i ( ) PDF http://moodle.sci.u-toyama.ac.jp/kyozai/ I +α II II III A: IV B: V C: III V I, II III IV V III IV 8 5 6 krmt@sci.u-toyama.ac.jp

8 5 6

i ( ) PDF http://moodle.sci.u-toyama.ac.jp/kyozai/ I +α II II III A: IV B: V C: III V I, II III IV V III IV 8 5 6 krmt@sci.u-toyama.ac.jp

ii I +α 3..................................................... 3. (order estimation).................................... 4.3.................................................... 4.3................................................... 4.3................................................. 5.4.................................................... 6.5..................................................... 7 9.............................................. 9.............................................. 9.3..............................................4.................................................... 3.5..................................................... 4 3 7 3.................................................. 7 3.................................................. 7 3.3................................................ 9 3.4.................................................... 3.5..................................................... 4 4 4..................................................... 4 4..................................................... 5 4.3......................................... 6 4.4.................................................... 8 4.5..................................................... 9 5 3 5..................................................... 3 5................................................. 3 5.3..................................... 3 5.4................................................ 3 5.5.................................................... 34 5.6..................................................... 35 6 37 6....................................................... 37 6........................................... 38 6.3............................................... 38 6.4..................................................... 39 6.5.................................................... 4 6.6..................................................... 4

iii 7 43 7............................................. 43 7................................................. 44 7.3............................................... 46 7.4.................................................... 48 7.5..................................................... 49 8 5 8............................................. 5 8............................................. 5 8.................................................. 5 8..3................................................ 53 8..................................................... 54 8.3..................................................... 55 II 57 9 58 9................................................... 58 9................................................. 59 9.3.................................................... 6 9.4..................................................... 6 6...................................................... 6...................................................... 63.3.................................................... 65.4..................................................... 66 68...................................................... 68...................................................... 69.3.................................................... 7.4..................................................... 7 74.................................................. 74.................................. 75.3.................................................... 77.4..................................................... 78 3 8 3.................................................. 8 3.................................................. 8 3.3.................................................... 83 3.4..................................................... 84 4 86 4.............................................. 86 4................................................. 86 4.3.................................................... 88 4.4..................................................... 89 5 9 5..................................................... 9 5................................................ 9 5.3 (grad), (div), (rot)..................................... 9 5.4................................................... 95 5.5..................................................... 96

iv 6 99 6................................................... 99 6........................................... 99 6.3.................................................... 6.4..................................................... 7 4 7................................................ 4 7...................................................... 5 7.3................................................. 6 7.4.................................................... 8 7.5..................................................... 9 8 8..................................................... 8....................................... 8.3........................................... 8.4.................................................... 4 8.5..................................................... 5 III A ( ) 7 9 8 9...................................................... 8 9.................................................. 8 9.3................................................ 9 9.4................................. 9 9.5.................................................... 9.5......................................... 9.5................................................. 9.6.................................................. 9.7................................................... 9.8..................................................... 4 8................................................. 8................................................. 8.3................................................... 3.4..................................................... 3 35................................................. 35.............................................. 36.3.................................................... 37.4................................................... 39.5..................................................... 4 46.............................................. 46............................................ 46.3........................................... 47.4.................................................... 47.5.................................................... 49.6..................................................... 5

v 3 δ δ 5 3. δ............................................... 5 3. δ............................................. 5 3.3.................................................... 54 3.4..................................................... 55 4 57 4.................................................. 57 4.................................................. 59 4.3.................................................... 6 4.4..................................................... 6 5 64 5........................................ 64 5.................................................. 66 5.3............................................... 66 5.4.................................................... 68 5.5..................................................... 69 IV B ( ) 7 6 73 6............................................ 73 6.................................................. 74 6.3................................................. 75 6.4.................................................... 76 6.5..................................................... 77 7 8 7................................................... 8 7...................................................... 8 7.3................................................ 8 7.3................................... 8 7.3.................................. 8 7.4.................................................... 83 7.5..................................................... 84 8 87 8..................................... 87 8.............................. 87 8.3..................................................... 88 8.4 ( )........................... 88 8.5 ( )....................... 89 8.6.................................................... 9 8.7..................................................... 9 9 97 9.............................................. 97 9.................................... 98 9.3.......................................... 99 9.4.................................................... 9.5..................................................... 3 6 3................................................. 6 3........................................ 8 3.3................................................... 3.4.....................................................

V C ( ) 3 3 4 3..................................................... 4 3..................................................... 4 3.3..................................................... 6 3.4................................................... 8 3.5..................................................... 9 3 4 3...................................... 4 3.............................................. 5 3................................................... 6 3.................................................. 7 3.3..................................................... 8 3.4................................................ 8 3.5.................................................... 3 3.6..................................................... 3 33 35 33............................................. 35 33...................................... 37 33.3......................................... 38 33.4................................................. 39 33.5............................................. 4 33.6................................................. 4 33.7.................................................... 43 33.8..................................................... 45 5

I +α

3. / ( ) ) 3.4 mm 4. 3 mm.34 cm 4.3 cm 3, :. +.78.98. ( ).. ±..78.78 ±..78.....98., :..7.84.8 ( ).. ±..7.7 ±. ( )..6.66.3.8.4.84..84 4 ).cm.. π 3.463659... 3.5 cm ( )

4. (order estimation) order estimation: a b a ( ) b. ( b (4. ) ) ) m gt, t /9.8..4 (s) ) l l kg H O 8. 3 5.6 (6 3 ) (5.6 ) 7 5 8 3) 5 34 m/s 34 (m/s) 5 (s) 7 (m) (km).3.3. m, kg, s, A MKSA (K) (mol) (cd) SI m : m (/9979458) 7 kg : kg ( ) s : ( 33 Cs) 996377 4 6 6 A : m 7 N A m, kg, s, A (C): [C] [A] [s] I dq dt

5 (N): [N] [kg] [m]/[s ] F ma....... ( x ) 5 P d T c 9 G 3 m 6 M 6 µ 3 k 9 n h p da 5 f.3. ) : λ (m) (J) λ (m) c (m/s) h (J s) ch/λ E hν hc/λ ) : m (kg) v (m/s) [J][N] [m] ([kg] [m]/[s ]) [m] [kg] [m ]/[s ] m v [J] mv (/)mv (/) 3) : T (s) R (m) G (N m /kg m 3 /s kg) M (kg) GMT R 3 3 ( ) ( ) GMT 4π R 3

6.4. ( ) (a) (l) (b) (m) (c) (n) (d) (o) (e) (p) ( ) (f) (q) (g) (r) (h) (s) (i) (t) ( ) (j) mol (u) C (k) (v). ( ) (a) (b) mol (c) mol g (d) (e) 3. m ( ), kg ( ), s( ), A ( ), K( ) (a) (e) (i) (b) (f) (j) (c) (g) (k) ( E) (d) ( B) (h) (l) 4. (a) (d) /(4πϵ ) (g) (b) (e) (h) ( ) (c) (f) (i) 5. (m) (c) 6.. mg ( ) (5m 5m.m) 7. cal/cm ( 5 km)

7.5. (a).4 (l) 9. 3 kg (b) 5 4 (m) 5 (c) 6 9 km (n) 3. 7 (d) 3.8 5 km (o) 864 9 4 (e) 4 4 km (p) 5 Hz (f) Å m (q) 5 5 T (g) 3 kg (r).6 9 C (h) 4 kg (s) cal/g C (i) 7 kg (t). 3 hpa. 5 Pa. 5 N/m (j) 3 g (u) 3.4 m/s (k).7 7 kg (v) 9.8 m/s. (a) (b) 6 3. 4 (c) g (d)..7 4 4 g (e) M kg M 3.7 4 M.59 7 3. (a) m (e) kg (i) s (b) A (f) J N m kg m /s (j) kg m (c) Hz /s (g) J/(kg K) (k) F qe [F/q] N/(As) (d) F q v B (h) V (l) C( ) [F/q v] W( ) IV A s N/(C (m/s)) [V] kg m /A s 3 kg /As Q CV [C] A s 4 /(kg m ) 4. (a) 6.7 [N m /kg ] (f) 6.6 34 [J s] (b) 4. [J/cal] (g) 3. 8 [m/s] (c).4 3 [J/K] (h) 9.6 5 [C/mol] (d) 9. 9 [N m /C ] (i) 8.3 [J/(mol K)] (e) 6. 3 [/mol]

8 5. 6. 7. [m] kg [c] m/s [ ] J kg m /s [mc ] kg m /s mc ( ) m E mc. mg. 6 kg c 3. 8 m/s (5m 5m m) mc (. 6 ) (3. 8 ) 9. [J] (5 ) (5 ) (. )..5 9 [g]. cal/(g K) 4. J/(g K) 9 4 [K] 4..5 9 4π (.5 8+3+ ).83 7 [cm ] cal/min 4./6.4 [J/s].83 7.4 4 6 [J/s] ( ) E mc m 4 6 (3. 8 ) 4 9 [kg/s] ( 3 kg)

9. a ( ), b, c θ a sin θ c a, cos θ b a, tan θ c b, cosecθ sin θ, sec θ cos θ, cot θ tan θ., a b + c, sin θ + cos θ, + tan θ cos θ. θ b c (rad) r θ h h θ h (rad) r ( ) 3 π 6, 45 π 4, 6 π 3, 9 π, 8 π O θ r (rad) π/6 π/4 π/3 π/ sin / / 3/ cos 3/ / / sin / 3 3. π/ xy ( ) x θ P P x y θ < π/ cos θ, sin θ

cos θ, sin θ P x y P y sinθ θ cos O θ x θ cos θ, sin θ tan θ tan θ sin θ cos θ sin( θ) sin θ, cos( θ) cos θ, tan( θ) tan θ, sin(θ + π ) cos θ, cos(θ + π ) sin θ, tan(θ + π ) tan θ, sin(θ + π) sin θ, cos(θ + π) cos θ, tan(θ + π) tan θ. tanθ cosθ.5 sin θ Π 3Π Π Π -.5 Π Π 3Π Π θ -

.3 sin(α+β) xy (x, y) α (x, y ) (x, y) (x, y ) ( x y ) ( cos α sin α sin α cos α ) ( x y ). β (x, y ) (x, y ) ( x y ) ( cos β sin β ) ( x ) sin β cos β y ( cos β ) ( sin β cos α sin α ) ( ) x sin β cos β sin α cos α y ( cos α cos β sin α sin β (sin α cos β + sin β cos α) sin α cos β + sin β cos α cos α cos β sin α sin β ) ( x y ). (x, y) α + β (x, y ) ( x y ) ( cos(α + β) sin(α + β) sin(α + β) cos(α + β) ( ) sin(α + β) sin α cos β + sin β cos α, ) ( x y cos(α + β) cos α cos β sin α sin β, tan(α + β) sin α cos β + sin β cos α tan α + tan β cos α cos β sin α sin β tan α tan β. sin θ sin(θ + θ) sin θ cos θ + sin θ cos θ sin θ cos θ, cos θ cos(θ + θ) cos θ cos θ sin θ sin θ cos θ sin θ cos θ sin θ, tan θ tan θ + tan θ tan θ tan θ tan θ tan θ. cos ), sin θ cos θ cos θ + cos θ tan θ cos θ + cos θ,, ( cos θ) sin θ sin θ ( + cos θ).

sin α cos β [sin(α + β) + sin(α β)], cos α cos β [cos(α + β) + cos(α β)];, sin α sin β [cos(α + β) cos(α β)]. α + β A, α β B sin A + sin B sin( A + B cos A + cos B cos( A + B P sin θ + Q cos θ ) cos( A B ), ) cos( A B ). [ ] P P + Q P + Q sin θ + Q P + Q sin θ P + Q [sin θ cos α + cos θ sin α] P + Q sin(θ + α) sin α Q P + Q, cos α P P + Q.

3.4., π 5, 7π 8 ( )., 75, 6 3. r θ r θ 4. sin π 6 sin π 4 sin π 3 3 5. sin( θ) sin θ cos(θ + π ) sin θ tan(θ + π) tan θ 6. sin(θ π ) cos θ cos(θ π ) sin θ 7. sin 7π 8. cos 5π 6 9. tan π 8. sin, cos. sin(ωt) + sin(ωt). sin( π λ x) + sin[π (x + d)] λ

4.5. π (rad) 36 (rad) 36 π π 36 (rad) 5 π 7π 36 (rad) 8 π 8 π ( 57.3 ) π 5 7 7π 8 7. π (rad) 36 π 36 (rad) π 8 75 75 π 36 (rad) (.75 (rad)) (rad) 5π (rad) 6 6 π 36 (rad) 6π 5 (rad) 3. r πr θ θ π πr θ π r θ 4. ABC A BC D ABD ADC 3 BAD CAD π 3 π 6 BDA CDA π π ADC AC BC DC AD AC DC sin π 6 DC AC, 3 AC sin π 3 AD AC 3 ABC AB AC ACB ABC π π 4 AC + AB BC AC BC π sin π 4 AC BC 4 A B B C π 4 π 3 A D π 6 π 3 C

5 5. (a) θ θ (x, y) (x, y) sin( θ) sin θ (b) θ θ + π (x, y) ( y, x) cos(θ + π ) sin θ (c) θ θ + π (x, y) ( x, y) tan(θ + π) tan θ y y y θ θ x θ + π θ x θ+π θ x (a) (b) (c) 6. θ θ π (x, y) (y, x) y sin(θ π ) cos θ cos(θ π x ) sin θ θ π θ 7. 8. sin 7π ( π sin 4 + π ) sin π 3 4 cos π 3 + cos π 4 sin π 3 + 3 + 3 cos 5π 6 cos ( π 6 + π ) cos π 6 3 9. θ π sin π 8 cos π 4 sin θ > cos θ >, cos π 8 + cos π 4 + tan π 8 sin π 8 cos π 8 + ( )( + ) + +. sin 3θ sin(θ + θ) sin θ cos θ + cos θ sin θ sin θ cos θ cos θ + ( cos θ ) sin θ (4 cos θ ) sin θ cos 3θ cos(θ + θ) cos θ cos θ sin θ sin θ ( sin θ) cos θ sin θ cos θ sin θ ( 4 sin θ) cos θ

6. sin(ωt) + sin(ωt) sin[ (ω + Ω) (ω Ω) t] cos[ t] ) Ω ω cos( t) sin[(ω )t] ω cos( t) ω. sin( π λ x) + sin[π λ (x + d)] sin [π λ x + π λ (x + d)] cos [π λ x π (x + d)] λ cos πd λ sin[π λ (x + d )] ) d n πd λ (n ) cos () () λ

7 3 3.,, 3,... n n + + + 3 a, a, a 3,..., a n,... {a n } a k {a n } k k a k k a 3 a n n, b n n, c n n {a n }, 4, 6,... {b n } {c n } : a, a + d, a + d, a + 3d,..., a + nd,... : a, ar, ar, ar 3,..., ar n,... : p, p, 3 p,..., n p,... 3. n a, a n+ a n + a ( )

3 8 : a a, a n+ a n + d : a a, a n+ ra n : a, a n+ [(a n ) /p + ] p a a, a n+ a n+ + a n a, a, a 3, a 4 3, a 5 5, a 7 8,... n a n [( ) n + 5 5 ( ) n ] 5 5 n,, 3 a n+ pa n + q (p ) (p ) a n+ c p(a n c) c pc q q c ( p) q q a n+ p{a n ( p) ( p) } q p {a n ( p) } p n q {a ( p) } a n p n q {a ( p) } + q ( p) pn a + ( pn ) q p a + (n )q (p ) a n+ pa n+ + qa n a n+ sa n+ r(a n+ sa n ) r + s p rs q r, s x px q (x r)(x s) a n+ sa n r(a n+ sa n ) r (a n sa n ) r n (a sa ) a n+ sa n r n (a sa )

3 9 3.3 {a n } p q (q p) Σ q a k a p + a p+ + + a q kp p ( p ) q ( ) n : a n n n S n k + + + n k S n + + + (n ) + n S n n + (n ) + + + : a n a + (n )d S n (n + ) + (n + ) + + (n + ) + (n + ) n(n + ) S n n(n + ) n n a k [a + (k )d] na + k k n(n + ) d nd na + : a n r n a n n S n a k r k a a ( + r + + r n ) k k : a n n S n a ( + r + + r n ) rs n a ( r + + r n + r n ) n(n ) d ( r)s n a ( r n ) S n a ( r n ) r a ( r < n ) r (k + ) 3 k 3 3k + 3k + k n n [(k + ) 3 k 3 ] n [3k + 3k + ] k k { 3 + + (n + ) 3 } { 3 + + n 3 } n n n 3 k + 3 k + k k k (n + ) 3 n 3 k n(n + ) + 3 + n k

3 n k k 3 : a n n a k k : a k n C k [ (n + ) 3 3 n(n + ) n k k(k + ) n k n(n + ) n ] ( k ) k + n(n + )(n + ) 6 ( ) + ( 3 ) + ( 3 4 ) + + ( n n + ) + ( + ) + ( 3 + 3 ) + ( 4 + + n ) n + (n ) n + n! k!(n k)! (n k,, n) n n a k nc k n k k n ( ) (x + y) n nc k x k y n k x y k π ( ) π 3.459 π 4 π 6 π 8 k k k ( ) k ( k + 4 3 + 5 ) 7 + ( k 6 + + 3 + ) 4 + ( (k + ) 8 + 3 + 5 + ) 7 + e.788 e k k! +! + 3! + 4! + 5! + γ.5775 γ lim n [ log e n + n k ] k

3 3.4. 5 (a) 7 (b) cos x x (c). a n+ a n+ + a n, a a 3. {a n } {b n } a n [( ) n + 5 5 ( ) n ] 5 a n+ a n b n, a, b n+ 3b n, b a n, b n 4. (a) n (k ) (b) k n k (4k ) (c) n nc k ( ) k k 5. p (i) p (ii) n n + (a) 4 n n < n! (b) a, a n+ a n + n a n (n n + ) (c) n k k 3 n (n + ) 4

3 3.5. a a a 3 a 4 a 5 a n (a) 9 6 3 3 7n 5 (b) π/ 3π/ 5π/ 7π/ 9π/ (n )π/ (c) /4 /9 /6 /5 /n. a n+ a n+ + a n r, s x x x ± 5 a n sa n r n (a sa ) a n ra n s n (a ra ) r s ra n rsa n r n (a sa ) sa n sra n s n (a ra ) (r s)a n r n (a sa ) s n (a ra ) r + 5, s 5 r s 5 a sa s + 5 a ra r 5 r s a n [r n s n ] [( ) n + 5 5 5 ( ) n ] 5 3. b n b n+ 3b n, b b n 3(b n ) 3 ((b n ) 3 n (b ) b n 3 n ( ) + 3 n + a n a n + b n a n + (3 n + ) a n + (3 n 3 + ) + (3 n + ) a + (3 + ) + (3 n + ) + 3n 3 n + 3n + (n )

3 3 4. (a) (b) (c) n k n n n n(n + ) (k ) k n n k k k (4k ) n n k (k )(k + ) ( k k ) k + ( 3 ) + ( 3 ( 5 ) + n ) n + n + n nc k ( ) k [ + ( )] n k 5. (a) n 4 4 6 < 4! 4 n 4 k k < k! n k + < k + k < k!(k + ) n k + 4 n n < n! (b) n a n (n n + ) a ( + ) n a n (n n + ) n k a n (n n + ) n k + a k+ a k + k (k k + ) + k (k + k + ) [(k + ) (k + ) + ] n k + n a n (n n + ) (c) n k 3 ( + ) n j k 4 n j + j+ k k 3 j k 3 + (j + ) 3 j (j + ) + (j + ) 3 k 4 4 (j + ) [j + 4(j + )] 4 (j + ) (j + ) n j + n

4 4 4. a ( ) n a n a a a (a n ) a n /a n a n a m (a n ) (a m ) a n + m a n+m a n a m a n m n m a n a n a n a n p a /p x p a a /p q a q/p r a r π s a s s {s n } a s n ( ) a s lim t s a t (t ). x a x x a a x a y a x+y a x a x (a x ) y a xy a e ( e n lim + ) n n e.788... e e e / x a x x p a a /p e ikπ (k,,,... p ) k

4 5 3 5 5 5-4 - 4 e e x 3 5 5 5-4 - 4 / (/) x sinh x ex e x, cosh x ex + e x, tanh x sinh x cosh x sinh cosh tanh 6 cosh x 4-3 - - 3 tanh x - sinh x -4 sinh x, cosh x, tanh x 4. y a x x y x a y x y y log a x a (a ) x r a x, s a y x log a r y log a s log a r + log a s x + y log a (a x+y ) log a (a x a y ) log a (r s). log a x + log a y log a (xy) ( ) log a log x a x log a (x y ) y log a x log a

4 6 x log a r y log a s t log r s r t s a y a y log a s log a r t t log a r tx log a s log a r y x t log r s e ln x log x a a x a x e x ln a a log a x log a x log e x/ log e a (/ ln a) ln x - 3 4 5 - -3-4 e ln x 4.3 f(x) x, y f(x + y) f(x)f(y) x y f() f( + ) {f()} f() n f(n) f(n )f() f(n )f()f() {f()} n f() f( n + n n ) f(n n )f( n ) {f( n )}n f() a (a > ) 4. g(x) g(x)g(y) ( g(x)g(y) (k + l n ) k n n k n n n! f(x) a x g(x) x k ) ( k! l x k y n k k!(n k)! k n y l ) l! x n n! k,l x k y l k! l! n! k!(n k)! xk y n k n n! (x + y)n g(x + y)

4 7 k + l n k, l g(x) g(x) e x e g() n ( + n) n n nc k ( k n )k + n n + n(n )! n n! n(n )(n ) + n 3! n + + n! 3 n! + +! ( n ) + 3! ( n )( n ) + + n! ( n )( n )... ( n n ) g(x ln a) e x ln a a x n n

4 8 4.4.. 4, 8, 6 3. 3, 64 4. 6 5. a n, m (a n ) m a nm ( 6. lim + n p,, 3, 4 n p n) 7. cosh x sinh x 8. sinh(x + y) sinh x cosh y + cosh x sinh y 9. 3 log.3, log 3.477 log 4, log 5, log 6, log 8, log 9. log(x y ) y log x

4 9 4.5. 4 6 8. 4 6, 8 56, 6 65536 3. 6 65536 3 ( 6 ) (65536) 4.9... 9 4.3 9 3 4.9... 9 64 ( 3 ) (4.9... 9 ).84... 9.8 9 4. 6 6 54 4 ( 5) 7 4 ( 5) 3 3 3 5 5 3 3 5 5. a n a n (a n ) m m (a n ) m (a a a) (a n ) (a a a) (a n ). (a a a) (a n ) a n m a nm 6. ( + ). ( + ) ( + 3) 3 ( + 4) 4 ( ) 3 9.5.3 4 ( ) 4 3 64.37....4 3 7 ( ) 5 4 65.44....4 4 56 7. cosh x sinh x ( e x + e x ) ( e x e x ) 4 (ex + + e x ) 4 (ex + e x ) 4 4 8. e x cosh x + sinh x, e x cosh x sinh x sinh(x + y) ex+y e (x+y) [ex e y e x e y ] (cosh x + sinh x)(cosh y + sinh y) (cosh x sinh x)(cosh y sinh y) [cosh x cosh y + sinh x cosh y + cosh x sinh y + sinh x sinh y cosh x cosh y + sinh x cosh y + cosh x sinh y sinh x sinh y] sinh x cosh y + cosh x sinh y

4 3 9. log 4 log log.3.6 log 5 log log log.3.7 log 6 log ( 3) log + log 3.78 log 8 log 3 3 log 3.3.9 log 9 log 3 log 3.477.95. log x s x e s x y (e s ) y e sy log(x y ) log(e sy ) sy ys y log x

3 5 5. ( ) ( ) a a... 5. R (a, b) (c, d) R R (c a, d b) y (c,d) ( ) R (c a) + (d b) (a,b) R O x 3 A (p, q, r) A n p + q + r n V (x, x,..., x n ) V x + x + + x n n x k k

5 3 3 5.3 a k k a k k a (a x, a y, a z ) k a (ka x, ka y, ka z ) k k k k a (a x, a y, a z ) b (b x, b y, b z ) ka a (k<) ka (k>) a + b (a x + b x, a y + b y, a z + b z ) a + b b a a b a+( ) b 5.4 a (a x, a y, a z ) b (b x, b y, b z ) a b a b a x b x + a y b y + a z b z (4. ) a a x + a y + a z a a a, b θ a b a b cos θ xy ( ) a (a x, a y, ) b (b x, b y, )

5 33 a x ϕ a b x ϕ b a x a cos ϕ a b x b cos ϕ b a y a sin ϕ a b y b sin ϕ b a b a x b x + a y b y a b (cos ϕ a cos ϕ b sin ϕ a sin ϕ b ) a b cos(ϕ a ϕ b ) a b cos(ϕ b ϕ a ) (5.) cos( x) cos x ϕ b ϕ a O b φb φ a a

5 34 5.5. ( ). q (a, b, c) d (k, l, m) 3. q (a, b, c) d (k, l, m) 4. a b a b cos θ a, b, c a b θ c a + b ab cos θ b c θ a 5. a (,, ), b (,, ) a b

5 35 5.6. O A, B P, Q, R OA a, OB b OQ k k k OQ k( OB + BA) k[ b + ( a b)] k ( a + b) AR l l OA + l AR a + l[ b a] ( l) a + l b OQ AR G k ( a + b) ( l) a + l b a, b k l /3 OG ( a + b)/3 BP m m OB + m BP b + m( a b) m a + ( m) b m /3 ( a + b)/3 OG A a P G Q O R b B. r r q + t d (t ) d r q O

5 36 3. r ( r q) d d r q q O r 4. a, b, c b a c b a a + b a b c a + b a b cos θ a b a b cos θ θ b c b - a a 5. a b θ a b a b cos θ a b (,, ) (,, ) ( ) + + a + + b ( ) + ( ) + cos θ / / θ π/3

37 6 6. x y e x (, ), e y (, ) θ e x, e y e x (cos θ, sin θ) e y ( sin θ, cos θ) a (a x, a y ) e x, e y a a x e x + a y e y a θ e y sinθ e y θ O cos θ sinθ θ e x e x cos θ a a x e x + a y e y a x (cos θ, sin θ) + a y ( sin θ, cos θ) (cos θ a x sin θ a y, sin θ a x + cos θ a y ) (a x, a y) ( a x a y ) ( cos θ ax sin θ a y sin θ a x + cos θ a y ) ( cos θ sin θ sin θ cos θ ) ( ax a y ) n m n m 3 3 6 3 3 3 A A A 3 A A A 3 A 3 A 3 A 33 v v v 3 A v + A v + A 3 v 3 A v + A v + A 3 v 3 A k v + A k v + A 33 v 3 A v 3 (A v) j A ji v i i

6 38 6. A ka k (ka) ij ka ij A A A 3 A A A 3 A 3 A 3 A 33 n m A ± B (A ± B) ij A ij ± B ij A A A 3 A A A 3 A 3 A 3 A 33 ± ka ka ka 3 ka ka ka 3 ka 3 ka 3 ka 33 B B B 3 B B B 3 B 3 B 3 B 33 A ± B A ± B A 3 ± B 3 A ± B A ± B A 3 ± B 3 A 3 ± B 3 A 3 ± B 3 A 33 ± B 33 AB (AB) ij 3 A ip B pj p ( ) ( ) ( a b p q ap + br aq + bs c d r s cp + dr cq + ds ) ( ) ( ) ( ) p q a b pa + qc pb + qd r s c d ra + sc rb + sd AB BA 6.3 A ij i j I I, I ij δ ij δ ij δ ij { (i j ) ( )

6 39 n n n n A A AI IA A, (AI) ij A ik δ kj (IA) ij δ ik A kj A ij. k k A B I B A A AA A A I ( a b A c d ) ad bc A ( d b (ad bc) c a ) ad bc x, y ax + by p, cx + dy q ( ) a b A, x c d ( x y ), v ( p q ) A x v A A A A x A v x A v x 6.4 A A (deta ) ( ) a b A A ad bc c d A 3 3 A A A A 3 A A A 3 A 3 A 3 A 33 A A A A 33 + A A 3 A 3 + A 3 A A 3 A 3 A A 3 A A A 33 A A 3 A 3

6 4 6.5. (a) ( ) ( ) (b) (c). σ ( ), σ ( i i ), σ 3 ( ) [A, B] AB BA {A, B} AB + BA a) [σ, σ ] {σ, σ } b) [σ, σ 3 ] {σ, σ 3 } c) [σ 3, σ ] {σ 3, σ } ( ) a b 3. (a) A A (a + d)a + (ad bc)i O (O c d ) (b) a b c d A n (n ) ( ) cos θ sin θ 4. R(θ) R(θ) sin θ cos θ 5. 3x + 5y 7 x 9y 6. (a) cos θ sin θ r sin θ r cos θ (b) (c)

6 4 6.6. (a) ( ) (b) 3 6 3 3 (c) ) v i k v k A ki. 3 [σ a, σ b ] iϵ abc σ c, {σ a, σ b } δ ab (a, b, c 3) c ϵ abc (abc) (3) ϵ abc (abc) (3) 3. (a) A ( ) ( ) ( a b a b a + bc ab + bd c d c d ac + cd bc + d (a + d)a ( a ) + ad ab + bd ac + cd ad + d (ad bc)i ( ad bc ) ad bc ) A (a + d)a + (ad bc)i O (b) a b c d A A + I O A A 4. R (θ) A n A A n A n A n n A n ( ( cos θ ( sin θ) cos θ ( sin θ) sin θ cos θ ) ) ( ) cos θ sin θ sin θ cos θ

6 4 5. ( 3 5 9 ) ( x y ) ( 7 ) 6. (a) (b) (c) ( 3 5 9 ( x y ) cos θ sin θ ) ( 9 5 37 3 ( 9 5 3 ( 9) 5 3 ) ( 7 ) ) ( 63 + 55 37 4 33 ( 9 5 37 3 ) ( 8 37 9 r sin θ r cos θ cos θ(r cos θ) ( r sin θ) sin θ r cos θ + r sin θ r + + 3 + + ) )

43 7 7. x f(x) f(x) (x, f(x)) (x + h, f(x + h)) f(x + h) f(x) h h f(x) x f(x) f(x+h) f(x) O x x+h df(x) dx lim f(x + h) f(x) h h df dx f (x) ( ) f f(x), g(x) d [f(x + h) ± g(x + h)] [f(x) ± g(x)] [f ± g] lim dx h h lim h [f(x + h) f(x)] ± [g(x + h) g(x)] h df dx ± dg dx d f(x + h)g(x + h) f(x)g(x) [f(x)g(x)] lim dx h h f(x + h)g(x + h) f(x)g(x + h) + f(x)g(x + h) f(x)g(x) lim h [ h ] f(x + h) f(x) f(g + h) g(x) lim g(x + h) + f(x) h h h df dx g + f dg dx d dx ( ) f g f(x+h) lim f(x) g(x+h) g(x) h h lim h h [ ] f(x + h)g(x) g(x + h)f(x) g(x + h)g(x)

7 44 lim [f(x + h)g(x) f(x)g(x) h g(x + h)g(x)h +f(x)g(x) g(x + h)f(x)] lim h g(x + h)g(x) g [ df dx g f dg dx [ f(x + h) f(x) g(x) h ] g(x + h) g(x) f(x) h g(x) c (c ) d dc df [cf(x)] f(x) + c dx dx dx c df dx f(x) f(x(t)) df dt x(t + h) x(t) a f(x(t + h)) f(x(t)) lim h h f(x(t + h)) f(x(t)) x(t + h) x(t) lim h x(t + h) x(t) h f(x + a) f(x) x(t + h) x(t) lim lim a a h h df dx dx dt ] 7. x n (n ) d (x + h) n x n dx xn lim h h lim h h [ lim h [nx n + (x n + nx n h + [ n ] lim h nc k x k h n k x n h k ] n(n ) x n h +... + h n ) x n n(n ) x n h +... + h n ] nx n x n (n ) x n x n x d dx d dx [(xn )(x n )] { } d dx (xn ) (x n ) + (x n ) d dx (x n ) nx n (x n ) + (x n ) d dx (x n ) d dx (x n ) nx n

7 45 sin h lim h h lim cos h h h OAB OA OB AOB h B OA OA C OC OD OB D OC OD cos h BC sin h OAB h/ OAB (/) sin h OCD (h/) cos h O B D C A h cos h sin h h h/ h + (h ) sin h lim h + cos h lim h + h sin h lim h + h h h a sin h sin( a) sin a sin h h (a > ) sin h lim h h sin a a sin a a cos h h ( + cos h) h lim h cos h cos h ( + cos h) lim h h h ( cos h) sin h lim lim h h h h lim h ( cos h) h lim h sin h h sin h d sin(x + h) sin(x) sin x cos h + cos x sin h sin x sin x lim lim dx h h h [ h lim sin x cos h + cos x sin h ] cos x h h h d cos(x + h) cos(x) cos x cos h sin x sin h cos x cos x lim lim dx h h h [ h lim cos x cos h sin x sin h ] sin x h h h d dx tan x d ( ) sin x cos x cos x sin x( sin x) dx cos x cos x cos x

7 46 f(x) ln x log e x e lim e lim x n ( + n) n n ( + x) x x /h [ ln lim h ( + h)(/h) ln( + h) a h a a lim a e a ] lim h ln( + h) h e a lim a a d e x+h e x dx ex lim h h d ln x lim dx h ln(x + h) ln x h e x e h lim e x h h lim h x ln( + h x ) h x x d dx ax d d dx log a x d dx dx ex ln a ln ae x ln a a x ln a ( ) ln x ln a x ln a x p (p ) y x p ln y p ln x x dy y dx p x dy dx y p x pxp 7.3 h f(x + h) f(x) + hf (x) f (x) f(x + h) > f(x) f(x + h) < f(x) x f(x) f (x) (y x 3 x ) x x

7 47 f(x) f (x) x x f (x) f (x) > f (x) < f(x) x a b f + + f f(a) f(b) y f(x) a b x

7 48 7.4. ω, h, v, g arctan x tan x (a) (d) (g) (j) d dx cos d x (b) dt esin t d (c) dt [e t sin t] ( ) d d (e) ln [ ] x d (f) [x ln x] dx tan x dx x + dx d sin(ωt) (h) d d sin(ωt) (i) dt [ arctan x dt [ dx d dt h + vt g t] d (k) dt h + vt g t] d (l) e dx x. P n (x) d n n n! dx n [(x ) n ] (n ) P (x), P (x), P (x) 3. U(r) a r b (r > a, b ) r r 4. f(x) n x n n! df dx 5. x < f(x) f(x) ( ) n x n df n dx + x n

7 49 7.5. (a) (b) (c) (d) (e) d dx cos x cos x(cos x) cos x sin x sin x d dt esin t e sin t cos t d dt [e t sin t] e t sin t + e t cos t e t ( sin t + cos t) ( ) d dx tan x [ ] d dx ln x x + (tan x) tan x cos x tan x sin x d [ ln x ln(x + ) ] dx x x x + x + x x(x + ) x x(x + ) (f) (g) (h) d dx [x ln x] ln x + x x ln x + d sin(ωt) ω cos(ωt) dt d dt sin(ωt) d dt [ω cos(ωt)] ω sin(ωt) (i) y arctan x tan y x x d dx tan y d dx x dy cos y dx dy dx cos y + tan y + x (j) (k) (l) d [h + vt g ] dt t v gt d [h dt + vt g ] t d [v gt] g dt d dx e x d dx [ xe x ] e x x( x)e x (x )e x

7 5. P (x)! (x ) P (x) d! dx (x ) x x P (x) d! dx [(x ) ] d 4 dx [(x )(x)] 8 4[(x ) + x(x)] (3x ) 3. U(r) d dr U(r) d [ a dr r b ] a ( b) (br a) r r 3 r r3 r a b r b a r b U + U U( a) b 4. f(x) n ( ) a U b x n n! ( a a b ) ( a b b ) a b 4a b b a b 4a df dx d [ ] + x + x dx! + xn n! + + + x! + + xn (n )! + n x n (n )! n k k x k k! f(x) ( ) n 5. f(x) x n n n df dx d [ ] x x dx + x3 3 + ( )n x n + n x + x + ( ) k x k + ( ) n x n ( x) + x n

5 8 8. 8.. f(x) F (x) d F (x) f(x) dx F (x) f(x) ( ) f(x)dx F (x) F (x) x F (x + x) F (x) lim x x F (x + x) F (x) + f(x) x F (x + x) F (x + x) + f(x + x) x f(x) F (x) + [f(x) + f(x + x)] x. F (x + n x) F (x) + [f(x) + f(x + x) +... + f(x + (n ) x)] x F (x + n x) F (x) [f(x) + f(x + x) +... + f(x + (n ) x)] x f(x) x f(x + x) x... n x w ( ) x, n f x O x... f(t) x t x+n x x+w F (x + w) F (x) x+w x f(t)dt f f(t)dt x

8 5 F (x + w) F (x) ( F (x + w) F (x) ) f ( ) F d x f(t)dt f(x), dx a d b f(t)dt f(x) dx x d F (x) f(x) 7. dx 8.. f(x) x p (p ) x sin x cos x e x a x (a > ) f(x), g(x) b a F (x) ( ) (p + ) xp+ ln x cos x sin x e x ln a ax d df [fg] dx dx g + f dg dx [ d b df dx [fg]dx a dx g + f dg ] dx dx [ b df [fg] b a a dx g + f dg ] dx dx b a df b dx g dx [fg]b a a f dg dx dx [ ] b a [ ] b a W (x) [W (x)] b a W (b) W (a) ) π (sin x)xdx [( cos x)x] π π ( cos π)π ( cos ) π + + π cos xdx ( cos x) d dx xdx π ( cos x)dx π + [sin x] π π + sin π sin π

8 53 t a ln x dx t a ln x dx [x ln x]a t t t ln t a ln a a t d x ln x dx a dx x d t dx ln x dx t ln t a ln a a t a dx t ln t a ln a (t a) t a t ln t t ln t f(x) ln x F (x) x ln x x x ln x x x ln x x x dx 8..3 ) df dx f x t x(t) d df dx F (x(t)) dt dx dt f(x(t))dx dt t F f(x(t)) dx dt dt ) + x dx x tan θ x θ π/4 π/4 + x dx d (tan θ)dθ ( + tan θ) dθ π/4 cos π/4 θ cos θ dθ dθ π 4 π sin θ ) dθ ( r < ) + r cos θ + r t cos θ dt sin θ θ π t dθ π sin θ π + r cos θ + r dθ dt ( + r cos θ + r dθ )dθ + rt + r ( dt) + rt + r dt [ ln( + rt + r ) ] r r [ln( + r) ln( r) ] ( ) + r r ln r

8 54 8.. (a) (c) (e) π x ln x dx sin xdx x n e x dx (n ) (b) (d) (f) π/ sin x cos x dx xe x dx dx x. n n. π/ ( ) n π/ sin n xdx sin n xdx n 3. x a + y (a, b ) b 4. r h

8 55 8.3. (a) (b) (c) (d) (e) I n x ln x dx π/ π [ x ln x ] sin x cos x dx sin x dx π x x dx x dx ( : lim x x ln x ) π/ [ ] x [ sin x dx ] π/ 4 cos x cos x dx [ x xe x dx [ e x ] x n e x dx (n ) I e x dx [ e x] 4 ] sin x π π I n [ x n ( e x ) ] nx n ( e x )dx n x n e x dx ni n I n ni n n(n )I n n!i n! (f) x sin θ dx cos θ x θ π/ dθ. I n π/ x sin n x dx I n π/ π/ sin θ cos θdθ π/ sin x sin n x dx [ ( cos x) sin n x ] π/ π/ + (n ) (n ) π/ I n + (n )I n (n )I n I n n n I n π/ cos x sin n x dx 4 π/ cos θ cos θdθ dθ π ( cos x)(n ) sin n x cos x dx ( sin x) sin n x dx (n )(I n I n )

8 56 3. y b 4. x a a b a x a dx x a sin t x a a t π/ π/ a b x π/ a a dx b sin t dx π/ π/ dt dt b cos t (a cos t) dt π/ π/ π/ ab cos + cos t t dt ab dt π/ π/ ab [t + ] π/ sin t πab ( : (. ) ) D D x ar cos t, y br sin t ( r, t π) ( x ) ( ) x J det r t a cos t ar sin t det abr b sin t br cos t D dxdy y r y t π π/ abrdtdr ab(π) rdr πab πab y d dy ( ) h y d r h πd dy π r h (h y) dy y h h π r h (h y) dy π r h [ (y h) 3 3 ] h πr h 3 h y d r

II

58 9 9. ( ) x t x x(t) O x(t) v(t) x t + t x(t + t) x(t + t) x(t) t t v(t) v(t) dx(t) dt x(t + t) x(t) lim t t v(t + t) v(t) t ( ) ( ) x(t+ t) x(t+ t) t x(t+ t) x(t) t t x(t + t) + x(t) x(t + t) ( t) t a(t) a(t) dv(t) dt F ma(t) m dv(t) dt m d x(t) dt m F x(t) x(t) (m) ( t : x(t ), v(t )) (F ) ( ) x(t)

9 59 9. t x(t) v(t) t + t t + t x(t + t) x(t) + v(t) t, v(t + t) v(t) + a(t) t v(t) + F m t x(t + t) x(t + t) + v(t + t) t x(t) + v(t) t + (v(t) + F ) m t t x(t) + v(t) t + F m ( t), v(t + t) v(t + t) + a(t + t) t v(t) + F m t + F m t t t (8.. ) ( : t t )

9 6 9.3. m t (a) : x vt (v ) (b) : z h gt (h t g ) (c) : z h + vt gt (v h, g ) (d) : (x, z) (x + v x t, z + v z t gt ) (t (x, y ), (v x, v z ) ) (e) : (x, y) R(cos(ωt), sin(ωt)) (R, ω ) (f) : x A sin(ωt + α) (A ω α ). m t z z(t) t z() z, dz/dt v g (a) (b) g ( ) dz(t) + mgz(t) (c) m dt 3. t d dt (x, y) (k ( cos(ωt)), k sin(ωt)) t t

9 6 9.4. (a) : dx dt v : d x dt (b) : dz dt gt : d x dt : md x dt g : md z dt mg (c) : dz dt v gt : d z dt g : z md dt mg (d) : d dt (x, z) (v x, v z gt) : : m d (x, z) m(, g) dt d (e) : (x, y) Rω( sin(ωt), cos(ωt)) dt d : dt (x, y) Rω (cos(ωt), sin(ωt)) : m d dt (x, y) ω (x, y) (f) : dx dt Aω cos(ωt + α) : d x dt : m d x dt ω x. (a) m d z dt mg d (x, z) (, g) dt Aω sin(ωt + α) (b) t : dz dt v gt : d z g dt ( ) ( ) m(v gt)( g) mgv + mgt (c) d m dt ( ) dz(t) + mgz(t) dt m dz(t) dt d z(t) dt + mg dz(t) dt mgv + mgt + mg(v gt) 3. (x(t), y(t)) t(k ( cos(ωs)), k sin(ωs))ds k(t ω sin(ωt), ω cos(ωt)) + (x, y ) (x(), y()) (, ) x, y k ω (x(t), y(t)) k (ωt sin(ωt), cos(ωt)) ω

6. f(x, x,..., x n ) f(x,..., x k + h,..., x n ) f(x,..., x k,..., x n ) f((x, x,..., x n ) lim x k h h d f(x,..., x n ) x k dx k f(x,..., x n ) df df f x dx + f x dx + f x n dx n n k f x k dx k n P n T V P nrt V (R ) dp P P P RT dn + dt + dv n T V V nr nrt dn + dt V V dv [ ] f(x, x,..., x n ) x, x,..., x n t, t,..., t n ; x i x i (t, t,..., t n ) t k (k,..., n) ( ) f t k n i f x i x i t k ) (x, y) (r, θ) ( r, θ π) x r cos θ, y r sin θ y r x + y, sin θ x + y, cos θ x x + y r x x x + y cos θ, sin θ y/ x + y x, y r y y x + y sin θ. cos θ θ x xy θ sin θ cos (x + y ) 3/ r cos θ θ y (x + y ) y / (x + y ) cos θ 3/ r,.

63 θ x sin θ r, θ y cos θ r f x f r r x + f θ θ x f f sin θ cos θ, r θ r f f r y r y + f θ θ y f f cos θ sin θ +. r θ r. 4 r r + dr θ θ + dθ dr R rdθdr + r dr, dθ θ r R θ π R dθ R R π r dθdr R ( π ) R r dθ dr π r dr πr f(x, x,..., x n ) dx dx... dx n x, x,..., x n k, k,..., k n x (k, k,..., k n ), x n (k, k,..., k n ),... x n (k, k,..., k n ) dx dx... dx n J dk dk... dk n x x x k k... k n x x x k J det k... k n.... x n k x n k... x n k n J J J 4

64 ) J det x r sin θ cos ϕ, y r sin θ sin ϕ, z r cos θ ( θ π, ϕ π) x r y r z r x θ y θ z θ x ϕ y ϕ z ϕ det sin θ cos ϕ r cos θ cos ϕ r sin θ sin ϕ sin θ sin ϕ r cos θ sin ϕ r sin θ cos ϕ cos θ r sin θ r cos θ sin θ cos ϕ + r sin 3 θ sin ϕ ( r ) cos θ sin θ sin ϕ ( r ) sin 3 θ cos ϕ r cos θ sin θ + r sin 3 θ r sin θ ( R ) ) e x dx π 4π e x dx I (I > ) I ( ) I π. π x +y +z R dxdydz R π R ( π r sin θdθdr π r dr 4πR3 3 ) ( e x dx e r rdθdr π R π π R ) e y dy R r [ cos θ] π dr r sin θdϕdθdr e r rdr π e (x +y ) dxdy [ e r ]

65.3. (a) ln [ ] x y [ x +y (d) y xy x] (b) ln [ ] x x ] x +y (e) y [ y x (c) sin( x x + y ) ] (f) x [ y x. (x, y, x) x r sin θ cos ϕ, y r sin θ sin ϕ, z r cos θ r, θ, r, θ, ϕ ϕ x, y, z 3. r x + y + z r [ ] x + y + z r 4. x a + y (a, b ) b (hint: x ar cos θ, y br sin θ ( r, θ π) ) 5. x a + y b + z (a, b, c ) c 6. < a < R, R < a R π π G, ρ, a G r + a ar cos θ ρr sin θdϕdθdr

66.4. (a) (b) (c) [ ] y ln x x + y [ ] x log x x + y y [ln x ln(x + y )] y x + y x [ln x ln(x + y )] x x x + y y x x(x + y ) x sin( x + y ) cos( x + y ) x x x + y cos( x + y ) x + y. (d) (e) (f) r θ ϕ x x r + y y r + z z r x x θ + y y θ + z z θ x x ϕ + y y ϕ + z z ϕ [ y xy x] x x x [ y y x] y x [ y x x] x [ ] y x y x 3 sin θ cos ϕ x + sin θ sin ϕ y + cos θ z, r cos θ cos ϕ x + r cos θ sin ϕ y r sin θ z, r sin θ sin ϕ x + r sin θ cos ϕ y 3. r x + y + z x r x (x + y + z ) / x [ (x + y + z ) 3/ (x) [ (x + y + z ) 3/ 3 (x + y + z ) 5/ (x)x ] ] r 3 + 3x r 5 y r r 3 + 3y r 5, z r r 3 + 3z r 5 [ ] x + y + z r 3 r + 3(x + y + z ) 3 r 5 3 r 3 + 3r r 5

67 4. D D x ar cos t, y br sin t ( r, t π) ( x ) ( ) x J det r t a cos t ar sin t det abr b sin t br cos t x r y r z r x θ y θ z θ D x ϕ y ϕ z ϕ dxdy y r y t π abrdtdr ab(π) rdr πab πab 5. D D x ar sin θ cos ϕ, y br sin θ sin ϕ, z cr cos θ ( r, θ π, ϕ π) a sin θ cos ϕ ar cos θ cos ϕ ar sin θ sin ϕ J b sin θ sin ϕ br cos θ sin ϕ br sin θ cos ϕ c cos θ cr sin θ abcr cos θ sin θ cos ϕ + abcr sin 3 θ sin ϕ + abcr sin θ cos θ sin ϕ + abcr sin 3 θ cos ϕ abcr (cos θ sin θ + sin 3 θ) abcr sin θ dxdydz 6. I I D R π π πgρ πgρ a πgρ a R R R π π 4πabc abcr sin θdϕdθdr πabc r dr 4π 3 abc G r + a ar cos θ ρr sin θdϕdθdr πgρ [ ] π r ar (r + a ar cos θ) / dr r { (r + a + ar) / (r + a ar) /} dr r( r + a r a )dr π R π r sin θdθdr r sin θ r + a ar cos θ dθdr r r R R < a r R < a r a a r I πgρ a R 4πGρR3 3a r{(r + a) (a r)}dr πgρ a ( G M ) (M 4πR3 a 3 ρ) R r dr < a < R I πgρ [ a R r{(r + a) (a r)}dr + r{(r + a) (r a)}dr a a πgρ [ a ] R r dr + ar dr πgρ [ a a a 3 a3 + a(r a )] 4πGρa 3 πgρ(r a ) πgρ( a 3 R ) ]

68. A B C B C C3 A θ r F ( r) F ( r) r F ( r) r F ( r) d r C r(t) (x(t), y(t), z(t)) t F ( r) d r F ( r(t)) d r C t dt dt ) ( µ) m (L, ) (, L) W y µmg L C F µmg d r d r C x -L O L F ( r) d r µmg d r d r µmg d r µmg d r d r dt dt C : r (L( t), ) ( t ) d r dt dt ( L, )dt Ldt W (C ) F d r µmgldt µmgl C C : r L(cos t, sin t) ( t π) d r π dt dt L( sin t, cos t)dt Ldt W (C ) F d r µmgldt µmglπ C C -L

69 s(t) (x(t), y(t)) (t t t ) C f(s)ds t t f(s(t)) ds dt dt, ) ds dt ( dx + dt ( ) dy dt (+ ) ) x + y r (r ) C : (x(t), y(t)) (r cos t, r sin t) ( t π) ) ( dx ds dx + dy + dt ( ) dy dt r dt sin t + r cos t dt rdt π ds rdt πr C. p, q R (θ, ϕ) ( θ π, ϕ < π) x R sin θ cos ϕ, x R sin θ sin ϕ, z R cos θ (x + y + z R ) z x dr q n dr p Σ y Σ r(p, q) f( r) Σ Σ r f( r) ds f( r(p, q)) p r q dpdq ) Q R Σ r r E Q r 4πϵ r n r r E n ds r ( ) Σ r θ (R cos θ cos ϕ, R cos θ sin ϕ, R sin θ), r ( R sin θ sin ϕ, R sin θ cos ϕ, ) ϕ r θ r ϕ (R sin θ cos ϕ, R sin θ sin ϕ, R sin θ cos θ)

7 Σ E n ds π π Q 4πϵ R R sin θ dθdϕ Q ϵ ( ) ( ) ( ) S A nds A dv V A d s ( A) n ds S

7.3. (a) (a, b) f(x, y) k x + y (k ) k (b) x (r, ) f(x, y) x + y (k ) µ (c) R f(x, y) π x + y (µ ). (P ) (V ) (T ) P V nrt (n R ) (a) A B C D P dv P P B C P A D O V V V (b) A B C P dv B C P P B P O A V V C V (c) (b) B C P V γ [ ] (γ ) A B C 3. a E E( r) k r S E nds ( n r/ r ) EdV S 4. a A A(x, y, z) (, ωx, ) (ω ) C A d s ( A) nds ( n (,, )) C V S

7.4. (a) C : (x(t), y(t)) (at, bt, ) (t ) ds a + b dt f(x, y)ds k (at) + (bt) a + b dt k(a + b ) tdt c k (a + b ). (b) C : (x(t), y(t)) (t,, ) (t r) ds dt r k f(x, y)ds c t dt k r (c) C : (x(t), y(t)) (R cos t, R sin t) (t π) ds Rdt π µ f(x, y)ds c π R cos t + R sin t Rdt µ (a) A B B C C D D A P RT V (V ) P ( ) V RT V (V ) P ( ) V P dv + P dv + + P dv V V P (V V ) + P (V V ) (P P )(V V ) (b) A B B C C A RT P V (V ) RT P ( ) V V RT P dv + V dv + V P dv RT ln( V ) + P (V V ) V V V RT ln( V V ) P (V V ) (c) (b) B C P V γ P V γ P V γ P P V γ P dv V V P V γ P V γ V dv P (V γ V ) V γ V γ γ P V P V γ V γ P (V V ) P (V V )

73 3. E n k r S r r k r ka E nds E k(x, y, z) 3k π π kaa sin θdθdϕ 4πka 3 V EdV a π π 3kr sin θdϕdθdr 3k 4πa3 3 4πka 3 4. C : (x(t), y(t)) (a cos t, a sin t) (t π) d s a( sin t, cos t, )dt C A s π ωa (, ωa cos t, ) a( sin t, cos t, )dt π + cos(t) dt ωπa π ωa cos dt A (,, ω) S ( A) nds a π ωrdθdr ωπa

74. f(x) (x a) n (a n ) ( ) c n f(x) c n (x a) n c + c (x a) + c (x a) + n () x a f(a) c () x f (x) + c + c (x a) + 3c 3 (x a) + x a f (a) c () x f (x) + c + 3 c 3 (x a) + 4 3(x a) x a f (a) c.. () x n dn dx n f(x) n (n ) c n + (n + ) n c n+ (x a) + x a f (n) (a) n!c n f(x) x a ( ) f(x) f(a) + f (a) (x a) + f (a) (x a) + f (a) 3! (x a) 3 +... n n! f (n) (a) (x a) n. a f(x) f() + f () x + f () x + f () 3! x 3 +... n n! f (n) () x n. ) e x + x + x + sin x x 3! x3 + cos x! x + n n n n! xn ( ) n (n + )! xn+ ( ) n (n)! xn

75 x ( ) x + x + x x n ( x < ) n ln( + x) x x + ( ) n 3 x3 + n n xn ( < x ) f(x, y) f(a, b) + f f (a, b)(x a) + (a, b)(y b) x y + f x (a, b)(x a) + f y (a, b)(y b) + f (a, b)(x a)(x b) +. xy. sin x, cos x x a sin, cos sin x cos x k k ( ) k (k + )! xk+ x x3 3! + x5 5! x7 7! +, ( ) k (k)! xk x! + x4 4! x6 6! +. sin x, cos x sin, cos ( ) e x k x k k! + x + x! + x3 3! +. x θ e iθ (iθ) k [ i m k k! m (m)! θm + ( ) m m (m)! θm + i m ] im+ (m + )! θm+ ( ) m (m + )! θm+ cos θ + i sin θ e iα e iβ (cos α + i sin α)(cos β + i sin β) (cos α cos β sin α sin β) + i(sin α cos β + cos α sin β) cos(α + β) + i sin(α + β) e i(α+β)

76 e A e B e A+B z, w e z e w k n e z+w z k k! n! j n k w j j! n k n k! (n k)! zk w n k (k + j n ) n! k!(n k)! zk w n k n! (z + w)n ( ) n

77.3. f(x) k k! f (k) () x k sin x, cos x. cos x x 4 x π 3 ( ) 3. x < x x x 4. a, r r a θ a + r ar cos θ r r 5. sin θ i (eiθ e iθ ) cos θ (eiθ + e iθ ) 6. N e ikx k

78.4. sin x, cos x n d dx sin x cos x, d sin x sin x,, dx d n { ( ) m dx n sin x sin x (n m ) ( ) m cos x (n m + ) d dx cos x sin x, d cos x cos x,, dx d n { ( ) m+ dx n cos x cos x (n m ) ( ) m+ sin x (n m + ). m x f (n) () ( ) m ( ) m sin x (m + )! xm+, cos x (m)! xm cos π 3 m ( π 3 m ) ( ) π 4 π + 4! 3 8 + π4 944.5796... ( π 3 ) 6.5.796....36% 6! 3. π6 5488.83... ( ) π 4 4! 3 x ( x) / f(x) f (x) (/) ( x) 3/ f (x) (3/4) ( x) 5/ z f() + f ()x +! f ()x + O(x 3 ) + x + 3 8 x + O(z 3 ) 4. a + r ar cos θ a [(a/r) cos θ (r/a) ] [(a/r) cos θ (r/a) ] x a + r ar cos θ [ + a x + 3 ] 8 x + O(x 3 ) [ + a [(a/r) cos θ (r/a) ] + 3 ] 8 [(a/r) cos θ (r/a) ] + O(x 3 ) [ + r r cos θ a a a + 3r cos ] θ + O(r 3 ) a a + r ( r 3 cos ) θ cos θ + + O(r 3 ) a a 3

79 5. e iθ cos θ + i sin θ e iθ cos( θ) + i sin( θ) cos θ i sin θ. sin θ i (eiθ e iθ ), cos θ (eiθ + e iθ ) 6. A N k e ikx A + e ix + e 4ix + e i(n )x (eix ) N einx e ix e ix einx (e inx e inx ) e ix (e ix e ix ) einx i sin(nx) e ix i sin x einx sin(nx) e ix sin x θ e iθ cosθ + i sin θ cos θ + sin θ A e inx sin(nx) e ix sin x sin (Nx) sin x ) N 5 5 5 5-6 -4-4 6

8 3 3. f(x) (f (x), f (x), f (x)...,) F (x, f, f, f,...) f f n n : d f(x) kf(x) (k ) dx : m d dt x(t) mω x(t) η d x(t) (m, ω, η ) dt n n (f(x ) f, f (x ) f, f (x ) f,...) : ( ) 38 9U( ) α 34 9Th( ) ( ) 45 p t N(t) t t + t pn(t) t N(t) N(t + t) N(t) N(t) pn(t) t t t d N(t) pn(t) dt N(t) dn N dt p

3 8 t dn N dt dt ( p)dt N dn ( p)dt ln N(t) pt + C (C ) N(t) N(t )e p(t t ) C t t N(t ) N(t) N(t )e p(t t ) ( ) τ / e pτ / / p p ln /τ / 45.5 5 8 6 3 6 3 (5 8 ) 3 6 : k x kx ( ) m m d x(t) kx(t) dt kx x x(t) A sin(ωt + B) (ω, A, B ) ω mω A sin(ωt + B) ka sin(ωt + B) k/m A, B 3. f(x, x,..., x n ) (x, y, z) t 3 3 r (x, y, z) x i x + y + z i ϕ( r) ρ( r) ( ρ( r) )

3 8 [ ] + µ ϕ( r) ρ( r) [ ] v t ψ( r, t) ρ( r, t) [ ] k t ψ( r, t) ρ( r, t) (v )

3 83 3.3. 6 88Ra.6 3 6 88Ra 6 88Ra /. d dt x(t) ω x(t) x(t) a) x() A, ẋ() b) x(), ẋ() v c) x() A, ẋ() v 3. v() g, k. dv(t) dt g kv(t) ( : e kt ) 4. f, g ξ(x, t) f(x vt) + g(x + vt) x ξ(x, t) v ξ(x, t) t

3 84 3.4. t ( ) t/(.6 3 ). t ln.6 3 ln t.6 3 ln ln.6 3.3.693 5.3 3 d dt x(t) ω x(t) x(t) a sin(ωt + b) ẋ(t) aω cos(ωt + b) a b a) x() a sin b A, ẋ() aω cos b b π/, a A x(t) A sin(ωt + π/) A cos(ωt) b) x() a sin b, ẋ() aω cos b v b, a v/ω x(t) v ω sin(ωt) c) x() a sin b A, ẋ() aω cos b v ( ) A sin b + cos b + a ( v aω ) a A + (v/ω) x(t) A + (v/ω) sin(ωt + θ) θ sin θ A/ A + (v/ω), cos θ (v/ω)/ A + (v/ω) 3. e kt kt dv(t) e dt ge kt kv(t)e kt ge kt v(t) d dt {ekt } kt dv(t) e + d dt dt {ekt }v(t) ge kt d dt [ekt v(t)] ge kt t e kt v(t) g k ekt + C (C ) v() C g k ( ) v(t) g k + Ce kt v(t) g k [ e kt ]

3 85 4. x ξ(x, t) v ξ(x, t) ξ(x, t) f(x vt) t ξ(x, t) g(x + vt) x f(x vt) f (x vt) ( v) v f(x vt) f (x vt) f (x vt) t v x f(x vt) v f(x vt) t x g(x vt) g (x + vt) v g(x + vt) t f, f af + bf (a, b ) x [af + bf ] a x f + b x f a v t f + b v t f v t [af + bf ] af + bf ( : ) f(x vt), g(x + vt) f(x vt) + g(x + vt)

86 4 4. r (x, y, z) r ( ) : ( : ) : r ( : ) r x y z r r r 3 r x y z R x y z 3 r i R ij r j j, 3 3 RR T R T R r i (R T ) ik r k r kr ki k k (R T R : (R T ) ij R ji ) (R: ) 3 3 3 : T ij...n (R ip )(R jq ) (R nu )T pq...u p q u ( : f xy ) 4. 3 a (a x, a y, a z ) b (b x, b y, b z ) ( ) a b a b (a y b z a z b y, a z b x a x b z, a x b y a y b x ) a b b a e x (,, ), e y (,, ), e z (,, )

4 87 e x e x e y e y e z e z e x e y e y e x e z y e e y e z e z e y e x y e z e z e x e x e z e y x e x a, b θ a b a b sin θ ( ) a (a x, a y, a z ) b (b x, b y, b z ) a b (a y b z a z b y ) + (a z b x a x b z ) + (a x b y a y b x ) (a y b z ) + (a z b y ) + (a z b x ) + (a x b z ) + (a x b y ) + (a y b x ) a y a z b z b y a z a x b z b x a x a y b x b y (a x + a y + a z)(b x + b y + b z) a xb x a yb y a zb z a y a z b z b y a z a x b z b x a x a y b x b y (a x + a y + a z)(b x + b y + b z) (a x b x + a y b y + a z b z ) a b ( a b) a b a b cos θ a b sin θ θ π sin θ a b a b sin θ 3 v x v y v z v 3 v (v, v, v 3 ) ϵ ijk ( a b) i 3 j,k ϵ ijk a j b k : (i, j, k) (,, 3), (, 3, ), (3,, ) ϵ ijk : (i, j, k) (, 3, ), (,, 3), (3,, ) : a c a (c ) ( a c a) i c c 3 j,k 3 k,j 3 j,k ϵ ijk a j ca k c ϵ ik j a k a j 3 j,k ϵ ijk a j a k (j k, k j ) ( ϵ ij k ) a j a k ( a c a) i a c a r p r p z

4 88 4.3. a) (,, ) (,, ) b) (,, ) (,, ) c) (,, ) (,, ) d) (v, v, ) (,, B) e) 3 (F, F, F ) (r,, r) f) (, p, ) (x, y, z). m d r dt F (a) r (b) p r p ( ) F r (c) F r ( F ) r p 3. 4. 3 A, B, C A ( B C) B ( C A) C ( A B) 5. 3 A, B, C A ( B C) ( A C) B ( A B) C 6. q B F q v B B (,, B) ( q m) r() (a,, ), v() (, v, ) 7. r() (a,, ), v() (, v, v z )

4 89 4.4. (a) (,, ) (b) (,, ) (c) (, 5, 3 ) (d) (vb, vb, ) (e) 3 (F r,, F r) (f) (pz,, px). (a) (b) r m d r dt r F d d r d p ( r p) p + r dt dt dt d r dt md r dt + r r md dt r r md dt r F (c) F r r F (b) d dt ( r p) r p 3. a b 3 3 a i a i R ij a j, a i b i R ij b j j j a b a b a b 3 3 3 3 3 3 3 a ib i R ij a j R ik b k R ij R ik a j b k i i j k i j k 3 3 3 δ jk a j b k a j b j j k j 4. A ( B C) 3 3 3 3 3 3 A i ( ϵ ijk B j C k ) ϵ ijk A i B j C k i j k i j k 3 3 3 3 3 3 ϵ kij A k B i C j ϵ ijk B i C j A k k i j i j k B ( C A) 3 3 3 3 3 3 ϵ jki A j B k C i ϵ ijk C i A j B k j k i i j k C ( A B) A, B, C

4 9 5. ( A ( B C) ) A (B C B C ) A 3 (B 3 C B C 3 ) (A C + A 3 B 3 )B (A B + A 3 B 3 )C (A C + A C + A 3 B 3 )B (A B + A B + A 3 B 3 )C ( A C)B ( A B)C ( A ( B C) ) ( A C)B ( A B)C, ( A ( B C) ) A ( B C) ( A C) B ( A B) C 6. m d r dt q v B qb(v y, v x, ) d r dt d v dt m dv x dt qbv y, d v x dt m dv y dt qbv x, ( qb m ) v x 3 ( A C)B ( A B)C 3 m dv z dt qb m ω v x(t) α sin(ωt+β) v x () β v x (t) α sin(ωt) x(t) α ω cos(ωt)+γ x() a γ α ω a y m dv x dt qbv y v y ω dv x dt αω cos(ωt) α cos(ωt) ω v y () v α v γ a + α ω a + v ω v y y(t) v sin(ωt) + δ y() δ ω z dv z dt v z(t) η v z () δ z() z(t) x(t) v ω [ cos(ωt)] + a, y(t) v sin(ωt), z(t) ω 7. z dv z dt v z() v z v z (t) v z z() z(t) v z t x(t) v ω [ cos(ωt)] + a, y(t) v ω sin(ωt), z(t) v zt

9 5 5. : t r(t) : t r ( E( r, t) B( r, t)) : : 5. ( x, y, z x, x y z x ỹ z R x y z x, y, z y, (R) xx (R) xy (R) xz (R)ỹx (R)ỹy (R)ỹz (R) zx (R) zy (R) zz ) z x x x x + ỹ x ỹ + z x z (R) xx x + (R) ỹx ỹ + (R) zx z x y y x + ỹ y ỹ + z y z (R) xy x + (R) ỹy ỹ + (R) zy z x z z x + ỹ z ỹ + z z z (R) xz x + (R) ỹz ỹ + (R) zz z R T R R R I x y z R ( ) x ỹ z ( x, y, x y z ) z

5 9 5.3 (grad), (div), (rot) (f( r)) ( A( r) (A x ( r), A y ( r), A z ( r)) (grad) : gradf( r) f( r) (div) : ( x f, y f, ) z f div A( r) A( r) x A x + y A y + z A z (rot) : rot A( r) A( r) ( A) i 3 j,k ( y A z z A y, ϵ ijk ( ) j ( A) k 3 j,k z A x x A z, ϵ ijk j A k x A y ) y A x grad M r m V ( r) G Mm r r r grad ( gradv ( r) G Mm ) ( ) Mm G r x + y + z ( ( ) ( ) Mm Mm G, G x x + y + z y x + y + z ( x GMm (x + y + z ), y 3/ ( ) GMm r r r GMm r (x + y + z ), ]; z 3/ (x + y + z ) 3/, ( )) Mm G z x + y + z ) r V ( r) V ( r) ( ) r

5 93 div a ρ E( r) 4πkρ r ( 6 3 grad ) div div E( r) E( r) x E x + y E y + z E z 4πkρ 3 ( x x + y y + z) 4πkρ z r Q E( r) kq (x, y, z) div r r dive( r) E( r) ( ) kq x r x + ( ) kq 3 y r y + ( ) kq 3 z r z 3 kq r 3 3kQ r 5 x + kq r 3kQ 3 r 5 y + kq r 3kQ 3 r 5 z (r ) ( ) div E( r) r div ( ) j( r) r rot A( r) (,, µ I 4π ln(x + y )) z z l x + y ( ).5.5 l A rot rota( r) (A x, A y, A z ) ( y A z z A y, ( µ I 4π y ln(x + y ), z A x x A z, µ I 4π z A y ) y A x ) x ln(x + y ), µ I πl ( y l, x l, )

5 94 z I ( ) A rot ( B A) A grad ( ) y x rot z

5 95 5.4. r (x, y, z) r r r r (a) r (d) r (g) r (b) (/r) (e) ( r/r 3 ) (h) ( r) (c) ( r) (f) ( )r (i) ( r). V ( r) k r (k ) V ( r) 3. B k (x + y ) ( y, x, ) ( x + y k ) B 4. m (,, ), A m r k B r 3 (r ) k r (x, y, z), r r B A 5. r (x, y, z) f f 6. r (x, y, z) A ( A) 7. r (x, y, z) A ( A) ( A) A

5 96 5.5. (a) r ( x + y + z ) ( x x + y + z, y x + y + z, ) z r x + y + z r (b) ( ) r ( x x + y + z, y x + y + z, ) z x + y + z ( ) x (x + y + z ), y 3/ (x + y + z ), z 3/ (x + y + z ) 3/ r r 3 (c) (d) (e) (f) ( r) 3 r x x + y y + z z 3 ( r/r 3 ) x x (x + y + z ) + 3/ y (x + y + z ) 3x 3/ (x + y + z ) 5/ + (x + y + z ) 3y 3/ (x + y + z ) 5/ ( )r y (x + y + z ) 3/ + z + (x + y + z ) 3z 3/ (x + y + z ) 5/ 3 (x + y + z ) 3(x + y + z ) 3/ (x + y + z ) 3 5/ r 3r 3 r 5 ( ) x + y + z x + y + z x y x + y + z + z x x + y + z + y x + y + z x (x + y + z ) 3/ + x + y + z y (x + y + z ) 3/ + x + y + z z (x + y + z ) 3/ 3 r (x + y + z ) r 3 r z (x + y + z ) 3/ z x + y + z

5 97 (g) (h) r ( r) ( z y y z, x z z x, y x ) x y ( x x + y + z, y x + y + z, ) z x + y + z ( z y x + y + z y z x + y + z, x z x + y + z z x x + y + z, y x x + y + z x y x + y + z ) yz ( (x + y + z ) + yz 3/ (x + y + z ), 3/ xz (x + y + z ) + xz 3/ (x + y + z ), 3/ xy (x + y + z ) + xy 3/ (x + y + z ) ) 3/ (i) ( r). ( ) V ( r) k r k (x + y + z ) k(x, y, z) k r 3. B k ( y x + y, ) ( x x + y, k xy (x + y ) xy (x + y ) + ) 4. A k m r r 3 5. k ( y, x, ) r3 A ( 3xz k r, 3yz 5 r, 5 r 3 ) [ 3 r 5 (r z ) k m ] 3( m r) + r r3 r 5 B ( A) ( f) ( y ( z f) z ( y f), z ( x f) x ( z f), x ( y f) y ( x f)) 6. ( A) x ( y A z z A y ) + y ( z A x x A z ) + z ( x A y y A x )

5 98 7. ( ( A) ) x y ( x A y y A x ) z ( z A x x A z ) y x A y + z x A z ( y y + z z )A x x ( x A x + y A y + z A z ) ( x x + y y + z z )A x x ( A) A x ( ( A) ) ( ( A) ) y z y ( A) A y z ( A) A z ( A ( A) A

99 6 6.?? n m ( n m ), 3 ( ) ( ) A A A n v A v + A v + + A n v n A A A n v A v + A v + + A n v n........ A k A k A kn v n A k v + A k v + + A kn v n A v n (A v) j A ji v i i n m n m s s 6. A A A A n ka ka ka n A A A n ka ka ka n ka k............ A k A k A kn ka k ka k ka kn (ka) ij ka ij n m A ± B A A A n A A A n...... A m A m A mn ± B B B n B B B n...... B m B m B mn

6 A ± B A ± B A n ± B n A ± B A ± B A n ± B n...... A m ± B m A m ± B m A mn ± B mn (A ± B) ij A ij ± B ij n m m k n m A m k B AB m (AB) ij A ip B pj m k m k m n m B B B k B s B B B k B. ( ) b b b..... k, B s bs. B m B m B mk B ms p AB (A b A b A b k ) 3 3 AB BA