2 2 2015/2/22 2

Copyright(C)2015 Amon. All rights reserved. 1

2 2 2015/2/22 2

1 11 1.1................... 11 1.2................. 12 1.3............................. 12 1.4.......................... 14 1.5.......................... 16 1.6......................... 16 1.7........................ 17 1.8............................. 18 1.9............................. 20 1.10............................... 22 1.11 3......................... 22 1.12.......................... 23 1.13........................ 24 1.14............................... 25 1.15 3........................ 27 1.16.................................. 28 1.17...................... 29 1.18.............................. 30 2 32 2.1.......................... 32 2.2........................... 33 2.3............................. 35 2.4................... 36 2.5...................... 37 2.6................................ 38 2.7...................... 40 3

2.8...................... 42 2.9............................. 43 2.10 1............................ 44 2.11......................... 45 2.12........................ 47 2.13 n............................. 48 2.14........................... 50 2.15.......................... 51 2.16............................... 52 2.17.............................. 53 2.18........................ 54 3 57 3.1............................... 57 3.2......................... 58 3.3......................... 59 3.4............................. 60 3.5............................... 60 3.6........................... 62 3.7................................... 63 3.8................................ 64 3.9............................. 65 3.10.............................. 65 3.11........................... 66 3.12......................... 67 3.13............................... 69 3.14........................... 70 3.15............................... 71 3.16........................... 73 3.17................................ 73 3.18........................ 75 3.19............................... 76 4

4 80 4.1............................. 80 4.2............................. 81 4.3................................ 82 4.4................... 82 4.5........................ 83 4.6............................... 84 4.7.................................. 86 4.8.............................. 87 4.9................................ 88 4.10............................. 89 5 92 5.1.............................. 92 5.2............................. 93 5.3............................. 94 5.4.......................... 95 5.5......................... 96 5.6..................... 97 5.7................... 99 5.8.............................. 100 5.9..................... 101 5.10.................................. 101 6 104 6.1................................ 104 6.2.............................. 105 6.3................................ 106 6.4.......................... 107 6.5...................... 108 6.6..................... 109 6.7........................ 109 5

6.8........................... 110 6.9............................. 111 6.10................................ 112 6.11............................. 112 6.12.................................. 113 6.13 2............................... 114 7 116 7.1.......................... 116 7.2............................... 117 7.3......................... 118 7.4............................. 120 7.5........................ 122 7.6.......................... 123 7.7........................... 124 7.8........................... 126 7.9......................... 127 7.10...................... 128 7.11 4............................... 129 7.12................................ 131 7.13............................. 131 7.14.................................. 132 7.15.......................... 133 7.16................. 135 8 138 8.1........................ 138 8.2............................... 139 8.3........................ 140 8.4.................... 141 8.5.............. 141 8.6...................... 142 6

8.7.......................... 144 8.8.......................... 145 8.9..................... 146 8.10...................... 147 8.11........................... 148 8.12............... 149 8.13.................................. 151 8.14............................... 152 8.15............................. 153 8.16.......................... 154 8.17................... 155 8.18.......................... 157 8.19............... 158 8.20................. 159 9 161 9.1......................... 161 9.2...................... 162 9.3........................ 164 9.4.................................. 166 9.5............................... 167 9.6............................. 169 9.7..................... 170 9.8.............................. 171 9.9.................................. 172 9.10............... 173 9.11.......................... 174 9.12............ 175 9.13........................... 177 9.14................................ 177 9.15............... 179 9.16........................... 180 7

9.17...................... 181 9.18 1.................... 182 9.19.................... 183 9.20................. 184 9.21............................. 185 10 188 10.1.................... 188 10.2................ 189 10.3................ 190 10.4................................ 191 10.5.............................. 193 10.6............................... 194 10.7......................... 195 10.8..................... 196 10.9......................... 198 10.10................... 199 10.11 3......................... 201 11 202 11.1........................... 202 11.2........................... 203 11.3.............................. 204 11.4........................ 205 11.5.................... 205 11.6.......................... 206 11.7.................... 207 11.8......................... 209 12 211 12.1........................ 211 12.2............................... 212 12.3......................... 214 8

12.4.............................. 216 12.5........................ 218 13 221 13.1........................... 221 13.2............................... 223 13.3.................................. 223 13.4.................................. 224 13.5................................ 225 13.6................ 226 13.7............................. 227 13.8............................. 228 13.9................................ 228 13.10.......................... 229 13.11............................... 230 13.12................................ 231 13.13............................... 232 13.14................................ 233 13.15......................... 233 13.16............................. 234 13.17.............................. 235 14 238 14.1.................................. 238 14.2................................ 238 14.3 SI............................... 239 14.4 SI............................... 239 14.5 SI................................ 240 14.6.............................. 240 14.7.............................. 241 14.8.............................. 241 14.9.............................. 242 9

14.10............................... 242 15 243 10

1 ( ) 1.1 N (x 1, x 2,, x N ) R N R N N R N N 2 P, Q (x 1, x 2,, x N ), (x 1 + x 1, x 2 + x 2,, x N + x N ) PQ s s 2 = N ( x i ) 2 = ( x 1 ) 2 + ( x 2 ) 2 + + ( x N ) 2, s 0 i=1 N N x i 1 s 2 = x i x i i ( ) s 2 = x j x j 11

1: 2 1.2 { 1 (i = j) δ ij = 0 (i j) δ ij A j = A i j N ɛ ij k = 1 ( ) 1 (N i, j, k 1, 2, N ) 0 ( ) ɛ ij k 2 ɛ ij k ɛ 12 N = 1 N 3 : ɛ ijk = ɛ kij = ɛ jki 1.3 12

N A det A = 1 N! ɛ i 1 i 2 i N ɛ j1 j 2 j N A i1 j 1 A i2 j 2 A in j N det(ca) = c N det A, det A T = det A c ( ) A T A (A T ) ij = A ji f i1 i 2 i N = ɛ j1 j 2 j N A i1 j 1 A i2 j 2 A in j N i 1, i 2,, i N = ɛ i1 i 2 i N f 12 N. f i1 i 2 i N ɛ j1 j 2 j N A i1 j 1 A i2 j 2 A in j N = ɛ i1 i 2 i N ɛ j1 j 2 j N A 1j1 A 2j2 A NjN. ɛ i1 i 2 i N ɛ i1 i 2 i N = N! det A = ɛ j1 j 2 j N A 1j1 A 2j2 A NjN, ɛ i1 i 2 i N det A = ɛ j1 j 2 j N A i1 j 1 A i2 j 2 A in j N 2 N A, B det(ab) = ɛ j1 j 2 j N (AB) 1j1 (AB) 2j2 (AB) NjN = ɛ j1 j 2 j N (A 1i1 B i1 j 1 )(A 2i2 B i2 j 2 ) (A NiN B in j N ) = A 1i1 A 2i2 A NiN ɛ i1 i 2 i N det B = det A det B. à ij = det A A ij = 1 (N 1)! ɛ ii 2 i N ɛ jj2 j N A i2 j 2 A in j N = (A i j ) ( 1) i+j A 1 A kj à ij = (N 1)! ɛ ii 2 i N ɛ jj2 j N A kj A i2 j 2 A in j N 1 = (N 1)! ɛ ii 2 i N ɛ ki2 i N det A 13

ɛ ii2 i N ɛ ki2 i N = (N 1)! δ ik A kj à ij = δ ki det A AÃT = det A δ ( δ ). à ij A ik = δ jk det A ÃT A = det A δ A (AB = BA = δ B) A 1 = 1 det A ÃT A à T A det A = 0 A ( ) ( ) A kj à ij = δ ki det A k = i = 1 det A = A 1j à 1j 3 a b c ( ) ( ) ( ) det d e f e f d f d e = a det + b det ( 1) + c det h i g i g h g h i = a(ei fh) b(di fg) + c(dh eg) 4 1.4 x i x i = x i a i ( a i ) x i x i = x i x i x i = x i x i = s 2 x i = Λ ij x j, Λ ij Λ ik = δ jk x i x i = Λ ij x j x i x i = Λ ij Λ ik x j x k = δ jk x j x k = x j x j = s 2 14

: Λ ij Λ ik = δ jk Λ T Λ = δ ( δ ) Λ (det Λ) 2 = 1 det Λ = ±1 det Λ = +1 det Λ = 1 ( ) 2 ( ) x i = kx i ( k 0 ) 2 3 2 x 1, x 2, x 3 2: ( ) ( ) ( ) 15

1.5 x i = Λ ij x j a i 2 s = x i x i A i = Λ ij A j A i x i x i = Λ ij x j x i A i (A 1, A 2,, A N ) A (A) i A = (A 1, A 2,, A N ), (A) i = A i A = B A i = B i ( i) 3 x i r (r) i = x i ( ) 1.6 M T ij k x i = Λ ijx j a i T ij k = Λ il Λ jm Λ kn T lm n T ij k M 1 0 16

Λ ij δ ij = Λ il Λ jm δ lm δ ij 2 2 A i, B i (A i B j ) = A ib j = Λ il Λ jm A l B m A i B j 2 (A i B i ) = A ib i = Λ il Λ im A l B m = δ lm A l B m = A l B l A i B i T ij k = det Λ Λ il Λ jm Λ kn T lm n T ij k M 1 0 (det Λ) 2 = 1 ɛ ij k = det Λ Λ il Λ jm Λ kn ɛ lm n N N det Λ = +1 1.7 2 A, B A B = A i B i 2 B A = A B. 17

A = A A A ( ) A 2 A 0 A 0 0 A Ā = A A 1 A 1 1.8 2 2 A, B Ā = (1, 0, 0,, 0), B = (a, b, 0,, 0), a 2 + b 2 = 1 ( 3) 3: 2 θ 2 C = { (x 1, x 2 ) x 1 = 1 t 2, x 2 = t, 0 < t < b } 18

θ = C ds = b 0 (dx1 ) 2 dt + dt ( ) 2 b dx2 = dt 0 θ a 0, b 0 dt 1 t 2 ( ) a = 1 2 a = 1 2 a = 0 2 π/2 π π = 2 1 0 dt 1 t 2 3.14159. R 2πR a = cos θ, b = sin θ, tan θ = sin θ cos θ cos 2 θ + sin 2 θ = 1. 4 4: Ā B = 1a + 0b = a = cos θ A B = A B cos θ. 19

A B 2 = (A B) (A B) = A A + B B 2A B = A 2 + B 2 2 A B cos θ 2 3 ( 5) A, B cos θ = 0 A B 2 = A 2 + B 2 ( ) 5: 1.9 2 6 3 p = (cos α, sin α), q = ( sin α, cos α), r = cos β p + sin β q = (cos α cos β sin α sin β, sin α cos β + cos α sin β) r = (cos(α + β), sin(α + β)) cos(α + β) = cos α cos β sin α sin β, sin(α + β) = sin α cos β + cos α sin β ( π ) ( π ) sin 2 θ = cos θ, cos 2 θ = sin θ. 20

6: cos(2α) = cos 2 α sin 2 α, sin(2α) = 2 sin α cos α, cos 2 α = 1 + cos(2α) 2, sin 2 α = 1 cos(2α) 2 sin θ dt θ = θ 1 t 2 1 = d sin θ dθ d sin θ d sin θ 0 d cos θ dθ dt = d sin θ 1 t 2 dθ 0 1 1 sin 2 θ d sin θ dθ = d ( π ) ( π ) dθ sin 2 θ = cos 2 θ = sin θ = cos θ ( ) : sin θ lim θ 0 θ = 1 ( ) ( ) : sin θ lim θ 0 θ f(x) f(0) = g(0) = 0 lim x 0 g(x) = lim f (x) x 0 g (x) = lim θ 0 cos θ = 1 21

1.10 2 A, B (A B) ij = A i B j 2 2 T ij A i (T A) i = T ij A j, (A T ) i = A j T ji B T A = B i T ij A j B T T A ( ) (A B) ij = A i B j. (A B)(C) = (B C)A (A B) C = A(B C) = A B C 1.11 3 3 2 A, B (A B) i = ɛ ijk A j B k = (A 2 B 3 A 3 B 2, A 3 B 1 A 1 B 3, A 1 B 2 A 2 B 1 ) i ( ) 2 : B A = A B. A = ( A, 0, 0), B = ( B cos θ, B sin θ, 0) θ 2 A B = (0, 0, A B sin θ) A B A B A B sin θ 0 < θ < π 3 A B ( 7) 3 A, B, C A 1 A 2 A 3 [A, B, C] = A (B C) = ɛ ijk A i B j C k = det B 1 B 2 B 3 C 1 C 2 C 3 22

7: ( ) 3 : [A, B, C] = [C, A, B] = [B, C, A] ɛ ijk ɛ ilm = δ jl δ km δ jm δ kl A (B C) = A C B A B C 1.12 3 dθ n dθ = dθ n dθ = dθ n A A da = A A A sin φ dθ n A ( 8) da = A sin φ dθ n A n A = dθ A = A sin φ dθ n A A sin φ 23

8: 1.13 N ( ) ξ i t i = r ξ i e i = t i t i e i = 1. 2 r = (r cos θ, r sin θ) ξ 1 = r, ξ 2 = θ t 1 = r r = (cos θ, sin θ), t 2 = r θ = ( r sin θ, r cos θ) t 1 = 1, t 2 = r e 1 = (cos θ, sin θ), e 2 = ( sin θ, cos θ) r = re 1 r 24

e i e 1 e 2 = 0 e i e j = δ ij x i, x i = Λ ijx j a i Λ ij x j = x i + a i Λ ik Λ ij x j = Λ ik (x i + a i ) x k = Λ ik (x i + a i ) ( ) r (t i) j = x = x j i x = Λ ij. i j t i t j = δ ij t i = 1 (e i ) j = Λ ij (A ie i) j = Λ ik A k Λ ij = A j = (A) j A = A ie i. A A = A i e i 2 T T = T ij e i e j 1.14 3 x i d 3 r = dx 1 dx 2 dx 3 ( ) : dx dy = dy dx 25

dx dx = 0 n n d 3 r 3 ξ i d 3 r = x 1 ξ i x 2 ξ j x 3 ξ k dξ i dξ j dξ k = ɛ ijk x 1 ξ i x 2 ξ j x 3 ξ k dξ 1 dξ 2 dξ 3 = det x ξ dξ 1 dξ 2 dξ 3. det( x/ ξ) x i / ξ j 3 det x [ ] ξ = ɛ x i x j x k r r r ijk = ξ 1 ξ 2 ξ 3 ξ 1, ξ 2, ξ 3 3 a, b, c V r = ξ 1 a + ξ 2 b + ξ 3 c (0 ξ i 1) 1 1 1 [ ] r d 3 r r r = dξ 1 dξ 2 dξ 3 = [a, b, c ] ξ 1, ξ 2, ξ 3 V 0 0 0 3 (2 ) (d 2 r) i = 1 2 ɛ ijk dx j dx k 2 ξ i (i = 1, 2) (d 2 r) i = 1 ( 2 ɛ x j x k r ijk dξ l dξ m = r ) dξ 1 dξ 2 ξ l ξ m ξ 1 ξ 2 2 a, b S r = ξ 1 a + ξ 2 b (0 ξ i 1) ( ) 1 1 d 2 r r = dξ 1 dξ 2 r = a b ξ 1 ξ 2 S 0 0 i 26

N x i N d N r = 1 N! ɛ i 1 i 2 i N dx i1 dx i2 dx in = dx 1 dx 2 dx N (N 1) (d N 1 1 r) i = (N 1)! ɛ ii 2 i N dx i2 dx in (N 2) 2 (N 3) 3 ( ) dx dy dxdy dx dy dxdy = dydx. ( 1) ( 1) = 1 1.15 3 3 r = (r sin θ cos φ, r sin θ sin φ, r cos θ), 0 r <, 0 θ π, 0 φ 2π 3 r, θ, φ ( 9) 3 r = r. r r = (sin θ cos φ, sin θ sin φ, cos θ) = r r, r = (r cos θ cos φ, r cos θ sin φ, r sin θ), θ r = ( r sin θ sin φ, r sin θ cos φ, 0) φ 27

9: 3 r θ r φ = (r2 sin 2 θ cos φ, r 2 sin 2 θ sin φ, r 2 sin θ cos θ) = r sin θ r, [ ] r r r = r r, θ, φ r (r sin θ r) = r2 sin θ R R π 2π [ ] r d 3 r r r = dr dθ dφ r, θ, φ = 0 R 0 dr 0 π π d 2 r = dθ = 0 π 0 0 dθ dθ 2π 0 2π 0 0 2π 0 dφ dφ r 2 sin θ = 4 3 πr3 r θ r φ r=r dφ R 2 sin θ = 4πR 2 1.16 x i i = / x i i = x i = x j x i 28 x j = Λ ij j

( ) i = i. φ(r) φ(r) A(r) A(r) 3 A(r) A(r) ( ) 2 ( ) 2 ( ) 2 = = i i = + + + x 1 x 2 x N 3 gradφ = φ, diva = A, rota = curla = A 1.17 N ( ) n V, (n 1) V, (n 1) α α = dα V V d ( ) d : dφ = φ dξ i, d 2 = 0, d(α β) = dα β + ( 1) s α dβ. ξ i ( φ 0 α s s 0 ) [ ] C A, B (0 ) φ C dφ = φ(b) φ(a) φ = φ(b) φ(a) C 29

n = 1 n 2 n V V V = { (ξ 1,, ξ N ) 0 ξ 1 1, π ξ p π, ξ q = 0 } V = { (ξ 1,, ξ N ) ξ 1 = 1, π ξ p π, ξ q = 0 } ( p = 2,, n, q = n + 1,, N ) ξ i ξ 1 p ξ p = π ξ p = π V ( ) (n 1) α = ξ k 1f(ξ 2, ξ N ) dξ 2 dξ n (k = 0, 1, 2, ) dα = kξ1 k 1 f(ξ 2, ξ N ) dξ 1 dξ 2 dξ n + ( dξ q ) V α V dα f(ξ2,, ξ n ) dξ 2 dξ n k = 0 k = ɛ +0 α dξ 1 dξ q V α V dα 0 V α V dα ( ) (n 1) α V V [ ] (* ) 4 1 n 1 n (S n ) V S n 1 2 r, θ ξ 1 = r/r, ξ 2 = θ R 3 r, θ, φ ξ 1 = r/r, ξ 2 = 2 arctan(tan(θ/2) cos φ), ξ 3 = 2 arctan(tan(θ/2) sin φ) R arctan tan 1.18 3 (0 ) φ = φ(r) dφ = dx i i φ, d(dx i φ) = dx i dφ = dx i dx j j φ = ɛ ijk d 2 x k j φ = ɛ ikj d 2 x k j φ, d(d 2 x i φ) = d 2 x i dφ = 1 2 ɛ ijk dx j dx k dx l l φ = 1 2 ɛ ijk ɛ jkl dx 1 dx 2 dx 3 l φ = δ il d 3 x l φ = d 3 x i φ. 30

d 3 x = d 3 r, d 2 x i = (d 2 r) i dx i dx j = ɛ ijk d 2 x k 1 C, 2 S, 3 V = dr, dr = d 2 r, d 2 r = d 3 r C C S S 2 S 10 S r r dξ 1 dξ 2 S S ξ 1 ξ 2 V 10 V ( ) V r r dξ 2 dξ 3 V ξ 2 ξ 3 V V 10: 3 ( ) 3 3 31

2 2.1 e = lim h 0 (1 + h) 1/h 2.71828 x exp x = e x (log) y = log x x = e y x > 0 ( 1) log(xy) = log x + log y, log(x y ) = y log x log(1 + h) ) lim = lim log ((1 + h) 1/h = log e = 1 h 0 h h 0 d dx log x = 1 x, d dx ex = e x cosh x = ex + e x 2, sinh x = ex e x 2, tanh x = sinh x cosh x 32

1: cosh 2 x sinh 2 x = 1, d dx cosh x = sinh x, d dx sinh x = cosh x. {(cosh θ, sinh θ) θ } cos θ, sin θ ( ) ln log y = log x x = 10 y Excel Excel VBA log log 2.2 x, y z = x + iy i i 2 = 1 33

z = x + iy x z Re z y z Im z (x, y) 2 z = x + iy e z = e x (cos y + i sin y) z, w e z e w = e z+w [ ] z = x + iy, w = u + iv e z e w = e x (cos y + i sin y) e u (cos v + i sin v) = e x+u( cos y cos v sin y sin v + i(sin y cos v + cos y sin v) ) = e x+u( cos(y + v) + i sin(y + v) ) = e x+u+i(y+v) = e z+w. [ ] θ e iθ = cos θ + i sin θ z = x + iy x = r cos θ, y = r sin θ r, θ z = re iθ ( 2) 2: r = x 2 + y 2 z ( ) z θ = arctan(y/x) z Arg z zw = z w, Arg(zw) = Arg z + Arg w 34

w = log z z = e w. n e i2πn = 1 r, θ e log r+i(θ+2πn) = re iθ log(re iθ ) = log r + i(θ + 2πn) ( ) z cos z = eiz + e iz, sin z = eiz e iz 2 2i cos z = cosh(iz), sin z = i sinh(iz). a z = e z log a a, z a z a w = a z+w, (a z ) w = a zw, log(zw) = log z + log w, log(a z ) = z log a [ ] 2 z 2 = i [ ] z = i 1 2 = e 1 2 log i = e 1 2 log eiπ/2 = e 1 2 i( π 2 +2πn) = e i( π 4 +πn) = ± 1 + i 2. [ ] 2.3 f, S ( ) z S lim f(z + z) = f(z) z 0 f S z z 0 f(z) ( ) f(z + z) f(z) z S lim = f (z) z 0 z 35

f f S ( ) f f u + iv = f(x + iy) u, v x, y 2 z = h ( h ) f(z + z) f(z) lim z 0 z = lim h 0 f(x + iy + h) f(x + iy) h = lim h 0 u(x + h, y) + iv(x + h, y) u(x, y) iv(x, y) h z = ih f(z + z) f(z) lim z 0 z = v y i u y = u x + i v x. f u x = v y u y = v x d dz za = az a 1, d d sin z = cos z, dz d dz az = a z d log a, dz log z = 1 z, cos z = sin z. dz ( ) 2 u, v u = 0, v = 0 = ( / x) 2 + ( / y) 2 2 φ = 0 2 φ u, v 2.4 f : f(z) = c 0 + c 1 z + c 2 z 2 + = c k z k. c k (k = 0, 1, 2, ) n z = 0 f n f (n) f (n) (0) = c n n! f(z) = k=0 36 f (k) (0) k! z k. k=0

f f(z) = e z f (k) (0) = 1 e z = k=0 1 k! zk. g(z) = f(z a) g(z) = k=0 g (k) (a) k! (z a) k g a 2.5 C dz f(z) C dz x, y dx + idy S f dz f(z) = 0 S S S [ ] z = x + iy, f(z) = u + iv d(dz f(z)) = dz df(z) = (dx + idy) (du + idv) ( u u v = (dx + idy) dx + dy + i x y x = dx dy ( i u x v x u y i v y ) dx + i v y dy S 0 : S α = S dα α = dzf(z) [ ] S f 3 2 C, C f ) 37

3: S g g(z) dz (z a) = 2πi n (n 1)! g(n 1) (a) (a S, n = 1, 2, ) S S [ ] dz g(z)/(z a) S a a z = a + re iθ (r 0) dz g(z) 2π z a = lim ire iθ dθ g(a + reiθ ) = 2πi g(a). r 0 re iθ S 0 a (n 1) [ ] 2.6 z (1 ) f(z) z = a a f lim (z z a a)n f(z) n z f(z) = i 3 (z i) 3 2 2 (z + 2) 2 f S a i (i = 1, 2,, N) S dz f(z) = 2πi Res(a) = 1 (n 1)! N i=1 lim z a Res(a i ), ( ) n 1 d (z a) n f(z) ( n a ) dz Res(a) 38

[ ] S dz f(z) S f ( 4 ) a f(z) = g(z)/(z a) n g(z) a [ ] 4: 2 [ ] : I = 0 dx x 4 + 1 [ ] R 5 C I dz = z 4 + 1 C 5: C C e iπ/4, e i3π/4 1 Res(e iπ/4 ) = z e iπ/4 lim z e iπ/4 z 4 + 1 = lim 1 z e iπ/4 4z = e i3π/4 3 4. 39

Res(e i3π/4 ) = e iπ/4 /4 ( ) e I i3π/4 = 2πi + e iπ/4 = π 4 4 2. R I dx π = R x 4 + 1 + ire iθ dθ 0 (Re iθ ) 4 + 1 R 2I, 0 I = 2I. [ ] : I = 2π 0 I = I 2 = π 2 2. [ ] dφ (1 + ɛ cos φ) 2 0 < ɛ < 1. [ ] z = e iφ dφ = dz/(iz), cos φ = (z + z 1 )/2 dz 1 I = C iz (1 + ɛ z + ) 2 = 4 z dz z 1 iɛ 2 ( C z 2 + 2 ) 2 2 ɛ z + 1. C 0 2 z 2 + (2/ɛ)z + 1 = 0 α ± = 1 ± 1 ɛ 2 ɛ α + C I = 4 z dz iɛ 2 C (z α + ) 2 (z α ) = 4 d z 2πi lim 2 iɛ2 z α + dz (z α ) 2 = 8π z α lim ɛ 2 z α + (z α ) = 8π α + α 3 ɛ 2 (α + α ) = 8π 2/ɛ 3 ɛ 2 ( 2 1 ɛ2 /ɛ ) 3 = 2π (1 ɛ 2 ) 3/2. [ ] 2.7 z = x + iy z = x iy z z z = z 2 0 ( z = 0 ). 40

2 z, w (z + w) = z + w, (zw) = z w. (e z ) = e z z 1, z 2, f f(z 1, z 2, ) = f(z 1, z 2, ) A (A ) ij = A ij A = A T (A + B) = A + B, (ca) = c A, (AB) = B A c ( N 1 ) a, b a b = a b = a i b i (a b) = b a, a a 0 ( a = 0 ) a = a a ( ) A = A A A A = δ A ( δ ) ( ) : ( a = Aa b = Ab A A = δ ) a b = a b. 41

2.8 N A Av = λv (v 0) A λ v A N λ k (k = 1, 2,, N) v k A(v 1, v 2,, v N ) = (λ 1 v 1, λ 2 v 2,, λ N v N ) = (v 1, v 2,, v N ) diag(λ 1, λ 2,, λ N ). B = (v 1, v 2,, v N ) v k diag(λ 1, λ 2,, λ N ) λ 1, λ 2,, λ N ( 0 ) B A = B diag(λ 1, λ 2,, λ N )B 1 A ( ) 0 1 [ ] 2 A = 1 0 [ ] ( ) ( ) 0 1 λ 0 Av = λv v = 1 0 0 λ v ( ) λ 1 v = 0. 1 λ ( ) λ 1 v 0 det = λ 2 1 = 0 λ = ±1. 1 λ ( ) 1 v = ( ) ±1 ( ) 1 1 A = B diag(1, 1)B 1, B =. [ ] 1 1 v B 42

2.9 A e A = n=0 1 n! An : (a + b) s = s n=0 s! n!(s n)! an b s n A, B e A e B = = n=0 m=0 s=0 1 n!m! An B m = 1 s! (A + B)s = e A+B s=0 s n=0 1 n!(s n)! An B s n e A e A A e ia A, B (BAB 1 ) n = BA n B 1 e BAB 1 = B e A B 1. diag(λ 1, λ 2,, λ N ) n = diag(λ n 1, λ n 2,, λ n N ) e diag(λ 1,λ 2,,λ N ) = diag(e λ 1, e λ 2,, e λ N ) ( ) 0 1 [ ] A = U = e iθa 1 0 [ ] U = e iθb diag(1, 1)B 1 = B e diag(iθ, iθ) B 1 = B diag(e iθ, e iθ )B 1 ( ) ( ) ( ) ( ) 1 1 e iθ 0 1 1 1 cos θ i sin θ = 1 1 0 e iθ =. [ ] 2 1 1 i sin θ cos θ ( ) 43

2.10 1 1 1 dy dx = f(x, y) dy dx = p(x)q(y) x, y dy q(y) = dx p(x) dy ( y ) dx = f x u = y/x dy [ ] dx = y y + x [ ] x u = y/x d(xu) dx = xu xu + x u + x du dx = u u + 1 x du dx = u2 u + 1 0 u = 0 u = 0 ( x) u 0 du u + 1 dx = log u 1 u 2 x u = log x + C C u = y/x y x = y log y Cy u = 0 y = 0 y = 0 x = y log y Cy [ ] 44

2.11 x D = d/dx φ N φ(d)y = f(x) N f N φ n a n (a N D N + a N 1 D N 1 + + a 1 D + a 0 ) y = f(x) a n x y 1 f(x) = 0 ( ) f(x) 1 φ(d)y = 0 ( ) 1 φ(d)y = 0 φ(d)y = 0 y = e λx φ(λ) = 0 N N λ 1, λ 2,, λ N y = e λ nx (n = 1, 2,, N) y = C 1 e λ 1x + C 2 e λ 2x + + C N e λ Nx C n N λ m (D λ)f(x)e λx = f (x)e λx (D λ) m f(x)e λx = f (m) (x)e λx e λx, x e λx,, x m 1 e λx [ ] y (4) y (3) 3y (2) + 5y (1) 2y = 0 y (n) x y n [ ] (D 4 D 3 3D 2 + 5D 2)y = 0 (D 1) 3 (D + 2)y = 0 (λ 1) 3 (λ + 2) = 0 λ = 1 (3 ) λ = 2 45

y = C 1 e x + C 2 x e x + C 3 x 2 e x + C 4 e 2x C 1, C 2, C 3, C 4 [ ] [ ] y 5y + 6y = x 2 + 1 [ ] 0 (D 2 5D + 6)y = 0 (D 2)(D 3)y = 0 y = C 1 e 2x + C 2 e 3x. y = y = 18x2 + 30x + 37 108 1 D 2 5D + 6 (x2 + 1) = 1 6 x2 + 5 18 x + 37 108 + C 1 e 2x + C 2 e 3x [ ] (D 2 5D + 6) 1 (x 2 + 1) (D 2 5D + 6) (x 2 + 1) 6 6: 1 φ(d) eαx f(x) = e αx 1 φ(d + α) f(x) [ ] D e αx f(x) = e αx (D +α)f(x) φ φ(d)e αx f(x) = e αx φ(d + α)f(x) φ(d)e αx 1 φ(d + α) f(x) = 1 eαx φ(d + α) φ(d + α) f(x) = eαx f(x) 46

[ ] [ ] y + y = cos(ax) A [ ] 0 (D 2 + 1)y = 0 λ 2 + 1 = 0 λ = ±i y = C 1 e ix + C 2 e ix. y = B 1 cos x + B 2 sin x B 1, B 2 cos(ax) = Re e iax 1 y = D 2 + 1 Re eiax = Re e iax 1 (D + ia) 2 + 1 1 = Re e iax 1 1 A 2 + 2iAD + D 2 1 = Re e iax 1 1 A 2 = cos(ax) 1 A 2 (A 1) Re e ix x 2i = x sin x 2 cos(ax) (A 1) 1 A y = B 1 cos x + B 2 sin x + 2 x sin x (A = 1) 2 [ ] (A = 1) ( ) 1 1 2.12 I = dx e x2 I 2 = dx e x2 dy e y2 = dx dy e x2 y 2 47

2π I 2 = dr dθ r e r2 = 2π dr r e r2. 0 0 0 s = r 2 I 2 = π dx e ax2 = dx e x2 = π. π a x > 0 Γ(x) = 0 dt t x 1 e t (a > 0) Γ(1) = 1, Γ(x + 1) = xγ(x) n Γ(n) = (n 1)! Γ(1/2) ( 1 Γ = 2) π. Γ(1 + ɛ) = ɛ Γ(ɛ) lim ɛ Γ(ɛ) = 1. ɛ 0 Γ(x) x 0 1 ( ) x > 0 x ( ) Γ(x + 1) = xγ(x) 0 Γ( 1/2) = Γ(1/2)/( 1/2) = 2 π. 2.13 n n I = dx 1 dx 2 dx n e (x2 1+x 2 2+ +x 2 n) 48

n I = π n/2. R n V n (R) = C n R n R n S n (R) = d dr V n(r) = n C n R n 1 I I = dr n C n r n 1 e r2 = n C n 0 2 = n C ( n n ) ( n ) 2 Γ = C n Γ 2 2 + 1 0 ds s n/2 1 e s ( n ) 2 + 1. 2 I C n = π n/2 /Γ V n (R) = πn/2 R n ( n ) Γ 2 + 1 V 2 (R) = πr 2, V 3 (R) = 4 3 πr3 R 4 V 4 (R) = 1 2 π2 R 4 ( ) 4 (x, y, z, w) x = r cos ψ, y = r sin ψ cos θ, z = r sin ψ sin θ cos φ, w = r sin ψ sin θ sin φ ( 0 r <, 0 ψ < π, 0 θ < π, 0 φ < 2π ) 4 4 1 1 x/ r x/ ψ x/ θ x/ φ det y/ r y/ ψ y/ θ y/ φ z/ r z/ ψ z/ θ z/ φ = r3 sin 2 ψ sin θ w/ r w/ ψ w/ θ w/ φ R 4 R π π 2π V 4 (R) = dr r 3 dψ sin 2 ψ dθ sin θ dφ = R4 0 0 0 4 π 2 2 2π = 1 2 π2 R 4. 0 49

2.14 Γ(x + 1) = 0 dx t x e t = 0 dx e f(t), f(t) = t x log t f(t) t ( ) f(t) t = x f(t) t = x f(t) = x x log x + 1 2x (t x)2 +. Γ(x + 1) e x+x log x dt e (t x)2 /(2x) x e x+x log x dt e (t x)2 /(2x) = 2πx x x e x x Γ(x + 1) 2πx x x e x 2 2 1.9190044 1.0422071 4 24 23.506175 1.0210083 6 720 710.07818 1.0139728 8 40320 39902.395 1.0104657 10 3628800 3598695.6 1.0083654 n log n! n log n 1 log(2πn) 2n n log n! n log n n 50

2.15 1 A p n A k n! P (k) = k!(n k)! pk q n k ( q = 1 p ) n, k, n k log P (k) n log n n k log k + k (n k) log(n k) + (n k) + k log p + (n k) log q = k log k (n k) log(n k) + k log p q + ( ) k = np + x x ( log P (k) (np + x) log 1 + x ) ( (nq x) log 1 x ) np nq + ( ) log(1 + x) = x (1/2)x 2 + 1/n ( ) log P (k) x2 2npq + ( ) P (k) ( ) exp (k np)2. 2npq 1 P (k) N (k; np, npq), N (x; µ, σ) = 1 ) exp ( (x µ)2 2π σ 2σ 2 N (x; µ, σ) µ σ ( 7) 7: 51

x N (x; µ, σ) x µ < sσ µ+sσ 2 s 0.6827 (s = 1) M(s) = dx N (x; µ, σ) = dt e t2 /2 0.9545 (s = 2) µ sσ π 0 0.9973 (s = 3) P (k) k np < s npq M(s) 2.16 x 1 δ(x) = N (x; 0, 0) = lim e x2 /ɛ 2 ɛ +0 πɛ δ(x) = 0 (x 0). δ(0) x = 0 : δ( x) = δ(x). 0 a δ(ax) = 1 a δ(x) 0 n dx xn δ(x) n 0 ( ) dx x n δ(x) = 2 dx x n 1 e x2 /ɛ 2 = ɛn n + 1 Γ (ɛ 0) πɛ π 2 0 dx x n δ(x) = δ n0 52

δ nm f(x), y f (n) (y) dx f(x)δ(x y) = dx (x y) n δ(x y) n! n=0 f (n) (y) = δ n0 = f(y) n! n=0 : n a nδ nm = a m δ(x y) δ nm : dk k δ(x) = lim 2. ɛ +0 2π eikx ɛ2 k ɛ +0 δ(x) = dk 2π eikx 2.17 k, x φ k (x) = 1 e ikx 2π : dx φ k(x)φ k (x) = δ(k k ), ( ) f(k) = dx φ k(x)f(x) dk φ k (x)φ k(x ) = δ(x x ) f(x) dk f(k)φ k (x) = = dk φ k (x) dx φ k(x )f(x ) dx δ(x x )f(x ) = f(x). 53

f(x) = dk f(k)φ k (x) dx f(x) 2 = dk f(k) 2 (* ) v k (k = 1, 2, ) v k v k = δ kk x (v k) x (v k ) x = δ kk. v k v k = δ (v k ) x (v k) x = δ xx k k 2.18 Z x D(x) = 1 e inx 2π n Z π π dx D(x) = 1, D(x + 2π) = D(x) cos(nx) sin( x 2 2πD(x) = 1 + 2 cos(nx) = 1 + 2 ) sin( x 2 ) = 1 + n=1 n=1 n=1 sin((n+ 1 2 )x) sin((n 1 2 )x) sin( x 2 ) = lim Λ sin(λx) sin( x 2 ) x = 2πn (n Z) ( ) D(x) = 0 (x 2πn). D(x) = δ(x 2πn) n Z 54

n Z, x ( π, π) φ n (x) = 1 2π e inx : π dx φ n(x)φ n (x) = δ nn, φ n (x)φ n(x ) = δ(x x ) π f(x) f n = π π n Z dx φ n(x)f(x) ( ) f(x) = n Z f n φ n (x) π π dx f(x) 2 = n Z f n 2 f(x) = x f n = i 2π ( 1) n n (n 0), f0 = 0 1 n = π2 2 6 n=1 ( ) n = 1, 2, ( ) φ n (x) φ = 1 ( ) ( ) 1 1 φn (x) = 1 ( ) cos(nx) n(x) 2 i i φ n (x) π sin(nx) φ 0(x) = φ 0 (x) = 1 2π 55

φ n(x) (n Z) x ( π, π) x (0, π) {cos(nx) n = 0, 1, 2, } {sin(nx) n = 1, 2, } (* ) (a, b) φ dx φ(x)f(x) = 0 x (a, b) (f(x) = 0) a ( ) b 56

3 3.1 a m a, r a, F a t ṙ a 2 r a (1) F a = 0 r a = 0 1 (2) F a = m a r a (3) a b F ab F ab = F ba // (r a r b ). 2 ( 1) (4) a F a F a = b F ab. 57

1: ( ) ( ) 3.2 r r = r V t V r = r 2: 58

? 3.3 r r = r R R r = r + R a F a = m a ( r a + R) F a m a R = ma r a m a R e i (i = 1, 2, 3) de i dθ de i = dθ e i ė i = ω e i, ω = dθ dt ω A = A i e i Ȧ = Ȧie i + A i ė i = A + ω A. A = Ȧie i r ṙ = r + ω r. r = d r + ω r + ω ṙ = r + ω r + ω r + ω ( r + ω r) dt = r + 2ω r + ω (ω r) + ω r a F a 2m a ω r a m a ω (ω r a ) m a ω r a = m a r a 59

2m a ω r a m a ω (ω r a ) m a ω r a ω ( ) 2 3.4 1 ( ) F g ab = Gm am b (r a r b ) r a r b 3 G G 6.674 10 11 m 3 /(kg s 2 ) F g ab = Gm am b r a r b 2 2 2 2 ( ) ( ) ( ) : F e ab = q aq b (r a r b ) ɛ 4πɛ 0 r a r b 3 0 8.854 10 12 s 2 C 2 /(kg m 3 )., ɛ 0 q a a C( ) 3.5 r d 3 r dm dm = ρ(r)d 3 r ρ(r) 60

b ρ(r) a F g ab = Gm m b (r a r b ) a = Gm r a r b 3 a d 3 r ρ(r)(r a r) r a r 3 b a r a = (0, 0, h) 3 r = (r sin θ cos φ, r sin θ sin φ, r cos θ) r 2 sin θ ( ) b F g ab = Gm a R 0 dr π 0 dθ 2π 0 dφ r 2 sin θ ρ(r)( r sin θ cos φ, r sin θ sin φ, h r cos θ) ( r2 sin 2 θ + (h r cos θ) 2) 3/2. R ρ(r) r ρ(r) φ b F g ab = 2πGm a(0, 0, 1) R 0 dr r 2 ρ(r) π 0 dθ sin θ (h r cos θ) (h 2 +r 2 2hr cos θ) 3/2. θ h 2 +r 2 2hr cos θ h r 2/h 2 b F g ab = 4πGm a(0, 0, 1) h 2 R 0 dr r 2 ρ(r) = GMm ar a r a 3 M = R 0 dr 4πr2 ρ(r) 2 ( r a R) b F g ab = m ag, g = g e, e = r a R, g = GM R 2 9.8 m/s 2 e g ( ) g G R 6380 km. g = GM/R 2 M 6.0 10 24 kg G 61

3.6 S P = a S m a ṙ a S Ṗ = m a r a = F a = F ab + F ab. a S a S a S b S a S 1 (F ab + F ba ) 2 a S b S Ṗ = F, F = a S F 0 Ṗ = 0 S S S P S b/ S J = a S m a r a ṙ a F ab b/ S S J = m a r a r a = r a F a = r a F ab + r a F ab. a S a S a S a S 1 (r a F ab + r b F ba ) = 1 (r a r b ) F ab = 0. 2 2 a S b S J = N, N = a S b S a S b S r a F ab N 0 J = 0 b/ S b/ S 62

3.7 S r G r G = a S m a r a /M, M = a S m a M P = m a ṙ a = Mṙ G a S Ṗ = F F = M r G F r a = r a r G m a r a = 0 a S J = a S m a r a ṙ a = a S m a (r G r a) (ṙ G ṙ a) = Mr G ṙ G + a S m a r a ṙ a 1 2 S F g = F g ab = m a g = Mg a S a S b S N g = r a F g ab = r a (m a g) = Mr G g = r G F g a S a S b 63

3.8 ( ) M, r G F g = Mg. F g = M r G Mg = M r G r G = g r G = g 2 t2 + vt + a. v, a v a r G = (x, y, z) z g = (0, 0, g). g (a = 0) v = (v x, 0, v z ) x = v x t, y = 0, z = g 2 t2 + v z t t z = g x 2 + v z x 2vx 2 v x ( 3) 3: M ( ) ( ) 2 2 1 64

3.9 0 v, M 0 α, β t r(t) t P (t) = M(t)ṙ(t). M(t) = M 0 αt t P (t + t) = M(t + t)ṙ(t + t) + α t ( ṙ(t) + β ) d (Mṙ) + α(ṙ + β) = 0 dt Ṁ = α M r = αβ r = αβ M 0 αt ṙ = β log(m 0 αt) + C. v C ṙ = v β log M 0 M 0 αt t t 3.10 2 A, B ( ) 2 A, B A B F AB = F ab a A b B F BA = F 65 AB

N, f f < µn µ 2 µ f f = µ N µ 2 µ µ < µ 3.11 4 ( ) 4: M, L, θ 4 F N 0 ( ) N = Mg, F = f, L 2 66 Mg sin θ = LF cos θ

tan θ < 2µ µ f < µn ( ) 4 3.12 ( ) ω α ṙ α = ω r α α m α J = m α r α (ω r α ) = m α ( r α 2 ω r α r α ω). α α J = I ω, I = ( ) m α rα 2 δ r α r α α I ρ(r) I = d 3 r ρ(r) ( r 2 δ r r ) = y 2 + z 2 xy zx d 3 r ρ(r) xy z 2 + x 2 yz, r = (x, y, z) zx yz x 2 + y 2 ( ) 67

5: ρ, R, L 5 (x, y, z) r, θ, s r = (x, y, z) = (r cos θ, r sin θ, s) [ ] r r r = ( (cos θ, sin θ, 0) ( r sin θ, r cos θ, 0) ) (0, 0, 1) = r r, θ, s R 2π I zz = d 3 r ρ (x 2 + y 2 ) = dr dθ = MR2 2. M = I xx = d 3 r ρ (y 2 + z 2 ) = = ρπr4 L 4 + ρπr2 L 3 12 0 R 0 dr 0 L/2 L/2 ds r ρ r 2 = ρπr4 L 2 d 3 r ρ = ρπr 2 L. 2π 0 dθ = M(3R2 + L 2 ) 12. L/2 L/2 ds r ρ (r 2 sin 2 θ + s 2 ) I yy = I xx, I xy = I yz = I zx = 0 (x, y, z) I I = I ij e i e j I ij e i I 68

J = N N = d (I ω). dt 1 e X, e Y, e Z Z ω = ωe Z. e x, e y, e z Z e z = e Z Z J Z = e Z (I ω) = e Z (I e Z )ω = e z (I e z )ω = I zz ω I zz 3.13 (X, Y, Z), z (x, y, z) y X-Y 6 θ, φ z ϕ 3 6: ω = ϕ e z + θ e y + φ e Z 69

ω = ϕ e z ϕ θ φ (x, y, z) I = I xx e x e x + I yy e y e y + I zz e z e z. J = I ω = I zz ϕ e z J = I zz ( ϕ e z + ϕ ė z ) e z = θ e y e z + φ e Z e z = θ e x + φ sin θ e y J = I zz ( ϕ θ e x + ϕ φ sin θ e y + ϕ e z ) M ( ) l, g N = le z ( Mg e Z ) = Mgl sin θ e y N = J θ = ϕ = 0, φ = Mgl I zz ϕ ( ) ( ) φ ϕ 3.14 a r a = (x 1 a, x 2 a, x 3 a) a ( a ) i = x i a. a b F ab = a U ab U ab 70

U g ab = Gm am b r a r b ( ) 1 a = r a r b i F g ab = au g ab = Gm am b a 1 r a r b x i a ( ra r b 2 ) 1/2 ( ra r b 2 ) 3/2 = 1 (x j 2 x a x j i b )(xj a x j b ) a ( 1 = 2 r a r b 3 2(xj a x j b )δ ij = r ) a r b r a r b 3 : F g ab = Gm am b (r a r b ) r a r b 3 U ab a, b 2 r a r b i 3.15 S K, U K = a S m a 2 ṙ a 2, U = 1 2 a S b S U ab dk = a S m a ṙ a dṙ a = a S m a dr a r a = a S F a dr a du = 1 ) ( a U ab dr a + b U ab dr b = 2 a S a S b S F ab dr a b S 71

de = dw, E = K + U, dw = a S F ab dr a E S dw S dw/dt (power) dw = 0 de = 0 r a = r a r G S b/ S K = a S m a 2 ṙ a 2 = M 2 ṙ G 2 + a S m a 2 ṙ a 2 ( 2 ) m α 2 ṙ α 2 + m a 2 ṙ a ṙ α 2 α α a α K I m α 2 ṙ α 2 = m α 2 ω r α 2 = m ) α ( r α 2 ω 2 r α ω 2 2 α α = 1 ω (I ω) 2 α K = M 2 ṙ G 2 + 1 2 ω (I ω) + K I K I U I E M = E K I U I ( ) 72

3.16 S ( ) U = U ab a S S du = a U ab dr a = F ab dr a = dw a S b/ S a S b/ S b/ S de = dw de = 0, E = E + U E de = 0 E E du = F g ab dr a = m a g dr a = d(mg r G ) a S a S b U = Mg r G. 0 r G = (x, y, z), g = (0, 0, g) U = Mgz 3.17 7 l m l O x = l sin θ, y = l cos θ g E = m 2 (ẋ2 + ẏ 2 ) + mgy = ml2 2 θ 2 mgl cos θ ( ) Θ E = mgl cos Θ 73

7: θ 2 = 2g (cos θ cos Θ) l t = 0 θ = Θ θ 0 t = l/g I(θ, Θ), I(θ, Θ) = 1 2 Θ θ dφ cos φ cos Θ T θ = 0 4 T = 4 l/g I(0, Θ) = 2T 0 π I(0, Θ), T 8 0 = 2π l/g 8: 74

Θ 1 cos x = 1 x 2 /2+ I(θ, Θ) arccos(θ/θ) θ Θ cos( g/l t), T T 0 g ( ) I(θ, Θ) (φ = Θ) cos φ = 1 2 sin 2 (φ/2) x = sin(φ/2)/ sin(θ/2) x I(θ, Θ) = 1 sin(θ/2)/ sin(θ/2) dx 1 x 2 1 sin 2 (Θ/2) x 2. x = cos ψ ψ I(θ, Θ) = arccos(sin(θ/2)/ sin(θ/2)) 0 dψ 1 sin 2 (Θ/2) cos 2 ψ I(0, Θ) = π/2 0 dψ 1 sin 2 (Θ/2) cos 2 ψ. 1 3.18 M, R, kmr 2 9 h (< R) O O v 0 v g O O ( ) O 0 9 v O M(R h)v 0 + kmr 2 v 0 R = MRv + kmr 2 v R 75

9: v 0 /R v /R v ( ) v h = 1 v 0. (1+k)R v 1 2 Mv 2 + 1 ( ) v 2 2 kmr2 + Mg(R h) = 1 R 2 Mv2 + 1 ( v ) 2 2 kmr2 + MgR R ( v v = v 2 2gh ) 1/2 v 1+k ( ( ) ) 2 h v = 1 v0 2 2gh 1/2 (1+k)R 1+k 3.19 M m M m 1 r ( ) r r r 1 E = m 2 ṙ 2 C C = GMm r, 76

J = mr ṙ J θ = π/2 3 (r, θ, φ) r = re r, ṙ = ṙe r + rė r = ṙe r + r φe φ E = m 2 (ṙ2 + r 2 φ2 ) C r, J φ = J mr 2, ṙ = φ dr dφ = J = J = mr 2 φ J mr 2 dr dφ E ( dr dφ ) 2 = 2mE J 2 r 4 + 2mC J 2 r 3 r 2 r = 1/u ( ) 2 du = 2mE dφ J 2 + 2mC J 2 u u 2 r = 1 u = l 1 ɛ sin(φ+α), l = J 2 mc, ɛ = (1 + 2EJ 2 ) 1/2. mc 2 α E > 0 ɛ > 1. E = 0 ɛ = 1. E < 0 ɛ < 1. ( 10) l ɛ 1 77

10: E < 0 J = mr 2 φ T = T 0 dt = m J 2π 0 dφ r 2 = ml2 J 2π 0 dφ (1 ɛ sin φ) 2 ( ) ( ) 3/2 T = ml2 2π 2π l = J (1 ɛ 2 ) 3/2 GM 1 ɛ 2 l/(1 ɛ 2 ) (1) 1 (2) (r 2 φ/2) (3) 2 3 78

( )??? 1687 ( ) 79

4 4.1 t q i (t) (i = 1, 2,, N) N S[q] : δs[q] = 0 q i q i S[q] = dt L(q, q). L(q, q) ( ) δl = L δq i + L ( L δ q i = d ) L δq i + d ( ) L δq i. q i q i q i dt q i dt q i 1 ( L δs[q] = dt d ) [ ] L L δq i + δq i q i dt q i q i δq i = 0 δq i (t) L q i d dt 80 L q i = 0

( ) ( ) 4.2 q i : δq i = ɛ a G ai (q, q) ɛ a G ai q i, q i δl = ɛ a Ẋ a (q, q) δl δl = d ( ) L δq i. dt q i 3 Q a = 0, Q a = L q i G ai X a : t = t ɛ q i (t ) = q i (t) = q i (t + ɛ) δq i (t) = q i(t) q i (t) = q i (t + ɛ) q i (t) = ɛ q i (t). δl = ɛ L E = L q i L q i 81

4.3 q i p i = L q i p i q i, p i 2N {(q, p)} : H(q, p) = p i q i L(q, q) ( ) ( ) H pj qj = q j + p j L ( ) qj L ( ) qj = L = ṗ i, q i q i p q i p q j q i p q j q i p q i ( ) ( ) H pj qj = q j + p j L ( ) qj L ( ) qj = q i p i p i p i q j p i q j p i q q i = H ṗ i = H p i, q i. q 1 q q 4.4 S K, U, U K = a S m a 2 ẋi aẋ i a, U = 1 2 a S b S U ab, U = a S x i a a U ab U ab = U ba ( ) a S K ẋ i a = m b ẋ i a 2 ẋj bẋj b = b S b S 82 b/ S m b ẋ j b δ ijδ ab = m a ẋ i a U ab

U x i a = 1 2 b S c S ( = 1 2 U x i a c S = b S U bc x i a U ac x i a c/ S U bc x i a = 1 2 + b S b S = b S U ba x i a c S ) c/ S ( δ ab U bc x i b = b S δ ab U bc x i b U ab x i a, ) U bc + δ ac x i c = c/ S U ac x i a S L = K U U L ẋ i a = K ẋ i a = m a ẋ i a, L x i a = U x i a U x i a = b U ab x i a L d L = 0 x i a dt ẋ i a b U ab x i a = m a ẍ i a a L = K U U S E = a S L ẋ i a ẋ i a L = a S m a ẋ i aẋ i a L = 2K L = K + U + U ( ) 4.5 L = K U = a m a 2 ẋ a 2 1 2 U ab ( A = A i A i ). ab 83

U ab = G m am b x a x b + q a q b 4πɛ 0 x a x b G ɛ 0 q a a U ab L : δx i a = ɛ i = ɛ j δ ji P j = a L ẋ i a δ ji = a m a ẋ j a L : δx i a = ɛ ijk ɛ j x k a J j = a L ẋ i a ɛ ijk x k a = a m a ẋ i aɛ ijk x k a = a m a ɛ jki x k aẋ i a L = m a 2 ẋ a 2 1 U ab 2 a U ab 2 ( ) ab 4.6 1 2 l 2 m g 84

1: 2 ( ) ( ) ( ) ( ) x1 l sin θ x2 l sin θ + l sin φ =, = l cos θ l cos θ l cos φ y 1 y 2 L = m 2 (ẋ2 1 + ẏ1) 2 + m 2 (ẋ2 2 + ẏ2) 2 mgy 1 mgy 2 ( = ml2 2 2 θ 2 + φ ) 2 + 2 cos(θ + φ) θ φ + mgl ( 2 cos θ + cos φ ) θ 1, φ 1 θ, φ 3 ( L = ml2 2 2 θ 2 + φ ) 2 + 2 θ φ + mgl (3 θ 2 12 ) φ2 L θ = ml2 (2 θ + φ), L φ = ml2 ( θ + φ), L θ = 2mglθ, L φ = mglφ { ( ) ( ) 2 θ + φ + (2g/l)θ = 0 2d 2 θ + φ t + 2g/l d 2 t θ + (g/l)φ = 0 d 2 = 0 t + g/l φ ( d) t θ = u cos(ωt) φ ( ) 2ω 2 + 2g/l ω 2 ω 2 ω 2 u = 0. + g/l d 2 t 85

u 0 0 ( ω 2 = 2 ± ) ( ) g 1 2 u l 2 ( ) ( ) ( ) ( ) θ 1 = A φ 1 cos(ω + t + α) + B 2 cos(ω t + β) 2 ω ± = (2 ± 2) g/l. A, B, α, β 2 2 4 2 ( ) 2 ( ) 2 2 1 4.7 y = y(x) y (x) I[y] = b a dx F (y, y ) x = a, x = b δy(x) δi[y] = 0 F y d dx F y = 0 F y y F = 2 86

4.8 0 O A OA 2: OA 2 y m, r = (x, y), g m 2 ṙ 2 mgy = 0 dr dt y = y(x) = 2gy. dr = dx 2 + dy 2 = dx 1 + y 2 T d 1 + y T = dt = dx 2 2gy 0 0 d A x T δt = 0 F F 1 dy y y F = = 2gy(1 + y 2 ) dx = ± C y y. C y x = ± dy y = C sin 2 θ C y. x = ±2C dθ sin 2 θ = ±C dθ ( 1 cos(2θ) ) ( = ±C θ sin(2θ) ) + D. 2 87

O D 0 θ = ±φ/2 x = R (φ sin φ), y = R (1 cos φ) R = C/2 R 1 R A 4.9 2 O, A 2 L L = A O dr = d 0 dx 1 + y 2. ρ U = A O dr ( ρgy) = d 0 dx ( ρgy) 1 + y 2 L U δl = 0 δu = 0. λ δu + λδl = 0 ( ) δ d 0 dx ( ρgy + λ) 1 + y 2 = 0 F F y y F = ρgy λ =. 1 + y 2 1 + y 2 y 1 1 + y 2 = αy + β α, β dy dx = ± (αy + β) 2 dy 1 x = ± (αy + β)2 1. 88

αy + β = cosh θ αy + β = cosh(αx + γ) cosh ± γ ( ) α, β, γ L O, A (* ) a, b 2 b = 0 a = 0 λ R (a + λb = 0) b = 0 a = 0 b 0 λ λ 4.10 3: 3 m ( ) n t φ(n, t) n (n+1) k 2 (φ(n+1, t) φ(n, t))2 (k > 0) L = ( m 2 φ(n, t) 2 k ) (φ(n+1, t) φ(n, t))2 2 n Z L φ(n, t) = m φ(n, t), L φ(n, t) = k(φ(n+1, t) + φ(n 1, t) 2φ(n, t)). 1 ( f(x + a) = n! f (n) (x)a n 1 = a d ) n ( f(x) = exp a d ) f(x) n! dx dx n=0 n=0 89

m φ(n, t) = k(e + e 2)φ(n, t), = n φ(n, t) = 4ω 2 0 sinh 2 2 φ(n, t), ω 0 = k m n Z, p ( π, π) {e ipn } ( ) φ(n, t) = π π dp c(p, t) e ipn c(p, t) = 4ω 2 0 sin 2 (p/2) c(p, t) ω(p) = 2ω 0 sin(p/2) c(p, t) = a(p) e iω(p)t + b(p) e iω(p)t φ(n, t) φ(n, t) = π π π ) dp (a(p) e ipn iω(p)t + b( p) e ipn+iω(p)t. p φ(n, t) b( p) = a (p) π ) φ(n, t) = dp (a(p) e ipn iω(p)t + a (p) e ipn+iω(p)t p, ω(p) = 2ω 0 sin 2 φ(n, 0)e ipn = 2π(a(p) + a ( p)), n Z n Z φ(n, 0)e ipn = 2πi ω(p)(a(p) a ( p)) a(p) a(p) = 1 ( φ(n, 0) + i ) 4π ω(p) φ(n, 0) e ipn. n Z 90

a(p) t = 0 0 0 v φ(n, 0) = 0, φ(n, 0) = vδn0 a(p) = iv/4πω(p). φ(n, t) = v 2π π π dp sin(ω(p)t pn) ω(p) 0 t φ(0, t) = v 2ω 0 F (2ω 0 t), F (x) = 1 π π 0 dp sin(x sin(p/2)) sin(p/2) F ( ) = 1 0 v/2ω 0 t = 0 0 [F ( ) = 1 ] q = x sin(p/2) F (x) = 2 π x 0 sin q dq q 1 (q/x) 2 1/ 1 (q/x) 2 q/x 0 dq sin q q = π 2, lim x 1 x n+1 x 0 dq q n sin q = 0 (n = 1, 2, ) f(z) = e iz /z 4 0 [ ] 4: 91

5 5.1 V V d 2 x i d 2 x i df j = d 2 x i T ij T ij 1: x 1 T 11 V V F j = d 2 x i T ij = d 3 x i T ij V i T ij V 92

N i = = V 3 ɛ ijk V V ɛ ijk x j df k = ɛ ijk x j d 2 x l T lk V d 3 x l (ɛ ijk x j T lk ) = d 3 x (ɛ ilk T lk + ɛ ijk x j l T lk ). i T ij N i = d 3 x ɛ ijk x j l T lk V V V ɛ ilk T lk = 0. T ij = T ji 2 5.2 x i x i = x i + u i (x) u i (x) j u i dx idx i dx i dx i = (dx i + j u i dx j )(dx i + k u i dx k ) dx i dx i = j u i dx j dx i + k u i dx i dx k = ( i u j + j u i )dx i dx j. ɛ ij = 1 2 ( iu j + j u i ) ( ) 0 u i ɛ ij 0 : T ij = E ijkl ɛ kl E ijkl 93

2 E ijkl = E jikl = E ijlk 1 ( ) ( ) E ijkl = E klij 3 6 6 21 E ijkl 3 4 = 81 21 [ ] δu i V δw = δu j df j = δu j d 2 x i T ij = d 3 x i (δu j T ij ). V V ( ) i T ij = 0 δw = d 3 x δɛ ij T ij = d 3 x δɛ ij E ijkl ɛ kl V V A V δa = δɛ ij E ijkl ɛ kl δ 2 A = E ijkl E ijkl = E klij. [ ] δɛ ij δɛ kl 5.3 λ, µ, µ E ijkl = λδ ij δ kl + µδ ik δ jl + µ δ il δ jk µ = µ E ijkl = λδ ij δ kl + µ(δ ik δ jl + δ il δ jk ) λ, µ 2 94

T ij = E ijkl ɛ kl ɛ ij = (1/2)( i u j + j u i ) T ij = λδ ij u + µ( i u j + j u i ) u = i u i ρ, ( ) f j ρü j = i T ij + f j. ρü j = (λ + µ) j u + µ u j + f j = 5.4 0 ρü j = (λ + µ) j u + µ u j : u j = a j sin(k x ωt) ρω 2 a j = (λ + µ)k j k a + µk 2 a j k ω k a = 0 ρω 2 = µk 2 v t = ω µ k = ρ k a 0 k j k a ρω 2 = (λ + 2µ)k 2 a j k j v l = ω k = λ + 2µ ρ ( ) 95

5.5 p x 1 p a i E = p a 1, ν = a 2 a 1 u 1 = a 1 x 1 + b 1, u 2 = a 2 x 2 + b 2, u 3 = a 3 x 3 + b 3 a i, b i a i x i : T ij = λδ ij u + µ( i u j + j u i ) T 11 = (λ + 2µ)a 1 + λa 2 + λa 3, T 22 = λa 1 + (λ + 2µ)a 2 + λa 3, T 33 = λa 1 + λa 2 + (λ + 2µ)a 3, T ij = 0 (i j) ( f j = 0) x 1 p T 11 = p, T 22 = T 33 = 0 λ + 2µ λ λ a 1 p λ λ + 2µ λ a 2 = 0 λ λ λ + 2µ a 3 0 a 1 4µ(λ+µ) p a 2 1 = 4µ a 2λµ 0 = (3λ+2µ) 3 2λµ 0 p 2µ(3λ+2µ) 2(λ+µ) λ λ. E = µ(3λ + 2µ) λ + µ, ν = λ 2(λ + µ) 96

(P ) (S ) P S 0.4 0.5 5.6 2 λ = 0 ( 0) 2: x 1 = x g x L ρü j = µ j u + µ u j + f j u 1 = u = u(t, x), u 2 = u 3 = 0 ρü = 2µu + ρg ü = c 2 u + g. x c = 2µ/ρ ( ) u = g c 2 u = gx2 2c 2 + Ax + B (A, B ) x = 0 u = 0 ( ) x = L u = 0 ( ) A, B u = gx (2L x) 2c2 97

gl 2 /2c 2 t = 0 0 x L { cos(nπx/l) n = 0, 1, 2, } u(t, x) = a(t) + a n (t) cos nπx L n=1 ü = c 2 u + g ( nπc ) 2 ä(t) = g, ä n (t) = an (t) L a(t) = gt2 2 + αt + β, a n(t) = α n cos nπct L + β n sin nπct L u(0, x) = 0 α = 0, β n = 0 u(t, x) = gt2 2 + β + n=1 α n cos nπct L nπx cos L. u(0, x) = (gx/2c 2 )(2L x) β, α n u(t, x) = gt2 2 + gl2 3c 2gL2 2 π 2 c 2 n=1 1 nπct cos n2 L cos nπx L : u (t, 0) = u (t, L) = 0 : ü(t, L) = g + 2g cos nπct L cos(nπ) = g + 2g ( cos nπ ct + L ) L n=1 n=1 ( = g + 2g π ( δ π ct + L ) ) L 2nπ 1 = g ( ) ct + L δ 2 2L n. n Z n Z cos(nx) = π δ(x 2nπ) 1 2 n Z n=1 δ(ax) = 1 a δ(x) 98

δ(x) ( ) t t u(t, L) = dt ü(t, L) = g dt ( ct ) + L δ 0 0 2L n n Z = 2gL (ct+l)/2l ds δ(s n) = 2gL [ ] ct + L c 1/2 c 2L n Z [x] x ( ) 0 < t < L/c t = L/c 2gL/c (ct L) g 5.7 ρ(t, x), v i (t, x) d 2 x i ρv i d 2 x i d d 3 x ρ = d 2 x i ρv i ρ + i (ρv i ) = 0 dt V V p i = ρv i P i (t) = d 3 x p i (t, x) V V t δt x i x i = x i + v i δt V δt P i (t + δt) = d 3 x p i (t + δt, x ) V x = δ ij + j v i δt x j det x x = ɛ ijk(δ i1 + 1 v i δt)(δ j2 + 2 v j δt)(δ k3 + 3 v k δt) = 1 + vδt 99 x i

P i (t + δt) = d 3 x(1 + vδt)(p i + ṗ i δt + j p i v j δt) V = P i (t) + d 3 x(ṗ i + j p i v j + p i v)δt = P i (t) + V V d 3 x(ρ v i + ρv v i )δt. ρ v i + ρv v i T ij, f j ρ v j + ρv v j = i T ij + f j T ij = pδ ij + λδ ij v + µ( i v j + j v i ) p µ, λ 2µ + 3λ = 0 ( ) 1 5.8 l µlv ρl 2 v 2 Re = ρlv µ 2 100

Re 10 3 ( ) 5.9 ρ ( ) ρ v i + ρv v i = i p + f i ρv v i = i p + f i f i = i φ φ ρv v i + i p + i φ = 0 v i v v 2 = 2v i v v i ( ) 1 v 2 ρv2 + p + φ = 0 (* ) µ 0 5.10 v i 2 ( ) v = 0, ρ v i = i p ρg i x 3 101

x 3 g v i (t, x) (ɛ ijk j v k ) 0 Φ(t, x) v i = i Φ ( Φ = 0, i Φ + p ) ρ + gx 3 = 0 3 ( ) x 3 = η(t, x 1 ) η h h 3: v 3 = 0 3 Φ = 0 (1) x3 = h Φ + (p/ρ) + gx 3 p 0 /ρ p 0 Φ + (p/ρ) + gx 3 = p 0 /ρ Φ + gη = 0 (2) x3 =η δt 4 v 3 x3 =η = η + ( 1η)v 1 3 Φ = η (3) x3 =η 102

(1) (3) Φ = 0 4: x 1 Φ = F (x 3 ) sin(kx 1 ωt) k ω Φ = 0 F (x 3 ) = k 2 F (x 3 ) (1) F ( h) = 0 A F (x 3 ) = A cosh (k(x 3 +h)) Φ Φ = A cosh (k(x 3 + h)) sin(kx 1 ωt) (2)(3) η h η = C cos(kx 1 ωt), C = Aω g cosh(kh), ω2 = gk tanh(kh) 0 v p = ω k = g tanh(kh) k kh 1 v p g/k 103

6 6.1 N R N R N N ( ) R N N 2 P, Q (x 1,, x N ), (x 1 + dx 1,, x N + dx N ) PQ ds ds 2 = g µν (x)dx µ dx ν g µν g µν = g νµ g µν / x λ = 0 g µν = δ ν µ δ ν µ g µν g µν g µν g νλ = δ λ µ. A µ A µ = g µν A ν A µ g λµ g λµ A µ = g λµ g µν A ν = δ λ ν A ν = A λ 104

6.2 2 (x, y) x = x ay, y = y (a ) (x, y ) ds 2 = dx 2 + dy 2 = (dx + ady ) 2 + dy 2 = dx 2 + (1 + a 2 )dy 2 + 2a dx dy x 1 = x, x 2 = y g 11 = 1, g 22 = 1 + a 2, g 12 = g 21 = a 3 (x, y, z) x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ 3 (r, θ, φ) ds 2 = dx 2 + dy 2 + dz 2 = (sin θ cos φ dr + r cos θ cos φ dθ r sin θ sin φ dφ) 2 + (sin θ sin φ dr + r cos θ sin φ dθ + r sin θ cos φ dφ) 2 + (cos θ dr r sin θ dθ) 2 = dr 2 + r 2 dθ 2 + r 2 sin 2 θ dφ 2 x 1 = r, x 2 = θ, x 3 = φ g 11 = 1, g 22 = r 2, g 33 = r 2 sin 2 θ ( 0) (µ ν g µν ) 0 (a 0) 105

3 r 2 2 3 (θ, φ) ds 2 = r 2 dθ 2 + r 2 sin 2 θ dφ 2 (r ) (θ, φ) 2 6.3 dx µ dx µ = x µ x ν dxν µ = / x µ µ = xν x µ ν T µν λ = x µ x ν x ρ x xκ T σ x λ ρσ κ T µν λ 1 dx µ ds A µν, B ν C µ = A µν B ν ν C µ = A µν B ν = x µ x ν x ρ x σ Aρσ xλ x B ν λ = x µ x λ x ρ x σ Aρσ B λ = x µ x ρ δλ σa ρσ B λ = x µ x ρ Aρσ B σ = x µ x ρ Cρ C µ C µ, B ν A µν g µν : g µν = xρ x µ x σ x ν g ρσ. 106

δ ν µ = xρ x µ x ν x σ δσ ρ δµ ν g µν 6.4 φ(x) µ φ : µφ = xν x µ νφ. A µ (x) : ( ) µa ν = xρ x ν x µ ρ x σ Aσ = xρ x ν x µ x ρa σ + xρ 2 x ν σ x µ x ρ x σ Aσ. 2 2 : µ A ν = µ A ν + Γ ν λµa λ. µ ( ) Γ ν λµ ( ) µ A ν ( µ A ν ) = µa ν + Γ ν λµa λ. x ρ x ν ( ) x µ x ρa σ = xρ x ν σ x µ ρ x σ Aσ + Γ ν x λ λµ x σ Aσ xρ x ν ( ρ A σ + Γ σ λρa λ) = xρ x ν x µ x σ x µ x ρa σ + xρ 2 x ν σ x µ x ρ x σ Aσ + Γ ν λµ x σ Aσ. 1 1 2 σ, λ x ρ x ν x µ x λ Γλ σρa σ = xρ 2 x ν x µ x ρ x σ Aσ + Γ ν x λ λµ x σ Aσ x λ 107

A σ x λ x σ Γ ν λµ = xρ x µ x ν x λ Γλ σρ xρ x µ 2 x ν x ρ x σ. x σ / x κ Γ ν κµ = x ν x λ x σ x κ x ρ x µ Γλ σρ xσ x κ x ρ x µ 2 x ν x σ x ρ 2 6.5 : : µ φ = µ φ. : µ (AB) = ( µ A)B + A µ B. A ν B ν A ν B ν µ (A ν B ν ) = µ (A ν B ν ). ( µ A ν + Γ ν λµa λ )B ν + A ν µ B ν. ( µ A ν )B ν + A ν µ B ν Γ ν λµa λ B ν + A ν µ B ν A ν µ B ν = 0. 2 ν λ A λ A λ µ B λ = µ B λ Γ ν λµb ν 2 C µν A µ, B ν C µν = A µ B ν : µ T νρ σ = µ T νρ σ + Γ ν λµt λρ σ + Γ ρ λµt νλ σ Γ λ σµt νρ λ. 108

6.6 0 0 1 λ g µν = 0 Γ λµν = 0 λ g µν Γ λµν ( ) 0 1 0 0 0 0 6.7 λ g µν = λ g µν Γ ρ µλg ρν Γ ρ νλg µρ = λ g µν Γ νµλ Γ µνλ 0 : λ g µν = 0 λ g µν = Γ µνλ + Γ νµλ. φ 2 ν φ = ν φ µ ν φ = µ ν φ Γ λ νµ λ φ. µ, ν ( µ ν ν µ )φ = (Γ λ µν Γ λ νµ) λ φ. 0 λ φ Γ λµν = Γ λνµ 2 109

( ) 2 Γ λµν = 1 2 ( λg µν + ν g λµ + µ g νλ ) ( ) 6.8 A ρ 2 µν µ ν A ρ = µ ν A ρ Γ λ νµ λ A ρ + Γ ρ λµ ν A λ = µ ( ν A ρ + Γ ρ λνa λ ) + Γ ρ λµ( ν A λ + Γ λ σνa σ ) + ( µν ) = µ Γ ρ λνa λ + Γ ρ λµγ λ σνa σ + ( µν ) = ( µ Γ ρ σν + Γ ρ µλγ λ νσ)a σ + ( µν ). µν ( µ ν ν µ )A ρ = R ρ σµνa σ, R ρ σµν = µ Γ ρ σν + Γ ρ µλγ λ νσ ( µν ) R ρ σµν 0 0 0 R ρσµν = g ρλ R λ σµν = g ρλ µ Γ λ σν + g ρλ Γ λ µτγ τ νσ (µν ) = µ Γ ρσν ( µ g ρλ )Γ λ σν + Γ ρµτ Γ τ νσ (µν ) 110

= 1 2 µ( ρ g σν + ν g ρσ + σ g νρ ) (Γ ρλµ + Γ λρµ )Γ λ σν + Γ ρµλ Γ λ νσ (µν ) = 1 2 µ ρ g σν + 1 2 µ σ g ρν Γ λρµ Γ λ σν (µν ). : R ρσµν = R ρσνµ = R σρµν = R µνρσ, R ρσµν + R ρνσµ + R ρµνσ = 0. n R ρσµν ρ, σ µ, ν (ρ, σ) (µ, ν) n(n 1)/2 n(n 1)(n 2 n + 2)/8 n = 2 1, n = 3 6, n = 4 21 n = 4 1 6.9 λ R ρ σµν = λ µ Γ ρ σν λ ν Γ ρ σµ + ( Γ ) λ R ρ σµν + ν R ρ σλµ + µ R ρ σνλ Γ 0 : λ R ρ σµν + ν R ρ σλµ + µ R ρ σνλ = 0. R µν = g ρσ R ρµσν, R = g µν R µν ( R µν = g ρσ R ρµνσ ) δ ρ µ g νσ µ G µν = 0, G µν = R µν 1 2 gµν R G µν 111

6.10 N dv = d N x = det g. g det g d N x = dx 1 dx 2 dx N d N x = det x x d N x = det x x dn x dv N x µ dv = d N x 6.11 g µν g νλ = δ λ µ δg µν = g µρ g νσ δg ρσ δ det g = det g g µν δg µν. δ = 1 2 δ det g = 1 2 det g g µν δg µν = 1 g µν δg µν 2 µ = µ g ρσ = 1 g ρσ µ g ρσ = 1 g ρσ (Γ ρσµ + Γ σρµ ) = Γ ρ ρµ g ρσ 2 2 A µ µ A µ = ( µ A µ + Γ µ νµa ν ) = µ A µ + ( ν )A ν = µ ( A µ ). 2 F µν µ F µν = µ ( F µν ). 112

2 T µ λ µ T µ λ = µ( T µ λ ) 1 2 λ g µν T µν 6.12 dx I = ds = dλ g µ dx ν µν dλ dλ λ 2 x µ (λ) δx µ (λ) : δi = 0 dx δ g µ dx ν µν dλ dλ = 1 dx (g ρ dx σ ) 1/2 dx ρσ δ (g µ dx ν ) µν 2 dλ dλ dλ dλ = 1 dλ ( λ g µν δx λdxµ dx ν 2 ds dλ dλ + 2g dx µ dδx ν ) µν dλ dλ = 1 2 dx µ dx ν dx µ dδx ν λg µν ds dλ δxλ + g µν ds dλ = 1 2 dx µ dx ν λg µν ds dλ δxλ d dx (g µ ) µλ δx λ + d ( ). dλ ds dλ δx λ dx (g µ ) µλ 1 2 dx µ dx ν λg µν ds dλ d dλ ds = 1 2 dx µ dx ν λg µν ds dλ dx ν dx µ νg µλ dλ ds g d 2 x µ µλ dλds = 1 2 ( λg µν ν g µλ µ g νλ ) dxµ dx ν ds dλ g d 2 x µ µλ dλds dx µ dx ν = Γ λµν ds dλ g d 2 x µ µλ dλds. 113

δi = d dλ (g 2 x µ λµ dλds + Γ dx µ λµν dλ dx ν ) δx λ ds δx λ (λ) d 2 x µ g λµ dλds + Γ dx µ dx ν λµν dλ ds = 0. g ρλ dλ/ds d 2 x ρ ds + dx µ dx ν 2 Γρ µν ds ds = 0 d 2 x ρ /ds 2 = 0 6.13 2 2 (θ, φ) ds 2 = r 2 dθ 2 + r 2 sin 2 θdφ 2 r θ = x 1, φ = x 2 g 11 = 1 g 22 = r 2, g 11 = r 2, g 22 = r 2 sin 2 θ, g 12 = g 21 = 0. 1 r 2 sin 2 θ, g 12 = g 21 = 0 0 1 g 22 = r2 sin 2 θ θ = 2r 2 sin θ cos θ Γ 122 = 1 2 1g 22 = r 2 sin θ cos θ, Γ 212 = Γ 221 = 1 2 1g 22 = r 2 sin θ cos θ 0 Γ 1 22 = g 11 Γ 122 = sin θ cos θ, 0 114 Γ 2 12 = Γ 2 21 = g 22 Γ 212 = cos θ sin θ

2 R 1212 ( ) R 1212 = g 11 R 1 212 = g 11 1 Γ 1 22 2 Γ 1 21 + Γ 1 1λΓ λ 22 Γ 1 2λΓ λ 12 2 3 0 4 λ = 2 ( R 1212 = g 11 1 Γ 1 22 Γ 1 22Γ 12) 2 = r 2 sin 2 θ. 0 2 R 11 = g 22 R 2121 = g 22 R 1212 = 1, R 22 = g 11 R 1212 = sin 2 θ 0. R = g 11 R 11 + g 22 R 22 = 2 r 2 d 2 x ρ ds + dx µ dx ν 2 Γρ µν = 0 θ = ds ds ( ) 2 dφ d 2 φ sin θ cos θ = 0, ds ds = 0 2 (θ = π/2) ( ) 2 G µν = R µν (1/2) g µν R 0 2 0 2 2 ɛ µν R ρµσν = R 1212 ɛ ρµ ɛ σν R µν = g ρσ R 1212 ɛ ρµ ɛ σν = R 1212 ( g 1 ) µν. g 1 g 1 g 1 = det(g 1 ) δ g 1 = (det g) 1 g. R µν = R 1212 det g g µν, R = g µν R 1212 det g g µν = 2R 1212 det g G µν = 0 0 3 115

7 7.1 1864 c 2.9979 10 8 m/s 1905 2 ( ) ( ) 2 2 2 ( ) 116

( ) 4 ( ) (1905) E = mc 2 : 7.2 c, µ 0 1 SI (MKSA ) SI ev ( ) m( ) = 5.0677 10 6 ev 1, s( ) = 1.5193 10 15 ev 1, kg( ) = 5.6096 10 35 ev, A( ) = 1.2441 10 3 ev, K( ) = 8.6173 10 5 ev c SI 1 ( ) 117

ev n n 1 1 0, 1 SI ri F = kma k ri kg 1 m 1 s 2 k SI k = 1 ( ) ri c = 1 ri F = ma? c = 1 ( ) c µ 0 7.3 4 x µ (µ = 0, 1, 2, 3) g µν : 1 0 0 0 0 1 0 0 η µν = 0 0 1 0 0 0 0 1 4 dτ 2 = g µν dx µ dx ν µν 118

τ dτ 2 = η µν dx µ dx ν = (dx 0 ) 2 (dx 1 ) 2 (dx 2 ) 2 (dx 3 ) 2 dx i = 0 (i = 1, 2, 3) dτ = dx 0 x 0 x 0 x 1, x 2, x 3 ( 1) i, j, k 1, 2, 3 1: x µ : v µ = dxµ dx 0 (4 ) v 0 = 1. v i v = v i v i dτ dx 0 = η µν v µ v ν = (v 0 ) 2 v i v i = 1 v 2 τ x 0 1 ( v 1) v = 1 dτ = 0 119

dτ = 0 dτ ( ) ( ) 7.4 : x µ = Λ µ νx ν ( x = Λ x) dτ 2 = dx T η dx = dx T η dx dx T Λ T η Λ dx = dx T η dx Λ T η Λ = η γ γv 0 0 γv γ 0 0 Λ = 0 0 1 0, γ = 1 1 V 2 0 0 0 1 1 < V < 1. γ Λ T ηλ = η x µ = (t, x, y, z) µ x = Λ x t = γ(t V x), x = γ( V t + x), y = y, z = z 120

: (1) x = 0 x = V t (2) x = L x = V t + γ 1 L (3) t = 0 t = V x (1) V ( ) (2) V γ 1 = 1 V 2 (3) 2 2 2 O A 2: 2 2 2 C D OC > OD D C E D E OD > OE 2 121

t = γ(t + V x ), x = γ(v t + x ), y = y, z = z V 2 cosh x = ex + e x 2, sinh x = ex e x 2, tanh x = sinh x cosh x cosh 2 x sinh 2 x = 1 x cosh θ sinh θ 0 0 sinh θ cosh θ 0 0 Λ = 0 0 1 0 0 0 0 1 < θ < V = tanh θ 7.5 3 3: 3 L c L g ( ) V 122

1 V 2 L c A, B ( ) L g 1 V 2 L c L g = 1 V 2 L c 1/ 1 V 2 1 V 2 L g? 2 A, B A, B B A t t = γ( t V x), t = 0, x = L g t = V L g 1 V 2 2 1 V 2 L g + V t = L c t L g = 1 V 2 L c (* ) 7.6 x V t = γ(t V x), x = γ( V t + x), y = y, z = z γ = 1/ 1 V 2 dx dt = dy dt = γ( V dt + dx) γ(dt V dx) = (dx/dt) V 1 V (dx/dt), dy γ(dt V dx) = 123 1 V 2 (dy/dt) 1 V (dx/dt)

v x = v x V 1 V v x, v y = 1 V 2 v y 1 V v x, (v x, v y, v z ) 1 V 2 v z v z = 1 V v x x θ (v x, v y, v z ) = ( cos θ, sin θ, 0) x V (v x, v y, v z) = ( cos θ, sin θ, 0) cos θ = cos θ + V 1 V 2 sin θ sin θ = 1 + V cos θ, 1 + V cos θ tan : tan θ 2 = sin θ 1 + cos θ tan θ 1 V 2 = 1 + V tan θ 2 ( ) 7.7 n m n, q n n x µ = x µ n(λ n ), τ n λ n S m = m n dτ n n 124

A µ (x) 4 F µν = µ A ν ν A µ µ A ν ν A µ = µ A ν Γ λ νµa λ ν A µ + Γ λ µνa λ = µ A ν ν A µ = F µν F µν S em = 1 d 4 x F µν F µν 4 = det g g S em 4 S q = q n dx µ n A µ (x n ) n 3 S = S m + S q + S em = m n dτ n n n q n dx µ n A(x n ) 1 4 d 4 x F µν F µν x µ n(λ n ), 4 A µ (x) : δs = 0 ( ) 125

7.8 S m S m = ( m n dλ n g µν (x n ) dxµ n dx ν ) 1/2 n. dλ n n dλ n x λ n ( δs m dτ n du µ ) n = m δx λ n g λµ + Γ λµν u µ n dλ n dτ nu ν n n u µ n = dxµ n dτ n u µ n vµ n = dxµ n S q ( δs q = δ ) dx µ n q n dλ n A µ (x n ) dλ n n = ( ) dδx µ q n dλ n n A µ (x n ) + dxµ n δa µ (x n ) dλ n n dλ n = q n dλ n ( δx µ dx ν ) n n ν A µ (x n ) + dxµ n δx ν n ν A µ (x n ) dλ n n dλ n = q n dλ n F νµ (x n ) dxµ n δx ν n δs q dx µ n = q dλ n n δx λ n F λµ n dλ n δs/δx λ n = 0 ( ) du λ m n n + Γ λ µνu µ dτ nu ν n = q n F λ µu µ n n Γ λ µν F λ µ ( ) x t (t = 1, 2, ) f(x 1, x 2, ) f(x) f(x)/ x t dx 0 n 126

x t t? x(t) t t x(t) f[x] x(t) f(x)/ x t δf[x]/δx(t) x t x s = δ ts δx(t) δx(s) = δ(t s). t dt df(x) = f(x) dx t δf[x] = dt δf[x] x t t δx(t) δx(t) 7.9 4 A µ (x) δs em = 1 d 4 x F µν δf µν = 1 d 4 x F µν ( µ δa ν ν δa µ ) 2 2 = d 4 x F µν µ δa ν = d 4 x µ ( F µν )δa ν δs em = µ ( F µν ). δa ν δs q δa ν (x) = n q n dx µ n δa µ (x n ) δa ν (x) = n q n dx ν n δ 4 (x x n ) δs/δa ν = 0 µ ( F µν ) = j ν q n dx µ n δ 4 (x x n ) j µ = n 4 j µ 4 δ 4 (x) = δ(x 0 )δ(x 1 )δ(x 2 )δ(x 3 ) F µν = F νµ µ ( j µ ) = 0 127

4 d 3 x 0 ( j 0 ) = d 3 x i ( j i ) = d 2 x i j i V V V V V j i = 0 0 Q = d 3 x j 0 = q n d 3 x δ 3 (x x n ) V n V x 0 n =x 0 V d3 x δ 3 (x x n ) x 0 n =x 0 n x 0 V 1 0 Q x 0 V ( ) µ µ F µν = j ν, µ j µ = 0 7.10 x µ = x µ ɛ µ (x) ( ) δg µν (x) = g µν(x) g µν (x) g µν(x ) = xρ x σ x µ x g ρσ(x) ν δg µν = g µλ ν ɛ λ + g νλ µ ɛ λ + ɛ λ λ g µν δg µν (y) δɛ λ (x) = g µλ(y) ν δ 4 (x y) + g νλ (y) µ δ 4 (x y) + δ 4 (x y) λ g µν (y). δs/δx µ n = 0, δs/δa µ = 0 δs δɛ λ (x) = d 4 y 128 δs δg µν (y) δg µν (y) δɛ λ (x)