Quz Quz
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- すずり なかじゅく
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1 III (1969). (1977). ( ) (1999). (1981). (199). Harry Lass Vector and Tensor Analyss, McGraw-Hll, (195)..
2 Quz Quz
3 II E B / t = Maxwell ρ e E = (1.1) ε E= (1.) B = (1.3) B = j (1.4) E B B - - Maxwell (1.) E= 1
4 E = φ (1.4) B (1.3) B = (1.1) E = ρe / ε B I B = B A B= A B B= A - (1.1) E = ρe / ε (1.3) B =
5 II q F= qe (.1.1) (.1.1) F E = (.1.) q q F= qv B (.1.3) v B magnetc flux densty 3
6 I (3) F v B v B F v B F = qvbsnθ (.1.4) θ v B v B v B 4
7 B (.1.3) B θ = π / (4) B v F F v B B = (.1.5) qv B MKSA B (5) F N B = (.1.6a) qv C(m/s) II N/C V/m F Vs B = T (.1.6b) qv m Vs/m teslat Vs WeberWb B Wb/m Wb B : T (.1.6c) m 5
8 (.1.1) (.1.3) q v F= q( E+ v B ) (.1.7) Lorentz force v = v + v j+ v k, B= B k x y z x, yz, v 1 4 m/s, 1 3 m/s, 1 3 x = vy = vz = m/s B = 1T x, yz, z E B E= E, B= ( B ) k y E v= vyj, vy = B q= e: e ( x, yz, ) 6
9 B B magnetc feld strength H B= H (..1) MKSA 1 π 7 = = 4 1 c ε m H (..) c : m /s, ε : C/(V m) s V m V s V s : m C (C/s) m A m H : / (..3) m Vs 1H = 1 (..4) A 7
10 (..1) (.1.6b) H A H : (..5) m H B H D D H A/m B q v v B F= qv B v N V N V V F F= ( N V)( qv B) (.3.1) V = S l : l : (.3.1) F= ( Nqv S l) B (.3.) Nqv S j e 8
11 F = j S l B (.3.3) e j S e F= I B l, I = j S (.3.4) e F ( F/ l) F= I B (.3.5) 9
12 (.3.5) N/m B z B= B + B j+ B k x y z, j,k xyz B = T, B = T, B = T x y z I = 1A F F = F z ( B= Bzk) ab, x ( x, y) θ 1
13 134 x a snθ I S( = ab) n m= ISn (.3.6) τ = m B (.3.7) m p q d p= qd p79 E τ = p E 11
14 - - II ( ) s q = λ s λ E j 1
15 E 1 q j R E () r = 4πε R R 1 λ s R = 4πε R R r P ( x, y, z ) j j j r Q ( x, yz, ) R P Q R = r r 13
16 N N 1 λ s R Er () = E () r = = 1 = 1 4πε R R N 1 λ R E ( x, yz, ) = ds 4πε R R j 14
17 Bot-Savart Law ( v = ω r) v ω r I 15
18 - - - ( ) P( x, y, z) s s I = I s, (3..1) s = s k (3..) 16
19 ( - - ) I Q ( x, yz, ) B I R (3..3) = 4π R R 17
20 R P( x, y, z ) Q ( x, yz, ) R = r r (3..4) r Q ( x, yz, ) r P( x, y, z) R P Q R = r r (3..5) (3..3) - (3..3) I B j I B I R I R I = I s P Q R I R θ I R = I R snθ = IRsnθ (3..3) I snθ I snθ B = (3..6) 4π R R I B R R 18
21 ( ) N N I s R B= B j = = 1 = 1 4π R R N I s R B = 4π (3..7) R R I Br () = e (3..8) π R θ R eθ (3..7) 19
22 r (,, z ) ( x, y) r( x, y,) R j ( R = x+ yj + z) k R = Re z e, R = x + y R z I p.17 r(,, z) I s = I zez R θ
23 r( x, y,) B RI z B = e, 3/ θ R = x + y 4 π ( R + z ) r( x, y,) B I Br () = e (3..8) π R θ + dz j = 3/ ( R + z ) R II p.3 B z R = x + y z = r( x, y,) z r( x, yz, ) z ( x, y) Br j () = Bx + By B ( / ), x = B y R B = B ( / ), x R B = I / π R, R = x + y y ( x, y) 1
24 ( ) a I P( x, y,) a er Q (,, z) z e z P Q R I = Ia θeθ R j I = Ia θeθ I R er, eθ, ez I Q
25 Iaz Ia B = dθe 3 R + dθe 3 z 4π R 4π R Q B B 1 a B(,. z) = I e z, R = a + z 3 R er er = cosθ + snθ j I S IS z R a z B(,, ) = e, 3 z = +, π R S = π a z m B(,, z) =, m= ISn 3 π z m n 3
26 I a B 4
27 ds I III j I I = jds e ds S s ds je 5
28 Ids= ( j ds) ds= j dv (3.3.1) e e dv = dsds ds ds r( x, yz, ) Br () - (3..7) Ids j e dv ( ) e Br () = dv 4π j r R (3.3.a) V R R r e( ) = 3 4π j r R V R Br () dv (3.3.b) R = r r r r R (3..7) (3..7) (3.3.) dv = dsds 6
29 j e = I / d (A/m) (3.3.3) d I I = jd (3.3.4) e ds ds 7
30 Ids= ( j d ) ds= j ds (3.3.5) e e ds ds = dds r( x, yz, ) Br () - (3..7) j e Br () ( ) j r R e = 4π S R R ds (3.3.6a) Br () e( ) = 3 4π j r R V R ds (3.3.6b) (3.3.) 8
31 P( x, y, z) I Q ( x, yz, ) B B () r = 4π R= r r r Q ( x, yz, ) r P( x, y, z ) R P Q R = r r I I I R I R I R I R θ I R = I R snθ = IRsnθ B B B = 4π I B R 9
32 q = I = - E = B = N Er ()= Br ()= N Er ()= Br ()= 3
33 B (a) I (I ) (b) I (I ) (a) (b) (a) (b) 31
34 I C A d r (4.1.1) C I (4.1.1) (4.1.1) z B C ( x, y) a B C 3
35 r( R, θ, z) I Br () = e (4.1.a) π R θ z R = x + y θ z C R = a I B( R= a) = e (4.1.b) π a θ C a dr I dr = adθe θ (4.1.3) C π I I π B dr = ( ad ) d C eθ θ θ = θ πa e π (4.1.4) B d r C I (4.1.5) = B C B 33
36 a B C B C Ampere s Law Ampere s Law of Crculaton) B d r = I = C S (4..1) C 34
37 C S I1, I,, IN I N I = I (4..) = 1 C j () e r C j () e r I = j () e r ds (4..3) S B dr = je () r ds (4..4) C S 35
38 (4..1) (4..4) B I A( x, yz, ) C A C S A ( A = rot A) A dr = A ds (4.3.1) C S (4..4) B dr = B ds= j ds (4.3.) C S S ( ) B j ds= (4.3.3) S S B = j (4.3.4) 36
39 I (4.1.5) (b) C1, C, C3 CC, 1, C, C 3 S1 S1 C1, C3 N (4.1.5) 37
40 a I P B I 38
41 z I e ( ) z R a je () r = π a (4.4.1) ( a< R) B z R θ z Br () = B( R) e θ (4.4.) P < R < a C R dr = Rdθ eθ (4..4) π ( ) B dr = B( R) eθ ( Rdθeθ) = πrb( R) (4.4.3) C R a I je() r = e z (4.4.4) π a ds= nrdrdθ S n= e (4.4.5) z R π I je() r ds= S e z e z θ π a I R π RdRdθ π a = ( RdRd ) 39
42 I π a = π R (4.4.6) (4.4.3) (4.4.6) I π RB( R) π π a = R (4.4.7) I π a B( R) = R (4.4.8) P a< R C R (4..4) B d r = π RB( R) (4.4.9) C a< R R a a π I jr () ds= ( ) S ez e zrdrdθ = I (4.4.1) π a π RB( R) = I (4.4.11) I π R BR ( ) = (4.4.1) 4
43 I Br () = e (4.4.13) π R θ j ( R) = j (1 R ) e a e e z I j e 41
44 x - z P (, a,) Q (, a,) z R(,, z) B ( z > ) ( z < ) 4
45 P Q R P R ( z > ) B P r = aj P Rr = zk R P P RP = r rp = zk aj R P = z + a (4.4.14) RP R p / Rp R zk aj = p R p z + a (4.4.15) x 43
46 IP = Ids= Ids (4.4.16) R p ( zk aj) Izds( k) Iads( j) Ids = Ids = (4.4.17) R p z + a z + a R p Izds( j) Iadsk Ids = (4.4.18) R p z + a P R 44
47 - (4..3) P R B P R I P R P BP R = B = yj Bzk (4.4.19) 4π RP RP Izds By = 4 π ( z + a ) 3/ Iads Bz = 4 π ( z + a ) 3/ (4.4.) (4.4.18) B P R yz, P Q R 45
48 Q P rq = aj P Rr = zk RQ RQ = r rq = zk+ ajr Q = z + a (4.4.1) Q P IQ = Ids= Ids (4.4.) Q R I R B B j Bzk (4.4.3) Q Q Q R = = y + 4π R Q R Q B, B (4.4.) y z (4.4.19) (4.4.3) B = B + B = B j (4.4.4) R P R Q R y R z < z > z > B = B + B = B j (4.4.5) R P R Q R y 46
49 - B B C ( C: C1 C C3 C4) (4..4) B dr = j() r ds (4.4.6) C C C C C3 C4 S B dr = B dr + B dr + B dr + B dr (4.4.7) 47
50 (4.4.7) C C B dr B dr = C 4 B dr C C 4 C B dr = B dr + B dr C C 4 4 C4 = ( Bj) ( drj) + ( Bj) ( drj) = B (4.4.8) (4.4.6) (3.3.3) j e (A/m) j () r d S = j e (4.4.9) S e B B = je (4.4.3) 1 = B j e (4.4.31) (4.4.31) B II B 48
51 R z < BP R B Q R B R - (4.4.5) 49
52 n I y x 5
53 R R P Q I P I Q R B B, B P R Q R B = B + B (4.4.3) R P R Q R BR y R y CC,, C C B dr = B dr + B dr + B dr + B dr (4.4.33) C C C C C C C B dr B dr = C C 4 B dr C 4 B dr C C 4 B dr = B dr + B dr C C C = ( Bj) ( drj) + ( B4j) ( drj) 4 = ( B B ) (4.4.34) 51
54 C jr () ds = (4.4.35) S 5
55 (4.4.34) (4.4.35) ( B B ) = (4.4.36) 4 B = B4 Bn (4.4.37) B B (4.4.37) B n C B dr = B dr + B dr + B dr + B dr (4.4.38) C C C C C C C B dr = C B d r C 4 B d r C C 1 3 B dr = B dr + B dr C C C = B dr + B 4 dr ( j) ( j) ( j) ( j) = ( B + B ) (4.4.39) 4 C jr () ds = (4.4.4) S (4.4.39) (4.4.4) ( B + B ) = (4.4.41) 4 53
56 B = B B (4.4.4) 4 ex B B B = (4.4.43) ex C B dr = B dr + B dr + B dr + B dr C C1 C C3 C4 (4.4.44) B dr = B dr + B dr 4 4 C C C4 (4.4.45) (4.4.37) (4.4.43) B = BnjB 4 = B ex j = C dr = dr j B dr = ( B j) ( dr j) = B C C n n (4.4.46) C C C C S 54
57 j () r d S (4.4.47) S n I j e = ni (4.4.48) 4 S j = j ds = d (4.4.49) e e d x L+ je() r S= S L e = d j d ni (4.4.5) (4.4.46) (4.4.5) Bn Bn = ni (4.4.51) B = ni n 55
58 B = ( ni) j (4.4.5) n Bext = (4.4.53) 56
59 a b I ) < r < a, ) a< r < b, ) b< r a,b a b j, j ) < r < r1 a, ) r 1 a< r < r 1 + a, ) r1+ a< r < r b, v) r b< r < r + b, v) r + b< r e e 57
60 Quz 1. C C S I1, I,, IN I I = C C j () e r C S j () e r 58
61 - r r q r I r d E () r = d B () r = R =, R = R =, R = N q1, q,, qn N I1, I,, In E d S = S B d r = C ρ () e r j () e r S E ds= dv B dr = ds V C S 59
62 - e( ) = 3 4π j r R V R Br () dv (5.1.1) R = r r R = r r (5.1.) r j ( r ) e r B dr = je () r ds (5.1.3a) C S B = j e (5.1.3b) II - E d r = (5.1.4a) C E= (5.1.4b) 6
63 C magnetc scalar potental (5.1.3a) (5.1.1)(5.1.3a) 5.1.3b B = (5.1.5) - (5.1.5) (5.1.5) B I B A B= A (5.1.6) I I 61
64 - - B (5.1.6) A - (5.1.1) R (5.1.) R = r r = ( x x ) + ( y y ) j+ ( z z ) k (5..1a) R = r r = ( x x ) + ( y y ) + ( z z ) (5..1b) II (4.3.6) 1 R = R 3 R (5..) R R 3 1 = R - (5.1.1)R / R 3 1 Br () = e( ) dv 4π j r (5..3) V R I j e 1 1 = ( je) + j e (5..4) R R R 6
65 je( r) r ( x, y, z) ( x, yz, ) j e = (5..5) (5..5) (5..4) 1 j e j e = (5..6) R R (5..3) je( r) Br () = dv 4π (5..7) V R ( x, y, z) ( x, yz, ) (5..7) je( r) Br () = dv 4π (5..8) V R (5..8)[ ] A je( r) A = dv 4π (5..9) V R B A 63
66 B= A (5..1) B = B - B B = (5..9) II (4.3.8) 1 ρ( x, y, z) φ( x, yz, ) = dv 4πε R (,, ) (,, ) e x y z A x yz = dv 4π j (5..11) V R V ρ( x, y, z ) j ( x, y, z ) (5..1) e R = r r = ( x x ) + ( y y ) + ( z z ) (5..13) (5..11) x jex( x, y, z) Ax( x, y, z) = dv 4π (5..14) V R 64
67 je( x, y, z) B= A I B B B A y z x = A z x y = z A y z Az x Ay Ax = x y (5..15a) (5..15a) (5..15c) (5..) 65
68 A (5.1.6) B= A (3.3.1) j dv e = Ids 66
69 (5.3.1) Ids A = 4π (5.3.1) C R d s R P r r d s ds = adϕ e ϕ (5.3.) R = r r R = ( r r ) ( r r ) = r r r + r (5.3.3) r = a R = r rr + r = r rr + a (5.3.4) P r r = a r a (5.3.5) Taylor rr = 1 + r rr + a R r r (5.3.6) Ids I 1 r r A 1 ds (5.3.7) = + 4π C R 4π C r r A I I ( ) d 3 4π r s 4π r r r s (5.3.8) = d + C C 67
70 d s = (5.3.9) C π π ( adϕ e ad ( sn cos ) ϕ = ϕ ϕ + ϕj = ) I A = ( ) d 3 4π r r r s (5.3.1) C P r ( x, y) R P zp = zk r = RP + z P (5.3.11) RP = RPer, RP = x + y z = zk P (5.3.1) (5.3.1) r r r ( x, y) z r = p r r = ( R + z ) r = R r = RacosΦ (5.3.13) p p p p Φ R P r Φ = ϕ ϕ A I IRpa π = Racos( )( ) cos( ) 3 p ϕ ϕ adϕ d 3 4 r C ϕ = ϕ ϕ ϕ 4 r ϕ π e π e (5.3.14) eϕ = snϕ + cosϕ j ϕ ( ) π cos( ϕ ϕ ) e d ϕ ϕ = πe ϕ (5.3.15) 68
71 IRp ( πa ) A= e 3 ϕ (5.3.16) 4π r (5.3.16) R = rsnθ (5.3.17) p I( πa ) snθ A= e ϕ (5.3.18) 4π r I( πa ) snθ Ar =, Aθ =, Aϕ = (5.3.19) 4π r (5.1.6) B= A (5.3.18) A I B r ( snθ Aϕ ) 1 1 A θ = ( A ) r = (5.3.) rsnθ θ r ϕ B θ 1 A 1 ( ) ( ) r raϕ = A θ = (5.3.1) rsnθ ϕ r r 69
72 B ϕ ( ra ) 1 θ 1 A r = ( A ) ϕ = (5.3.) r r r θ (5.3.19) B r ( sn A ) ϕ I( a ) 1 θ 1 π sn θ = = rsnθ θ rsnθ θ 4π r (5.3.3) B θ 1 ( raϕ ) 1 I( πa ) snθ = = r r r r 4π r (5.3.4) B ϕ = (5.3.5) B B r θ I( πa ) cosθ = 3 4π r (5.3.6) I( πa ) snθ = 4π r (5.3.7) B ϕ = (5.3.8) 7
73 cos( ϕ ϕ ) = cosϕcosϕ + snϕsnϕ e = snϕ + cosϕ j ϕ cos( ) cos cos sn sn sn ϕ ϕ eϕ = ϕ ϕ ϕ ϕ ϕ + cosϕ cos ϕ j+ snϕsnϕ cosϕ j 1 = cosϕ sn ϕ snϕsn ϕ + cosϕcos ϕ j+ snϕ sn ϕ j 1 π 1 cosϕ sn ϕ dϕ = π π snϕ sn ϕ dϕ = snϕ sn ϕ dϕ = π snϕ π π d π cosϕ cos ϕ j ϕ = cosϕ cos ϕ dϕ = π cosϕj 1 snϕ sn ϕ jdϕ = π ϕ ϕ e d ϕ ϕ = π ϕ+ π ϕ= π ϕ+ ϕj cos( ) sn cos ( sn cos ) π cos( ϕ ϕ ) e d ϕ ϕ πe ϕ = 71
74 + q, q P ( x, yz, ) 1 pr φ() r (4.3.9) = 3 4πε r r P r = x+ yj+ zk p p= qd, d= dk (4.3.1) q d qd 1 p 3 r( p r) Er () = πε r r 7
75 P ( x, yz, ) Ar () m r 4π r = 3 m m = IS, S= Sn I S m= IS S n 73
76 a z S = π a, n= k (5.3.18) I( π a ) snθ m r Ar () = e () ϕ Ar = 3 4π r 4π r φ v ( φv) = φ( v) + ( φ) v m r 1 1 Ar () = ( ) ( ) 3 = π r 4π m r m r r r I ω = ωk, ω = const. ( ω r) = ω ω m ( m r) = m A [ ] 1 m ( ) 3 m r = 3 r r A [ ] 1 3 r, 3 = r = r 4 r r r I 74
77 a ( b c) = ( a c) b ( a b) c [ ] 1 3m 3( m r) ( ) 3 m r = + r 3 5 r r r B() r m 3( m r) 4π r r r = 3 5 Br, Bθ, B (5.3.6) ϕ (5.3.8) 75
78 76 = p : q : d = m : I : S ( ) 1 () 4 φ πε = r : r ( ) () 4 π = Ar : r () φ = Er [ ] 1 () 4πε = Er = B A [ ] () 4 r π = B
79 r ( x, y, z ) je( r ) = je( x, y, z ) r( x, yz, ) 3 je( r ) = j e( x, y, z ) e( x, y, z ) Br () = dv 3 4π j R V R R = r r R = r r Br () = 4π C Ids R 3 R 77
80 je( r ) = j e( x, y, z ) e( x, y, z ) A ( x, yz, ) = dv 4π j V R B= A A = 4π C Ids R (1/ R 3 ) (1/ R ) 78
81 B = Α (5.5.1) = Α ( Α) A (5.5.) j = j = e e A = (5.5.3) B= A (5.5.4) B = j e (5.5.5) A = j e (5.5.6) Ax = Ay = Az = j, j, j ex ey ez (5.5.7) 79
82 ρ φ = e ε A = e( x, y, z ) A( x, yz, ) = dv 4π j V R R = r r R = r r 1 A= e e dv 4π j V R ( x, yz, ) ( x, y, z ) j e = ( x, y, z ) = + j + k x y z 1 1 = R R φ v ( φ v) = φ( v) + v ( φ) 8
83 je 1 1 = j e + j e R R R j = e je A = dv 4π V R V V j = e V V j j e dv = e dv V R V R A 4π j ds R e = S S V j = e A = 81
120 9 I I 1 I 2 I 1 I 2 ( a) ( b) ( c ) I I 2 I 1 I ( d) ( e) ( f ) 9.1: Ampère (c) (d) (e) S I 1 I 2 B ds = µ 0 ( I 1 I 2 ) I 1 I 2 B ds =0. I 1 I 2
9 E B 9.1 9.1.1 Ampère Ampère Ampère s law B S µ 0 B ds = µ 0 j ds (9.1) S rot B = µ 0 j (9.2) S Ampère Biot-Savart oulomb Gauss Ampère rot B 0 Ampère µ 0 9.1 (a) (b) I B ds = µ 0 I. I 1 I 2 B ds = µ 0
50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........
i
009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3
Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e
7 -a 7 -a February 4, 2007 1. 2. 3. 4. 1. 2. 3. 1 Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e z
1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
![1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1](/thumbs/94/121802164.jpg "1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1")
1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2
2.5 (Gauss) (flux) v(r)( ) S n S v n v n (1) v n S = v n S = v S, n S S. n n S v S v Minoru TANAKA (Osaka Univ.) I(2012), Sec p. 1/30
2.5 (Gauss) 2.5.1 (flux) v(r)( ) n v n v n (1) v n = v n = v, n. n n v v I(2012), ec. 2. 5 p. 1/30 i (2) lim v(r i ) i = v(r) d. i 0 i (flux) I(2012), ec. 2. 5 p. 2/30 2.5.2 ( ) ( ) q 1 r 2 E 2 q r 1 E
2.4 ( ) ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) Minoru TANAKA (Osaka Univ.) I(2011), Sec p. 1/30
2.4 ( ) 2.4.1 ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) I(2011), Sec. 2. 4 p. 1/30 (2) Γ f dr lim f i r i. r i 0 i f i i f r i i i+1 (1) n i r i (3) F dr = lim F i n i r i. Γ r i 0 i n i
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