11.11) = 1 n! f (n) (t)(x t) n x 1 a + (n 1)! = 1 n! f (n) (a)(x a) n 1 + (n 1)! x a x a f n (t)(x t) n 1 dt f n (t)(x t) n 1 dt = = 1 n! f (n) (a)(x a) n 1 (n 1)! f (n 1) (a)(x a) n 1 1 (n 2)! f (n 2) (a)(x a) (n 2) + +f(x) f(a) (11.12) f(x) f(x) = f(a) + (x a)f (a) + + 1 n! f n (a)(x a) n + 1 n! 154 x a f (n+1) (t)(x t) n dt (11.13)

11.2 f(x) y = f(x) (11.14) x y f x y y x x = f 1 (y) (11.15) x y y = f 1 (x) (11.16) x = f(y) (11.17) x = f 1 (f(x)) (11.18) x = f(f 1 (x)) (11.19) 11.2.1 1 x y = x 2 (11.20) x = ± y (11.21) x 1/2 (± x) 2 = x (11.22) (x 2 ) = x (11.23) 155

2 y = e x (11.24) 3 x = log y (11.25) e logx = log e x = x (11.26) y = sin x (11.27) x = sin 1 y (11.28) sin(sin 1 (x)) = x (11.29) sin 1 (sin(x)) = x (11.30) 11.2.2 y = f 1 (x) (11.31) x = f(y) (11.32) 1 = d dx x = d dx f(y) = f (y) dy dx (11.33) 156

y dy dx = 1 f (y) y=f 1 (x) (11.34) y = log x (11.35) x = e y (11.36) 1 = e y dy dx (11.37) dy dx = 1 e = 1 y x (11.38) y = sin 1 x (11.39) x = sin y (11.40) 1 = cos y dy dx (11.41) dy dx = 1 cosy = 1 (1 x 2 ) 1/2 (11.42) 157

1 L T M 2 3 x 1 (t) = (1, t + 3t 2, 4t), x 2 (t) = (5 sin 2t, 5 cos 2t, 0), 4 m g cv c > 0 5 m (r, θ) (r,θ) r(t) d r(t) dt 158

1 L T M 2 3 t x 1 (t) = (t, t, 4t + 4.9t 2 ), x 2 (t) = (5 sin t, 5 cos t, 0), 4 m g cv c > 0 (1) 5 m (r, θ) (r,θ) r(t) d r(t) r(t) dt 159

1 L T 2 a, b, c, d n sin ax, cos bx, exp(cx + d), log x, x n, sin 1 x sin x 3 x 1 (t) = (1, t + 3t 2, 4t), x 2 (t) = (5 sin 2t, 5 cos 2t, 0), 4 f p+1 (x) f p (x) 1 x n! a f (n+1) (t)(x t) n dt (11.43) = 1 n! f (n) (t)(x t) n x a + 1 (n 1)! x a f n (t)(x t) n 1 dt = 1 n! f (n) (a)(x a) n 1 (n 1)! f (n 1) (a)(x a) n 1 1 (n 2)! f (n 2) (a)(x a) (n 2) + + f(x) f(a) (11.44) n! = n (n 1) (n 2) (n 3) (n 4) 3 2 1 (11.45) 5 [A change in motion is proportional to the motive force impressed and takes place along the straight line that force is impressed.] 160