Ps22Pdf

Transcription

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2 ,,,,,, (CIP) / :, 00 ISBN CIP (00)07498 : : : (, 00084) / : : : / 6 : 5 : 53 : : ISBN / TN7 : : 4 00

3 865,,,,,,,,,,,,,,,,,,,, 4, 3,,,,,,,,,, 5,, 6 8,,, 0 4, 0 60,,, 9,

4 , ( ),,,, 0 60,,,,,,,, ;,,, 5, 9, 0,,, 3, 4, 6, 7, 8,, 000 8

5 E 9

6 D

7 B A 4 3 A H, m, r B H M A

8 A 67 6 Am m e m

9 ( TM ) ( TE )

10 ( )

11 ( )

12 ,, E D B H,,, ( ),,,, A, ( ) A A,, ^, ^u, ^u = ^u = A/ A A = A^u ( ) ^x, ^y, ^z, x, y, z,,, 6 A o,

13 o A ( Ax, Ay, A z ), A = Ax ^x + Ay^y + A z^z ( ) Ax, Ay, A z A x, y, z A A = A x + A y + A z 3 / ( 3) P, r, r P, 3r P ( x, y, z ), r = x^x + y^y + z^z ( 4) 3 4, P ( x, y, z ), r= x ^x + y ^y + z ^z,, P PPP R ( 4 ), R P P, r+ R = r, R P P, R R = ( x - R = r - r ( 5) x ) ^x + ( y - y ) ^y + ( z - z ) ^z ( 6) R = ( x - x ) + ( y - y ) + ( z - z ) ( 7) A, B, C, : ( ) A + B = B + A ( 8) ( ) A + ( B + C) = ( A + B) + C ( 9) (3 ) AB = BA = ABcos( 0 ) ( 0) (4 ) A( B + C) = AB + AC ( ) (5 ) ^u A, B, A B = ABsin^u ( ) (6 ) AB = - BA ( 3) (7 ) A( B + C) = AB + AC ( 4)

14 (8 ) A = A ^u + A ^u + A3 ^u3, B = B ^u + B ^u + B3 ^u3, ^u ^u ^u3 A B = A A A3 ( 5) B B B3 (9 ) A( BC) = B( CA) = C( AB) ( 6) (0) ( AB) CA( BC) ( 7) () A( BC)( AB)C ( 8) () A( BC) = ( AC) B - ( AB) C ( 9),, 3,, A, A = A( x, y, z ), A x, y, z A x = lim x 0 A y = lim y0 A z = lim z0 A( x + x, y, z ) - A( x, y, z ) x A( x, y + y, z ) - A( x, y, z ) y A( x, y, z + z ) - A( x, y, z ) z A( x, y, z ) = A x ( x, y, z ) ^x + Ay ( x, y, z ) ^y + A z ( x, y, z ) ^z, A d A A x = Ax z ^x +Ay ^y +A x x x ^z A y = Ax z ^x +Ay ^y +A y y y ^z A z = A x z ^x +A y z ^y +A z z ^z d A = d Ax^x + d Ay^y + d A z^z d Ax = A x x d x +A x y d y +A x z d z 3

15 d Ay, d A z, d A = A x x ^x +A y x ^y + A z x ^z d x + A x y ^x +A y y ^y +A z y ^z d y f + A x z ^x +A y z ^y + A z z ^z d z = A x d x + A y d y +A d z ( 0) z d f = f x d x +f y d y +f d z ( ) z,,, f, d f = f x d x +f y d y +f z d z A d r, A = x^x f f +f + ( ) y^y z^z d r = d x^x + d y^y + d z^z Ad r = f x d x + f y d y + f z d z ( ), A f, A f, A = grad f ( 3),, = AdS ( 4) S S, V, V0, AdS S lim V 0 V = div A ( 5), A, 3 4 div A = A x x +A y y +A z z ( 6)

16 , A, ^n, A ^n S c (^n ) S, Ad r c lim S0 S = ( rot A) ^n ( 7) rot A A,, rot A = ^x ^y ^z x y z Ax Ay A z ( 8) 3 3,,,,,,, = ^x + ^y + ^z x y z = x^x + + ( 9) y^y z^z,, f x f, f x,, f = A = A = x^x f + + f = +f +f y^y z^z x^x y^y z^z x^x + + y^y z^z Ax ^x + Ay^y + A z^z = A x x +A y y + A z z = x^x + + y^y z^z Ax^x + Ay^y + A z^z ^x ^y ^z x y z A x Ay A z 5

17 grad f = f f + +f = f ( 30) x^x y^y z^z div A = A x x +A y y + A z z rot A = ^x ^y ^z = A ( 3) = A ( 3) x y z A x Ay A z, A A,,,,, AB = BA, A A ;, C = A B C A, B, A( AB) = 0, A A, AA0,,,, 3 T () T () = T ( ^x) + T ( ^y) + T ( ^z) ( 33) x y z T ( ^x), T ( ^y) T ( ^z) T (), ^x, ^y, ^z ( ) T ( ) = = x^x + + y^y z^z ( ) T ( ) = f = x^xf + + =f +f +f y^yf z^zf x^x y^y z^z (3 ) T ( ) = A = ( ^xa) + ( ^ya) + x y z ( ^za) = A x x +A y y +A z z (4 ) T ( ) = A = ( ^xa) + ( ^ya) + x y z ( ^za) = A z y - Ay ^x + A x - A z ^y + Ay z z x x - Ax ^z y T (),,,, T () T (), a P + a P ( a a, P P ), 6 T a P + a P = a T ( P ) + a T ( P )

18 T ()T ( ), T () : T (),, ( ), c, : T () ( ), T (),, ;,, ( fa) = ( fc A) + ( f Ac ) ( fca) = x ( ^xf ca) + y ( ^yf ca) + z ( ^zf ca) = f ( ^xa) + f ( ^ya) + f x y z ( ^za) = f A ( fac ) = x ( ^xf A c ) + y ( ^yf A c ) + z ( ^zf A c ) = A f + Af + Af x^x y^y z^z = A f ( f A) = f A + A f ( A B) = ( Ac B) + ( A Bc ) ( 34) A ( BC) = ( AC) B - ( AB) C = B( AC) - C( AB) ( Ac B) = A( B) - ( Ac ) B = A( B) - ( A ) B ( A Bc ) = ( Bc ) A - B( A) = ( B ) A - B( A) ( 34 ) ( A B) = A( B) - ( A ) B + ( B ) A - B( A) : T () P 3 3, T ( ), f,, A, A T (, ), T (, ), : P, T ( P, ),, T ( P) ; T ( P) P, T ( ), T (, ), 7

19 T ( P) = T ( P, ) = T ( P, ^x) + T ( P, ^y) + x y z T ( P, ^z) T (, ) = T () = x T ( ^x) + y T ( ^y) + z T ( ^z) = T ( ^x, ^x) + T ( ^x, ^y) + T ( ^x, ^z) x x y z + T ( ^y, ^x) + T ( ^y, ^y) + T ( ^y, ^z) y x y z + T ( ^z, ^x) + T ( ^z, ^y) + T ( ^z, ^z) z x y z = x + xy T ( ^x, ^x) + y T ( ^y, ^y) + z T ( ^z, ^z) T ( ^x, ^y) + T ( ^y, ^x) + ( T ( ^y, ^z) yz + T ( ^z, ^y) ) + ( T ( ^z, ^x) + T ( ^x, ^z) ) ( 35) zx, ( 35 )T (, ) f ( 35 ), ( ^x^x) =, ( ^y^y) =, ( ^z^z) = ( ^x^y) = 0, ( ^y^z) = 0, ( ^z^x) = 0 f = x ( ^x^x) f + y ( ^y^y) f + z ( ^z^z) f = f x + f y,, A ( 36) + f z ( 37) f ( 36), A ( 37 ), f A, A = A x A A = Ax ^x + Ay^y + A z^z,, A = A = + Ay x Ax x + Ay y + Ax y + Ay z + A y + A x z + A z ( 38) ^x ^y + Az x + A z y + Az z = Ax^x + Ay^y + A z^z ( 39) ^z 8

20 3 4 T (), T (, ), ( ) ( f + g) = f + g ( 40) ( ) ( f g) = fg + gf ( 4) (3 ) ( f/ g) = ( gf - fg)/ g ( g0) ( 4) (4 ) ( A + B) = A + B ( 43) (5 ) ( fa) = f A + fa ( 44) (6 ) ( AB) = B ( A) - A( B) ( 45) (7 ) f = f ( 46) (8 ) ( A) = 0 ( 47) (9 ) ( A + B) = A + B ( 48) (0) ( fa) = f A + f A ( 49) () ( AB) = ( B ) A - B( A) - ( A ) B + A( B) ( 50) () f = 0 ( 5) (3) ( A) = ( A) - A ( 5) (4) ( AB) = ( B ) A + ( A ) B + B( A) + A( B) ( 53) (5) ( f g) = 0 ( 54) A, A f, A f, ( ^x, ^y, ^z ), f, A?, 4 4 A = A x^x + Ay^y + A z^z, : A = x^x + + y^y z^z A x^x + Ay^y + Az^z = A x x ^x^x +A y x ^x^y + A z x ^x^z +A x y ^y^x +A y y ^y^y + A z y ^y^z +A x z ^z^x +A y z ^z^y +A z z ^z^z = A x x ^x + A x y ^y + A x z ^z ^x + A y x ^x +A y y ^y +A y z ^z ^y 9

21 + A z x ^x + A z y ^y +A z z ^z ^z 3, 3, 9 9,, D D = AB ( 55) A, B,,,,,,,, ( 5 ), d V o d H0 d H0 = r(d V ) v, v = r H0 H0 = ( rv )d V V = [ r( r) ]d V ( 56) V, v r, r A ( B C) = B( AC) - C( AB) ( 56 ) H0 = r - r( r ) d V ( 57) V 5 o r,, r( r ) ( rr)( rr), ( rr), ( 57) H0 = q ( 58) q, I q = V [ r I - rr]d V, 3, 3 3,, ( 55 ), A = A ^u + A ^u + A3 ^u3, B = B ^u+ B ^u+ B3 ^u3 ^u, ^u, ^u3 ^u, ^u, ^u3 A B, 0

22 , A B D ( 55 ), D = AB = A B ^u ^u+ A B ^u ^u+ A3 B ^u3 ^u+ A B ^u ^u + A B ^u ^u+ A3 B ^u3 ^u+ A B3 ^u ^u3+ A B3 ^u ^u3+ A3 B3 ^u3 ^u3 = D ^u ^u+ D ^u ^u+ D3 ^u3 ^u+ D ^u ^u+ D ^u ^u+ D3 ^u3 ^u + D3 ^u ^u3+ D3 ^u ^u3+ D33 ^u3 ^u3= D ^u + D ^u + D3 ^u3 ^u + D ^u + D ^u + D3 ^u3 ^u+ D3 ^u + D3 ^u + D33 ^u3 ^u3 = D ^u+ D ^u+ D3 ^u3 ( 59) D = ^u D ^u+ D ^u+ D3 ^u3+ ^u D ^u+ D ^u+ D3 ^u3 + ^u3 D3 ^u+ D3 ^u+ D33 ^u3= ^u D+ ^u D+ ^u3 D3 ( 60) ( 59 ) ( 60 ),, D = D^u, D = D^u, D3 = D^u3 D= ^u D, D= ^u D, D3= ^u3 D ^u = ^u, ^u = ^u, ^u3 = ^u3 D D, D D, D3 D3,, A D DA ( 6) A D, DA, D, C = C ^x + C ^y + C3 ^z F = DC = [ ( D ^x + D ^y + D3 ^z ) ^x + ( D ^x + D ^y + D3 ^z) ^y + ( D3 ^x + D3 ^y + D33 ^z) ^z]( C ^x + C ^y + C3 ^z) = D C + D C + D3 C3 ^x + ( D C + D C + D3 C3 ) ^y + ( D3 C + D3 C + D33 C3 ) ^z = F ^x + F ^y + F3 ^z,, F D D D3 C F = D D D3 C ( 6) F3 D3 D3 D33 C3, F= C D, F F= F3 D D D3 D D D3 D3 D3 D33 C C C3 ( 63) 4 :

23 ( ),, ( ) C C D = DC ( 64) (3 ),, (4 ) D D, D = AB, D= A B D D= ABA B= A( BA ) B= ( BA ) AB,, D D = ABA B= ( AA ) ( BB ) ( 65) (5 ), C D = C AB = ( C A) B ( 66) DC = ABC = A( BC) ( 67) (6 ) (7 ) (8 ) D D= ABA B= A( BA ) B ( 68) D D= AB A B= ( A A ) ( BB ) ( 69) D = ( D ) ^x + ( D ) ^y + ( D3 ) ^z ( 70) D = ( D ) ^x + ( D ) ^y + ( D3 ) ^z ( 7) (9 ) D = D - D ( 7) 4 3 :

24 ( ) 0 C,, 0, 9 ( ) C0 = 0 C = 0 ( 73) D,, D (3 ) C, C D = DC ( 74) C D = DC ( 75) D, (4 ) C, D (5 ) C D = - DC ( 76),, (6 ) D, I D I = I D = D ( 77), I ( ) (7 ) D, D D, D - I = ^x^x + ^y^y + ^z^z ( 78) D D= D D = I ( 79),,, AB C = BC A = CA B, C D, C, ABC = - BA C = A BC C C C, C, C3, AB C = - BA C = A BC AB C = - BA C = A BC AB C3 = - BA C3 = A BC3 ^u, ^u, ^u3, AB C ^u = - BA C ^u = A BC ^u 3

25 AB C ^u = - BA C ^u = A BC ^u AB C3 ^u3 = - BA C3 ^u3 = A BC3 ^u3 AB C ^u + C ^u + C3 ^u3 = - BA C ^u + C ^u + C3 ^u3 = A BC ^u + C ^u + C3 ^u3 AB C = - BA C = A B C ( 80) 5 5 V S, A V, ^n S V Ad V = S AdS = S A^nd S ( 8) 5 S C, A S, Ad r = C ( A)dS = S ( A) ^nd S ( 8) S C : S, S, C, S, A, CAd r = 0,, S AdS = 0 C S, S AdS = 0, A = 0 A = R xy C, M N R x, y,, C 4 C ( Md x + Nd y) = R N x - M y d xd y ( 83)

26 , S R, A, Md x + N d y = ( M^x + N^y)( d x^x + d y^y) = Ad r A = M^x + N^y, r = x^x + y^y A = - N N ^x +M ^y + z z x - M y A^z = N x - M y ( 83 ), d R = d xd y C Ad r = R ( A) ^zd R ( 84) ^z 5 4, V + () d V = S ()ds ( 85), V : V - d V = S (- )ds ( 86) 5, A =, ( 8) ( 44 ) ( ) d V = S ()^nd S = S ()ds V ( ) = ( ) + = + ( + ) d V = V S ()ds, ( 85), ( 85 ), V [ + ( )( ) ] d V = S ()ds 5

27 V ( - ) d V = S (- ) d S 5 5 B = B, ( A B ) = B( A) - A( B ) ( A B) = B A - A( B) 5 V ( A B) d V = V [ A B - A( B) ] d V ( 87a)A B, = S ( A B)dS ( 87a) [ B A - B( A) ] d V = V S ( B A)dS ( 87a)( 87b), [ B( A) - A( B) ] d V V ( 87b) = S ( A B - B A) d S ( 88) , [ B A - ( B) A]d V V = - S B( A ) + ( B) A ds ( 89) 5 7 ( ) V Bd V = S ( ^nb) d S = Sd SB ( 90) : 6 A = BC,, C,

28 V ( B C) d V = S ( BC)^nd S ( 90a) ( B C) = C B - B C = C B ( 90b)( 90c) ( 90a), C, C,, ( 90b) ( B C) ^n = B( C ^n) = C( ^n B) ( 90c) C( B) d V = V S C( ^n B)d S C Bd V = C V S^nBd S Bd V = V S^nBd S ( ) V d V = S^nd S ( 9) : A = C, C, V (C) d V = S C^nd S C, (C) = C( C = 0) V Cd V = S C(^n) d S C d V - S ^nd S = 0 V d V V = ^nd S S (3 ) S B( AdS) = V [ ( A ) B + B A] d V ( 9) : B 3, ( 44 ), B = Bx ^x + By^y + Bz^z ( Bx A) = A Bx + Bx A ( Bx A) d V = V S Bx AdS 7

29 Bx ( Ad S) = S [ A Bx V + Bx ( A) ] d V By ( AdS) = S [ A By + By ( A) ] d V V B z ( AdS) = S [ A Bz + B z ( A) ] d V V B( Ad S) = S [ A Bx + Bx ( A) ] ^x V + [ A By + By ( A) ] ^y + [ A B z + Bz ( A) ] ^z d V =[ ( A Bx ) ^x + ( A By ) ^y + ( A Bz ) ^z] + B A d V V ( A Bx ) ^x + ( A By ) ^y + ( A Bz ) ^z = ( A ) B B( AdS) S =[ ( A ) B + B A] d V V 5 8, : C ( r),,,, C( r) = D( r) + F( r) ( 93) D = 0, F = 0 ( 94), D( r), F( r) ( r), A( r), D( r) = A( r) ( 95) F( r) = - ( r) ( 96) D( r) F ( r) ( 94 ), ( 96 ) ( 95 ), = = - C ( r) ( 97) ( A) = C( r) ( 98) 7 4 3,, A,, 8 = C d V ( 99) 4 V R

30 A = C d V ( 00) 4 V R A C( r), D( r)f( r), :, 6,,,, 6, P0 ( x0, y0, z0 ) x = x0, y = y0, z = z0 f( x, y, z ) = u ( 0) u ( 0), u f ( x, y, z) = f ( x, y, z) = f 3 ( x, y, z) = u u u3 ( 0),, P, P u, u, u3 ( 6 ), u, u, u3 6 P( u, u, u3 ) P ( u + d u, u + d u, u3 + d u3 ), r r + d r, d r, P ui d li, 3 d r = d li = h d u ^u + h d u ^u + h3d u3 ^u3 ( 03) i = d li = hi d ui^ui ( 04) i ui, d li, u i, hi, hi, u, u, u3 z,, z, h = h3 =, 9

31 , h, h =, ui =, d lj, d lk, d Si = d lj d lk = hj hk d uj d uk^ui ( 05) d V = d li ( d lj d lk ) = hihj hkd uid ujd uk ( 06), ( ) (,, z) P = = z = ( 7 ) ^u = ^, ^u = ^, ^u3 = ^z ( 07) P r r = ^+ z^z ( 08) P, d d r = d ^, d d r = d ^, d z d r = d z^z d r 7 d r = d ^+ d ^ + d z^z ( 09) h =, h =, h3 = d V = ddd z ( 0) x = cos y = sin z = ^= cos^x + sin^y ^ = - sin^x + cos^y ^z = ^z ^ = ^, ^ = - ^, ^z = 0 ^ = ^ = ^z = 0 ^ z = ^ z = ^z z = 0 z ( ) ( ) A Ax, Ay, A z, A, A, A z, 0

32 A x = Acos- Asin Ay A z A= = Asin+ Acos = A z Ax cos+ Aysin A= - Ax sin+ Aycos ( 3) ( 4) ( ) ( r,, ) Az = A z P r = = = ( 8 ) P r, P, d r d r = d r^r, d d r = rd ^, d d r = rsin d ^d r d r = d r^r + rd ^+ rsind ^ ^u = ^r, ^u = ^, ^u3 = ^ ( 5) r = r^r ( 6) ( 7) d V = r sind rd d ( 8) h =, h = r, h3 = rsin ( 9) x = y = z = rsin cos rsin sin rcos ^r = sin cos^x + sin sin^y + cos ^z ^= cos cos^x + cos sin^y - sin ^z ^ = - sin^x + cos^y ^r r = ^ r = ^ r = 0 ^r = ^, ^ = - ^r, ^ = 0 ^r = sin ^, ^ = cos ^, ^ = - sin ^r - cos ^ 8 ( 0) ( ) ( ) A Ax, Ay, A z, A r, A,

33 A, Ax = A rsin cos+ Acos cos- Asin Ay = Arsin sin+ Acos sin+ Acos A z = Ar cos- Asin A r = A xsin cos+ Aysin sin+ A z cos A= A x cos cos+ Ay cos sin- A= - Ax sin+ Ay cos Az sin ( 3) ( 4) 6 ( ), = f ^u + f ^u + f 3 ^u3 f, f, f3 d r d r = h d u ^u + h d u ^u + h3 d u3 ^u3 ( 5) d= d r = ( f ^u + f ^u + f 3 ^u3 )( hd u ^u + h d u ^u + h3 d u3 ^u3 ) = h f d u + h f d u + h3 f 3 d u3 ( 6) d= d u u ( 6)( 7 ), + d u u + d u3 ( 7) u3 f = h, f = u h, f 3 = u h3 u3 = h ^u u + h ^u u + h3 ^u3 ( 8) u3 ( ) A A A = ( A ^u + A ^u + A3 ^u3 ) = ( A ^u ) + ( A ^u ) + ( A3 ^u3 ) ( 9) ( A ^u )

34 ( A ^u ) ( 30) u = ^u/ h, u3 = ^u3/ h3 u u3 = ^u ^u3/ h h3 = ^u/ h h3 ( A ^u ) = ( A h h3 u u3 ) ( 30) ( fa) = f A + f A ( A h h3 u u3 ) = ( A h h3 )( u u3 ) + A h h3 ( u u3 ) ( u u3 ) = u3(u ) - u( u3 ) = 0 ( A ^u ) = ( A h h3 ) ^u h h3 = ^u ( A h h3 ) + ^u ( A h h3 ) + ^u 3 ( A h h3 ) ^u h u h u h3 u3 = h h h3 h h3 ( A h h3 ) ( 3) u ( A ^u ) = ( A3 ^u3 ) = h h h3 h h h3 ( 3)( 33 )( 9),, A = (3 ) A h h h3 ( A h3 h ) ( 3) u ( A3 h h ) ( 33) u3 ( A h h3 ) + ( A h3 h ) + ( A3 h h ) ( 34) u u u3 A ( 35) A = ( A ^u + A ^u + A3 ^u3 ) = ( A ^u ) + ( A ^u ) + ( A3 ^u3 ) ( 35) ( f A) = f A + f A ( A ^u ) = ( A h u ) = ( A h ) u + A h u = ( A h ) ^u/ h = ^u h ( A h ) u + ^u h ( A h ) u + ^u 3 h3 ( A h ) u3 ^u h 3

35 = h3 h ( A h ) u3 ^u - h h ( 35) ( A ^u ) = ( A3 ^u3 ) = h h h h3 ( A h ) u ( A3 h3 ) u ^u3 - ^u - ( A h ) u h h3 h h3 ^u3 ( 36) ( A h ) u3 ( A3 h3 ) u ( 36)( 38 )( 35), A ^u ( 37) ^u ( 38) h ^u h ^u h3 ^u3 A = h h h3 u u u3 ( 39) A h A h A3 h3 f = h h h3 u h h3 h A = = h h h3 f u h h3 + h3 h u h ^u u h3 h f u ^u u + h h u3 h3 h h ^u3 u3 3 ( A3 h3 ) - u ( A h ) u3 f u3 ( 40) ( 4) = h h3 h ( A h ) - u3 ( A3 h3 ) u 3 = h 3 h h ( A h ) - u ( ) ( A h ) u f = f ^+ f ^ + f ( 4) z^z A = (A) + A +A z z ( 43) A = ^ ^ ^z = A z z - A ^ z A A A z 4 + A - A z z ^ + (A) - A ( 44) ^z

36 f A A = f f = f + + z ( 45) A- A - A ^ z f + A- A A + A ^ + A z^z ( 46) A = - A = ^+ ^ + 3 ^z ( 47) = A + A z z + A + A - A - = A z z + A + A + A 3 = A z + A z z + A z + A z A - z A A = ^+ ^ + 3 ^z ( 48) A A z = - A z + A z z - A 3 = - A z ( ) - A z A + z + + A - A A + A A - + z + z + A - z f = f r^r + f r ^+ f rsin ^ A = r r ( r Ar ) + A = = ^r r sin r A r + + A r + A - r rsin ( sina ) + r^ rsin ^ A r ra rsina A rtan - A rsin A rsin ^r + f A A A + A z Ar rsin - A - r A ^ r ( 49) ( 50) Ar r ^ ( 5) 5

37 f = f r + f r r + f r + = A r - A r r = A+ A r r - 3 = A+ A = Ar r + r + r A = ^r + ^+ 3 ^ - cot r A- A r sin - f f r tan + r sin ( 5) A r - A r sin cos A r sin A r r sin - r sin A + cos A r sin A = ^r + ^+ 3 ^ A r r - A r r - A r - A r + = Ar r r + Ar r - 3 = - + A + r Ar A r tan + A rtan r A rsinr - A r sin A r sin + A r tan A r sin - rsin r + A r r sin + A A 6 = r = 3 = + - r sin + r sin cosa r sin cos A r sin A A = ^r + ^+ 3 ^ A r + A r - Ar r A rtan r + A A r - rtanr r tan A r r sin A + A rsin r + A r sin A r sin + cos A r sin - r sin - A r r + A r r r - A r A r rsinr - r A A A - r r - r ( 53) ( 54) ( 55)

38 - A r tan + A r sin + A r sin - cosa r sin f = ln r, f f = / r, f 3 r n = nr n - r 4 ( / r ) = 0 ( r0 ) 5 ( r/ r 3 ) = 0 6 (f) = 0 7 ( A) = 0 8 ( AB) = ( B ) A + ( A ) B + B( A) + A( B) 9 ( ) A ; ( ) ( A ) B ; (3 ) A B, A B = ( A ) B 0 S, r o P, : ^r^n ( ) o S, S r d S = 0 ^r^n ( ) o S, S r d S = 4 A D = DA [ I] = ( ) 3 ( f/ g) = ( gf - f g)/ g ( g0 ) 4 ( ) f( r ) = d f d r + r ( ) f ( r), f ( r) = 0 5 ( f f ) = 0 d f d r 6 ( 83 ) C B^nd S = R Bd R, B M, N 7 f ( 40 ) 8 r = I, r = 3, r = 0 7

39 , :,,,,,, q q, R( ), q q F qq,, q q r ; q, r, q : F q qq ; R ; F q, R= r- r F q F q q = k qq R R = k qq R 3 R ( ) k, k = 4 0 0, 0= F/ m ( ), q,, q, q q 8 F qq= qq R qq = R ( 3) 4 0R 4 0R 3 R = r - r = - R

40 F q q= - F q q ( 4) ( ),, (Maxw ell) ( P limton ) ( W illiams), R 0-0 m, 0-9,, E ( ),,,, q ; q q q,,,,,, q q, q, q q q, E E = q R ( 5) 4 0R q F = qe = qq R 4 0R,, q, N F = F q q = =, E N N = qq R 4 0R ( 6) q E( r) = R 4 0R ( 7) =,,,,,, 0-4 m,,, = lim V 0 q V = dq dv ( 8) C/ m 3 dv,, E 9

41 E( r ) = V ( r ) dv R ( 9) 4 0R, S = lim S 0 q S = dq ds ( 0) S C/ m S ds, E E ( r) = S S ( r ) ds R ( ) 4 0R, l = lim l 0 q l = dq dl ( ) l C/ m l dl, E E( r ) = l l( r ) dl R ( 3) 4 0R l,,, z, E, o z ldz P de de = ldz R = r 4 0R de r + z de z R = r + z, z z - z P z, P r, de r = r R de E = + - r E r = r lr dz 4 0 r + z 3/ = l 0r r 3,,, E ds, q, ( 5), 30 E = q R 4 0R

42 E S E ds = S qr ds = 4 0R q 4 0 S R ds R ( 4) R ds R, R ds ds q( 3) d, E ds= = q 4 0 S R ds = R q 4= q q d 4 0 S ( 5) ( 5), S, S E S,, E S N, ( 5) S E ds = N q 0 = R 3dS q,, S V, ( 5) S E ds = dv ( 6) 0 V, E ( 5), S ( ) : E ( ),, ( 6), S E ds = V EdV = 0 V dv,, E = / 0 ( 7) ( 7), ( 7), E,, E, E, E,, r,, E= r E r, 3

43 r,, L, E r E ds = L 0 E = 0 E rrd dz = r E r = r l 0r,,, ds= r r d dz, rler = ll 0,,,,, E,, E q E P P, q P P P W = F dl = P q E dl = P P qe dl ( 8) P Edl E l, E P P P P P, 4,, 4 P P q P P, P,, :, P P Edl P P, P P, P l P, l P, P P P L E dl = L P l E dl + P l P P E dl = l P E, 3 E dl - P l P P E dl

44 L E dl = 0 ( 9) ( 9),,, ( 9) C E dl = S ( E ) ds = 0 C S, S ( E ) ds= 0, E = 0 ( 0) ( 0) E,,, ( 0) 0,,, E = - ( ) ( ),, E,, E ( 5), E( r) = - - R = R R q 4 0 R ( ) ( ), = q = - 4 0R q 4 0R ( ) ( 3), ( r), ( ) E ( r ), ( 3), : N ( r ) = = q 4 0R ( r ) = V ( r ) = S ( r ) = l ( 4) ( r ) dv ( 5) 4 0R S ( r ) ds ( 6) 4 0R l ( r ) dl ( 7) 4 0R ( ), E, E, E, 33

45 C, C= 0, - = - ( + C) = E, + C E,, C, C,,, = 0,,,,, 3 d, l,, 5, z,,,, z R= [ + ( z- z ) ] /, dl = dz, ( 7) ( ) ḏ l ( r ) = l 4 0R dl = l 4 0 = E= - = - = - l 0 d dz [ + ( z - z ) ] / l ln ( z + d ) + [ ( z + d ) ( z - d ) + [ ( z - d ) + ] / ] / - ^ - z z + ( z - d ) + ( z - d ) + z ( z - d ) + + ( z - d ) ( z - d ) + z + ( z + d ) + ( z + d ) + ( z + d ) + + ( z + d ) ( z + d ) a, S,, 6, ( 6) 34 = S ds 4 0 S R

46 , ds = RdRd, -, R ( x - a ) + y = a x = Rcos, y= Rsin, R= a cos, R 0acos, = a c os d 0 S dr = S a 0 3,,,,,,,,, 3,,,,,,,,,,,,,,, ;, ( 7),,,,,,,, P 7 P 8 35

47 P, C/ m N, p 0, ( 8), dv dp ( ) P = N p 0 ( 8) N p 0 = p / N ( 9) = dp = PdV ( 30), V, V dv, dp = 8, r d = ( r ) = V P ( r PdV dp R = P ( r ) R dv 4 0R 4 0R ) R dv = 4 0R 4 0 V P( r ) R, R= r - r, R = R, r R dv ( 3) P R = R P + P R ( 3), ( 3) - ( r) = = P R = - P R V V P R P dv + R 4 P 0 V R dv P dv + R 4 P n 0 S R ds ( 3) P C/ m 3, P n C/ m, ( 5), ( 6), b = - P ( 33) S b = P n = P ( 34) b S b ( ) ( ) ( 3), ( r ) = b 4 0 V R dv + 4 S b 0 S R ds ( 35) ( 35),,, ( 33) ( 34) b S b, ( ) 36

48 ( 33) P, E,, P, P = 0, b = 0,,, Q b= V ( - P ) dv + S P n ds = S - P n ds + S P n ds = 0, 3 D ( 35) ( ),, : ( r ) = E( r) = 4 4 ( 0 V R ( f + 0 V f + b) dv R b) R dv + S + S ( S f + S b ds R ( 36) S f + S b) R ds R ( 37) ( 36) ( 37) f, S f, b, S b, b S b P, P,,, ( ) D E = f/ 0, b, b, E = E = ( f + b = - P b) / 0 ( f - P ) / 0 ( 0E + P ) = ( 38) ( ) D D = f ( 38) 0E + P ( 39) D = f ( 40) ( 40),, D f 37

49 D ds V = DdV V = fdv = Q f ( 4) S ( 4) D, D, E,, D, E 4 4, P E, P = P( E), P E, P x = E x + Ey + 3Ez + E x + E xe y + P y = E x + E y + 3 E z + E x + E x E y + P z = E x + E y + 3 E z + E x + E xe y + ( 4) P E,, P E,, P E P x = 0 x xe x + x ye y + x z E z P y = 0 y xe x + y yey P z = 0 z xex P x P y P z + y ze z + z ye y + z z E z xx xy x z = 0 yx y y y z z x z y z z E x E y E z ( 43) ( 44) j (, j x, y, z),, j E P E,,, P E ( 4) 4, E,,,, P E, ( 44) j = 0( j ), x x= yy = 38

50 z z,, P E P = e 0E ( 45) e, E, ( 39), D = ( + e ) 0E = r 0E = r = + e E ( 46),, r,,, e 0, r ( 46),, D E ( 46), P E ( 43), j 0( j ), P E,, ( 43) P x = 0 x xe x + x ye y + x z E z P x E x, E y E z, P E 3, 9,, P E,, , = 0,, ( L I H ), D = P = e 0E E, r = + e e, r, q, re,,, E, P, D, P, D, ( 4), q, ( 46) 4 r D = q ( 47) D = E = P = 0 ee, e = qr ( 48) 4 r qr ( 49) 4 0 rr r - 39

51 P = ( r - ) q r 4 rr ( 50) ( 49),, :,,, q, b S b b, b= - P, ( 50) P = ( r - ) q r 4 0r = 0 ( r 0) b= 0,,, q,, Q S b= = - lim r0 4 r ( P n ) = - lim r 0 4 r ( P r ) r - q r Q = q + qs b = q - r - q = r, b= 0,, r, P = P = D - 0E = r - r = - f = - r D = r - D r b r - r q r D,, f= 0, b = 0, 5 5,,,,,,,,,, 40

52 ,,,,,,, ( ),, Q Q C, C = Q, a,, Q = 4 0a ( 5) C= 4 0a ( 5) 5, :,,,,, N ( 9 ), q, q,, qn,, = p q + p q + + p N qn = p q + p q + + p N qn N = p N q + p N q + + p N N qn ( 53) p j,,, 9 N 0 4

53 ( 53), = p j N q q [ p j ] [ j ], q qn qn q = j N ( 54) ( 55) [ j ] = [ p j ] - ( 56), ( 55) q = N - ( - ) - 3( - 3) - - N ( - N ) = C + C( - ) + C3( - 3) + + CN ( - N ) q qn = C ( - ) + C + C3( - 3) + + CN ( - N ) = CN ( N - ) + CN ( N - ) + CN 3( N - 3) + + CN N N C = N N = j = ( 57) j ( 58) C j = - j ( j ) ( 59) C, C j j ( 57), C C j, 0, C C,,, = 0, C= 0, ( 57) q = q = - C C ( 60) C j = Cj, q= - q, C, ( 60) q, q, q = C + C( - ) q = - C( - ) + C ( 6), ( ), qq, C, 4

54 , 6,,,, E D E D,,,, 6, ( ), h0,, S, S D S f S f, n, n, n, n n = - n, n = n, ( ) D ds= n D S + n D S = n ( D - D ) S = S f S n ( D - D ) = S f ( 6) D - ( 6) ( 63) D = S f ( 63),, n ( D - D ) = 0 ( 64) D = D ( 65) ( 65),, D 6, 43

55 ( ), l,, h, h0, l, E l n, t, N, N, t t, N, n, t = Nn, E dl = - E lt + E lt = 0 - E t l + E t l = 0 E t = E t ( 66) n ( E - E ) = 0 ( 67) ( 66) ( 67), E,,,, D E?,, E = 0, D = 0,,, ( 63) ( 66) D = S f ( 68) E t = 0 ( 69),,,, 3 E E,,,, 3,, D = D, E = E E t= E t, 3 t an = t an E t/ E = E E t/ E E =, 44

56 6 3, 4, n,,, h, h 0, h n,,, - = E dr,, n, - = E dr = E h 4 E E h 0,,, S, - = 0 S = S ( 70), S = S S, ( 68) ( 69), ( 7) S = ( 7) = - S S f ( 73),,, S, 7, ( 4) ( 7),,,,, 45

57 7 E= -, ( - E = / 0 ) = - = / 0 = - = 0, / 0 ( 74) = 0 ( 75) ( 66) D = ( E) = ( - ) = f,, D = - = f = - f/ ( 76) ( 76),, 0 f= 0, ( 75) = 0 ( 77),, 7,,,,,,,,, ;,,,,,,, 5 d, 46

58 ( 5),,, E = = d = 0 ( x < - d / ) dx d = - / 0 ( - d / x d / ) dx = d = 0 ( x > d / ) dx = A x + B = - = A 3x + B 3 x + A x + B 0,,, x = 0, = 0, B= 0 x = 0,, x = 0, A = 0 x = - E = - = - d dx = x - A 0 x = 0 = 0 x = d,, x = d d = = x =, 0 x =, 0 0 x 0 x d = d 8 0 A 3 + B 3, A 3 = - d 0 d = d A + B, A = d 0 B = d, B 3 = d ,,, = d x + 0 d, E = x x = - d x 0 47

59 = - = - x, E = - 0 d x + 0 d, E = x x = xx 0 x x = d x 0,,,,,, 6 a, Q, ( 6), d d d d = - = - Q ( r a ) 0a d d d d = 0 ( r > a ) Q + A ln + B ( r a ) 4 0a = A ln + B ( r > a ), 0,, A = 0,, = 0, = 0, B= 0 = a,, =,, = - = 3, 0 - = 0 Q = A lna + B, A = - Q, B = - 0 Q, E = - 4 0a Q ln a 0 - Q Q a = A 0 a = Q, E = - = 4 0 Q lna 0 Q 0a Q 0 6 6,,,,, r, 48 7 ( 7) a,

60 Q,, r r d dr d dr r r d = - dr, = 3Q 4 a 3 Qr = - + A + 8 0a 3 B r = A r + B ( r a ) 0 d = 0 ( r > a) dr, r, = 0, B = 0 r0,, r = a, - = A = 0, 0 = 0 r r Q 8 0a + B = A a, A = Q 4 0 = - = E = - E = - B = 3Q 8 0a Qr + 8 0a 3 Q 4 0r = = 3Q 8 0a Q r 4 0a 3 Q r 4 0r 3 8,,,,,, 49

61 ,,,,, 8,, l, 8 r l P, P = r a = r + r = + r a = q r b l r l r a + rlcos + l r cos - / l 4r + l r cos l 4r + l r cos r l, l / r ( 78), r r a r r b = - + ( 78), l r cos + l 4r = + l r cos + l 4r P = 3cos - 3cos - 8 ql cos ( 79) 4 0r p = ql ( 80) l, p ( 79),, E = - = - P = p r ( 8) 4 0r 3 r r + r + r sin ^ 50

62 E r = - E = - E = 0 r = p cos 4 0 r 3 r = p sin 4 0r 3 ( 8) ( 8) ( 8),, r 8 N q, q,, qn V( 9 ), o V, r, r,, r N, P r ( 4), N q ( r) = 4 0R = R = r - r R = r + r / - r r cos ( 83) r > r, r / r <, ( 83) R = r - r / ( 84) r cos + r r, r r, R r r - + r r - r r cos + ( r ) ( 4), ( r) = N 4 0rq + = N cos + r r r r 9 N r r cos + r r ( 3cos - ) ( 85) q r 4 0r q r cos + 4 0r 3 ( 3 cos - ) + ( 86) = ( 86), / r,, M( r) ; / r N =,, D( r ) ; / r 3,, Q( r), ( 86) ( 86) ( r ) = M( r ) + D( r) + Q( r ) + ( 87) ( r ) = N 4 P l ( cos ), 0 l = 0 r l + P 0( cos ) = N q r l P l ( cos ) ( 88) = 5

63 P ( cos ) = cos P ( cos ) = ( 3cos - ) 9, ( 86) cos P 3( cos ) = ( 5cos 3-3cos ) cos = r r = r r r r r = lx + my + z r ( l, m, ) P, ( x, y, z ) q ( 86), ( ) M ( r) = N 0r 4 N q = Q = Q V,, o, M ( r ) = = Q 4 0r ( 89) q ( 90) ( 9) ( r) r, V,, Q V = ( r ) dv ( ) D( r) = N 4 0r q r cos ( 9) = N N q r cos = q lx + my + z = = = lq x + mq y + q z = r N q r ( 93),,, ( 9) D ( r) = p = N q r ( 94) = p r = p r ( 95) 4 0r 4 0r 3 ( 8), N =, D ( r), 5 =

64 ( 3) Q ( r) =,, p V = ( r ) r d V ( 96) N q r 4 0r 3 ( 3cos - ) ( 97) = r ( 3cos - ) = 3( r r ) - r l + m + =, = 3 lx + my + z - r l + m + r ( 3cos - ) = l 3x - r + m 3y - r + 3z - r q r ( 3cos - ) = l + 6lmx y + 6m y z + 6 lz x ( 98) q 3x - r + mlq 3y x + + lmq 3x y + l q 3x z m q 3y - r + m + lq 3z x + m q 3z y + q 3y z q 3z - r ( 99) ( 99) 9, 9, l, m,, P ; 9 Qj k = j, k x, y, z, j k = N q 3j k - r j k ( 00) = ( j = k) 0 ( j k) N Qx x = q 3x - r = N Qx y = q 3x y = Q j k, Qj k = Q kj ( j k) ( 0) ( 0),, 6 9 Q ( r ) = l 8 0r 3 Qx x + lmq xy + l Q x z + m Q y z + m Q yy + Qx z ( 0), x + y + z = r, 5 Q x x + Q y y + Q z z = 0 ( 03),, 0, q x = y = 0, 53

65 ( 00) ( 0), Q ( r ) = = = Q j k = Qkj = 0 ( j k) Q x x = - qr = - qs Q y y = - qr = - qs Q z z = - q 3s - r = 4qs - qs l - qs m + 4qs 8 0r 3 - qs ( - ) + qs 4 0r 3 qs ( 3 - ) = 4 0r 3 qs ( 3cos - ) 4 0r 3 0,,,, ; ;,,, ( ),,, q = 0, ;, 9 9,,,,,,,, N,?,,,, q q,,, W, 54 W = q q

66 ,, W = q = q q 4 0 r + q3 r qn r N q, q, W = q = q q3 4 0 r 3 + q4 r qn r N,, N,, W = W + W + W3 + W N = q 0 + q 4 0 r + q3 r qn r N + q q3 4 0 r qn r N + + qn ( ) 4 0 = q 0 + q 4 0 r + q3 r q N r N + q q 0 r q3 r qn r N + q3 q r q r 3 = q + q + q qn r 3N qn + + qn q r N 4 0 N = N = + q r N + q 3 r N q ( 04) ( 04),,, q, ( 04) q dv,, dv,, W = V dv ( 05),,,, S, ( 05) 55

67 W = S S ds ( 06) S,,, ( 06) S S ds = N W = S S ds = q N q ( 07) = ( 04), ( 04), q, q,, ( 07) q,,, ( 07) W = q q, C C= q/, W = q = C = q C ( 08) 9 ( 05) ( 07),,,, = 0 E ( 05),, ( 09), W = 0 ( E) dv ( 09) V ( E) = - E + ( E ) = E + ( E ) W =, W = 0 V E dv + 0 V E dv + 0 V ( E) dv 0 S E ds ( 0),,, ( 0) 56 W = ( E) ds = 0 S 0 E dv V = V 0E dv ( )

68 ( ), E 0,, w = 0E ( ) 9 3,,, SdS,, S ds SdS E E = S n ( 3) 0 n, ( 3) SdS, ( 3) S ds SdS,, S ds/ ( 0 ), S / ( 0), ds E = df = S n S - n 0 = S n 0 ( 4) 0 f = E S ds = df ds = S 0 S 0 ( 3), f E,, f = df ds = ds n ( 5) ( 6) 0 E ( 7),, n,, S,, ( 7) ds dx, w, dw = wdx ds,, 57

69 ( 7) dw = w dxds = w = f w = 0 E f ds dx 9 4,,, T = pe ( 8) p, p = ql, T,,,,,, l, p = ql, E, x F x = - qe x + q E x + E x x l x + E x y l y + E x z l z, = E x x p x + E x y p y + E x z p z ( 9) F y = E y x p x + E y y p y + E y z p z ( 0) F z = E z x p x + E z y p y + E z z p z ( ) F = F x x + F yy + F z z = p x x + p y y + p z z E x x + E yy + E zz = ( p ) E ( ), N ( N ), 58

70 F = ( P ) E ( 3) P, P = 0 ee = ( 3) x F = ( P ) E = ( - 0) ( E ) E = ( - 0) E x F x x + E y = ( - 0) E x E = 0, ( 5) F x, ( - 0) E y + E z z E x x + E yy + E zz E x x + E E x y y + E E x z z E x y = E x z = E y x E z x F x = ( - 0) x E F y = ( - 0) y E F z = ( - 0) z E, F = ( 4) ( 5) ( - 0) E = r - r E ( 6) E, E, E, 9 5,,,,,, (, ),,, + = 59

71 f dg + dw e = dw ( 7) f, g,,, ;, ;, ;, ( 7) dw,,, ( 7) dwe, N dw = dwe = = dq ( 8) N = dq ( 9),,,, dg dr, f dg = dwe ( 30) f dr = dwe f dr = f xdx + f ydy + f zdz ( 3) f x = We x, f y = We = y =, f z = We z = ( 3),,, dq = 0 dw = 0, ( 7) f x = - f dg + dwe = 0 f dr = - W e, f y = - x q= dwe W e, f z = - y q= We z q= ( 33),,,, ( 3) ( 33) 8 3, A, d, x 60 We = CU = q C

72 3 8 U, q, C, ( 3) f x = W e x = x CU = U C= 0A, x f x = U C x 0A = - x x U 0A = - x 0 E A, q, ( 33) f x = - = - We x = - x q C = - q q 0A = - C U 0A = - U 0A = - x x C 0 E A 0, - E, x, 9 r > ( 4 ), U, y f y z, We =, U 0 ( l - y ) d + d U d f y = W e y = yd U d ( - 0) = ( - 0) E d f y > 0 y,,, 4 9 6

73 ,, S - S, d a,, l, 3 a, 4q - q d,, ( ) 5 a, S, S,, S = - S S = S 6a q, Q Q, q 7 a Q 8 a e,, 9 d ( ) 0 q a q d ( d > a ), : a ( ), q = q 4 0a p Q( ) 3 6

74 , ( ) R, E= Ar r ( ) a, L, E= A ( 3), a, E = A ( x x + yy ) A 6 a, 7 a,, = 0 ( r a ) a 0 r ( r > a ) 8p j, a, b c, d e 9 a ( r), E r = A r 4 ( r a ) Ar - ( r > a ) ( r ) S 0, a, b, U 0 ( ), C br ( r ) = qe-, q, r b, = 0cos x ( - a x a ) a 0 ( x > a ) 3 a, 0,, 63

75 b, d ( 4), a,, 0( 5), 5, a b, q, - q, 6, a b, ( r ) = 0a / r, C 7 a b, d, ( 0a / ), U 0, 8, 0< <, ( 6), d E 0, E 0 ( 7), = / 4 30, d ( 8), U 0, ( g, ), h 3 a, p = r r,,, : ( ) ; ( ) ; ( 3) 64

76 8 9 3 U 0, ( 9), 33V, V S, S =, V p r 34: p = 4 0 r 3, r 0 35, a q, b,, b b, q ( b- b) / ( 8 0bb) 36, 0 a ( b- b) / ( b- a ) ( b- a ), 37 a, E,, 38 a, a a = a a 65

77 3,,,,,,,,,,, 3 3, :, ;, ;,,,,, 3 :,,, = - / = 0, ;,, S V,, = - / = - / = -, V 66 = (- ) = - = 0

78 ( 85 ), = =, + ( ) d V = V ( ) d V V S V, ^n S = S ^nds ( 3 ) n S,,, S = S - S = 0, (3 ) ( ) d V = 0 V ( ),, = 0, = - =,, + C,, n S, = 0 (3 ) n = n ( ) d V = 0 V, = C( ),,,, S = - = 0, S - S n = n - = 0, ( 3 ) n ( ) d V = 0 V, = C( ),,,, ( ),,,,, 3 ( ) q, 67

79 , q,,,,, ( ),,, q 3 q, d, 3 x > 0,, (0, y, z ) = 0 ( 3 ) x > 0 = q ( x - d, y, z ) ( 3 3) 0 ( - d, 0, 0) q x > 0 ( 3 3 ) (3 ) q qx > 0 = q 4 0 R + q R = 4 0 q ( x - d ) + y + z / + (3 ),, x > 0 = = q 4 0 R - R q q ( x + d ) + y + z / ( 3 4) d= d, q= - q ( 3 5) ( x - d) + y + z / - ( x + d) + y + z / ( 3 6) (3 6) (3 3 ), (3 ),, x > 0, (3 5 ) q 68, ( 3 6 )

80 Ex Ey E z = - x = = - y = = - z = q 4 0 qy 4 0 qz 4 0 ( x - d ) ( x - d ) + y + z 3/ - ( x - d) + y + z 3/ - ( x - d) + y + z 3/ - ( x + d ) ( x + d) + y + z 3/ ( x + d ) + y + z 3/ ( 3 7) ( x + d ) + y + z 3/ x = 0, Ey = Ez = 0, Ex En, En = Ex ( 0, y, z) =, S = 0 En =, S, qs = - - S d yd z = - qd - qd 0 d + y + z 3/ ( 3 8) - qd d + y + z 3/ ( 3 9) - - d yd z d + y + z 3/ = - q, q,,,,,,, ( 3 6 )x > (a), q, d, d, q 3 3, 3 ( b ), q = - q, q = - q, q3 = q, q q, 69

81 q, q3 q, F F F3 = F = F + F + F3 = - ^y = - ^x q 4 0 ( d ) q 4 0 ( d ) q 4 0 ( d ) + ( d ) 3/ ( d ^x + d ^y) 3 q a, d o ( r,, ), 3 3 r > a, r = a, ( a,, ) = 0 (3 0) q d, d< a, = q 4 0 R + q R = 4 0 q r + d - rdcos / q + r + d - rd cos / (3 ) 3 3 (3 0 ), q a + d - adcos / + a q + d - ad cos / = 0 qd,, qd, q a + d - ad cos= q a + d - adcos, 70 q ( a + d ) = q ( a + d ) q d= q d d= a d, q= - a q (3 ) d d= d, q= - q (3 3)

82 (3 3), d< a, (3 ) q < q, q, (3 )( 3 ), = E = -, E = q 4 0 q ( r + d - rdcos ) / R 3 ( r - dcos ) - r + a d ( a/ d ) ( r - d cos ) R 3 ^r + a/ d - r d R 3 - R= ( a/ d) R, r = a E= 0, Er = En = - qs = - = - = - q( d - a ) 4a q( d - a ) a a d cos / (3 4) ( a/ d ) d R 3 sin ^ (3 5) q( d - a ) 4 0 a a + d - adcos 3/ (3 6) S = 0 En = 0 Er q( d - a ) a a sindd a + d - adcos 3/ d(cos) a + d - adcos 3/ d ( d - a ) = - a q = q (3 7) d q, q F = - adq 4 0 ( d - a ) ^z (3 8),, q 3 4,,, q= - q= ( a/ d ) q, q o, 3 4, = q 4 0 a = q 4 0 d (3 9), q, q 7

83 o 3 3 a, l d, 3 5,,, o ( r, ), r > a, r = a, ( a, ) = 0 (3 0),,, 3 5, lo doa C ( 3 5 ), P = P= 0, l ln - l R ln 0 d - a - l R ln 0 a - d R = a + d - adcos, R = a + d - ad cos a + d - adcos / d - a = - l ln a + d - ad cos / a - d (3 ) l= - l, d= a / d (3 ), ( 3 ) (3 ), = E = - = l ln d R 0 ar l R - 0 R (3 3) R R (3 4) R = a d R, S = 0 ( ^ne) = l ^n^r R = - l ar ( d - a ) - ^n^r R ^n 7

84 qs = Sd S = - l ( d - a ) a = - l ( d - a ) a 0 d z 0 0 ad R ad a + d - adcos = - l = l (3 5),,,, 3 4 3, q d,, 3 6 (a) q,, q,,,,,,,, 3 6 ;,,, =, ( 70 )( 65 ), = ^nd ^n = ^nd, : q q ( 3 6 ( b ) ), 73

85 , = q 4 R + q 4 R (3 6) D = q^r + q ^R (3 7) 4R 4R : q ( q )q, = ( 3 6(c) ) = q + q (3 8) 4 R D = ( q + q ) ^R 4R (3 9) R = R= R (3 30) (3 6 )(3 8) ( 70), ( 3 30)( 70) q + q = q + q (3 3) (3 7 )(3 9) ( 65 ), (3 30 ) ( 65) (3 3 ) ( 3 3 ) = = E = E = q - q= q + q (3 3) q= - q= q - + q 4 R + q 4 R + q ^R 4 R - ( + ) R (3 33) (3 34) (3 35) + ( - ) ^R ( + ) R (3 36) q ^R (3 37) ( + ) R,, :, ,

86 ,,,,,,,, 3 3,, CR W ( z ) = W ( x + j y) = u ( x, y) + j v( x, y) u x = v y u y = - v x (3 38) CR,, u x v x + u y + v y = v xy - v yx = 0 = - u yx + u xy = 0 (3 39),,,,, u v (,, ), ( ), u v, u, v, v ( x, y), u ( x, y)?? u = ^x u x + ^yu y u v = v = ^xv x + ^yv y u v x x +v u y y = - u u x y +u u x y = 0 (3 40) (3 40 ), u = v = v, v =,, u = v, u ( x, y ), 3 7, z, z, A B 75

87 B = B Dd S = A A D( d l^z) B = D[ ( d x^x + d y^y) ^z] A B = D( d y^x - d x^y) A = B ( D xd y - Dyd x ) A B = u A y d y + u x d x Dx = Ex = - v x = u y Dy = Ey = - v y = - u x = B A d u = [ u( B ) - u ( A) ] (3 4), z ( x y ) u,, u, AB, z B Q = A Sd S = B, A DdS = [ u( B) - u ( A ) ] (3 4),,, v, u,, W ( z ),,,,,, z = re j, W ( z ) = A ln z + B + j B (3 43) W ( z) = Aln( re j ) + B + j B = A ln r + B + j( A+ B ) = u + j v (3 44) u = Aln r + B, v = A+ B Z u = r =, v =, W ( z ) u, 76

88 ( 3 8 ) W ( z ), ( 3 9 ),,, r = r, =, r = r, = = 0,, ( 3 44), = u = Aln r + B (3 45), r = r u = = A ln r + B r = r u = = A ln r + B = 0 A = ln ( r/ r ), B = - ln r ln( r/ r ) (3 45 ), = ln( r/ r ) (ln r - ln r ) (3 46) (3 44), = 0, v = 0, B = 0, ( 3 4 ), Q = A = C = Q = - ln( r/ r ) ln( r/ r ), (3 47) (3 48) W ( z ) = arccos z (3 49) z = cos W = cos( u + j v) = cos u cosj v - sin u sinj v = cos u ch v - jsin u sh v = x + j y x = cos u ch v, y = - sin u sh v x ch v + y sh v = cos u + sin u = (3 50) 77

89 x cos u - y sin u = ch v - sh v = (3 5), v =, ( 3 50 )Z, u =, (3 5) Z,, 3 0, (3 49), ; (3 49 ), 3 3, W ( z ) Z W,, W ( z )d W/ d z0, Z W, W Z,,,, d W/ d z z 0 z0 d z W d W, Z W, W, :f( x, y)z, u, v, W,, : W ( z ) = u + j v = f ( z ) = f( x + j y), ( x, y) ( u, v), ( x, y) = x + y, ( u, v) 78 x = u x u + u x y = u y u + u y = + = x y + v y x = u x u + v x v u + v x v + v x u + v y v + v y u x v + x u + u v x x +u v y y + v y + u y uv v + u v x x uv v + u v y y uv + u v u + x v x + v y v (3 5) (3 53) (3 54)

90 CR (3 54 ), = x + y = u x = v y, u y = - v x f ( z ) = u x + jv x u x + v x u + v = f ( z ) u + v (3 55), f ( z) 0, ( x, y) ( u, v) ( x, y) ( u, v) x + y = 0 u + v = 0 x + y = - u + v = - (3 56) ( u, v) = f ( z ) - ( x, y) (3 57),, ( u, v )d ud v = f ( u, v) ( x, y) = u v x y - u v y x = ( z ) - ( x, y) ( u, v) ( x, y) u x ( u, v) d ud v = + v x d xd y = f ( z ) ( x, y) d xd y (3 58) f ( z )0,,,, Z W, E ( Z ) = f ( z ) E ( W ) (3 59),,, 79

91 , d W = f ( z ), Z W d z f ( z ), f ( z ),,, C, S Z, S W, S, QS = D ( Z) n S d S = E ( Z ) n S d S = n E ( W ) S d W d z d S d W d z = D ( W ) n d S= QS (3 60) S 3 C, a, b a, b ( 3 ( a) ), 3 3 0, = 0, c c = a - b = a - b (3 6) W = arc cos z/ k z = kcos W (3 6) z = x + j y, W = u + j v,, 80 x = kcos u ch v, y = - ksin u sh v (3 63) x k ch v + y k sh v = cos u + sin u = (3 64) x k cos u - y k sin u = ch v - sh v = (3 65)

92 , v =, ( 3 64 )Z, u =, (3 65) Z a = k ch v, b = k sh v (3 66) a - b = k (ch v - sh v ) = k = c (3 67) k = c ch v =, a - b v = arch v a a a - b (3 68) ch v = a a - b, v = arch a a - b, z = kcos W, Z W v = v (3 69) v = v ( 3 ( b) ), Z W,, = 0 v - v ( v - v) (3 70) Z (u, v x, y ) C,, W, Z W,, C = A d (3 7) W = 0 = 0 d v - v, A Z D W D, u = 0, x = c ch v, y = 0, W Z xc, u = /, v0, x = 0, y = - c sh v, W u = /, v0 Z, u = 3 /, v0 Z, u = Z x c, W u 0, Z W, (3 7), C = v - v (3 68 ) (3 69) v, v, C = arch a a - b - arch a a - b = ln a + b a + b (3 7), 8

93 3 3 ( 3 ( a) ),, l,,,, W = z / (3 73) z = re j, W = r e j, (3 73) W = r e j = ( re j ) / (3 74) r= r /, = Z = 0 =, W, Z (3 74 )W, Z l,, W ( 3 ( b) ), r0= r 0 /, 0= 0 W, a = r e j - (3 74 ), Z a = r / e j / - r P = l ln a / a (3 75) r0 e j 0, a = r e j - 0 / e j (3 75 ) / 0, a = r / e j r0 e - j 0 / - r 0 / e - j / 0,, 3 4,, : 8

94 , ; ;,,,,, 3, [ 7] 3 4,,,, (3 77 )( 3 76 ),, = x + y + z = 0 (3 76) ( x, y, z ) = f ( x) g( y) h ( z ) (3 77) f f d x d + g d g d y + h d h d z = 0 (3 78) (3 78),, f g h d f d x = - k (3 79) d g d y = - k (3 80) d h d z = - k 3 (3 8) k + k + k 3 = 0 (3 8) ( 3 76) ( 3 79 ) ( 3 80) ( 3 8 ) ( 3 76 ), (3 79 ) (3 80) ( 3 8 ), k, k, k3, f ( x ), g ( y), h ( z ) k, k, k3,, ( x, y, z ),, 3 ( x, y, z ), x, y, z,, m n = - k l = k m + k n ( l, m, n =,, 3; lmn) 83

95 3 f( x ) g( y) h( z ) k > 0 k = 0 k < 0 k > 0 k = 0 k < 0 k 3 > 0 k 3 = 0 k 3 < 0 cos k x cos k y ch z sin k x cos k x sin k y ch 3 y cos k3 z sh z sin k x ch 3 x cos k y sh 3 y sin k 3 z cos k 3 z cos k x sh 3 x sin k y ch k y sin k 3 z z sin k x ch k x cos k y sh k y z x sh k x sin k y y z, , ( x, y), = 0, z x = 0, 0 y b, = U0 ( 3 83 ) x = a, 0 y b, = 0 y = 0, 0 < x a, = 0 y = b, 0 < x a, = 0 f f d x = - k, g d ( 3 84 ) ( 3 85 ) ( 3 86 ) d g d y = - k ( 3 87 ) k + k = 0 k = - k , f ( x )g( y) k = k = 0, f( x )g( y) ; k > 0, k < 0, f ( x ), g( y); k < 0, k > 0, f( x ), g( y), y,, ( x, y) = ( C sh k x + C ch k x ) ( D sin k y + D cos k y) (3 88) (3 85 ), D = 0, ( x, y) (3 86 ), 84 ( x, y) = D sin k y( C sh k x + C ch k x) (3 89) k = n ( n =,,3, ) (3 90) b

96 (3 84 ), (3 83 ), ( a, y) = n = ( x, y) = n = Dn sin n n n y sh a + Cn ch b b b a = 0 Cn = - th na b Dn sin n n na y sh x - th b b b = Dn sin n sh n = b y = n = n na x ch b b An sin n n( a - x) y sh b b An sh na n = b ch n b x - sh na b ch na b ch n b x (3 9) sin n b y = U 0 (3 9) An,, sin m y, y 0 b b, b 0 b A m sh ma b n m sin y sin b b y d y = Am = ( x, y) = b = U 0 sin m 0 b y d y = 4 U0 msh ma b n =, 3, 4 U0 0 ( n m) b/ ( n = m ) U 0 b( - cos m) m ( m =, 3, 5, ) (3 93) sh n sin n b y n( a - x) b sh na b (3 94),, , 3 4 ( a), x = 0, x = a,,, = a + b, a b a, y = 0, b, 0 < x a, a = 0 x = a, 0 y b, a = 0 85

97 b, a, b 3 4 x = 0, 0 y b, a = U ( y) y = 0, b, 0 x < a, b = 0 x = 0, 0 y b, b = 0 x = a, 0 y b, b = U ( y) 3 4 ( b), = a + b, a y = b, 0 x, a = U0 y = 0, 0 x, a = 0 x = 0, 0 y b, a = U 0 b y b x, 0 y b, a = U 0 b y y = b, 0 x, b = 0 y = 0, 0 x, b = 0 x = 0, 0 < y < b, b = U( y) - x, 0 y b, b = 0, a b 3 4 ( b), x, x = 0 x, U 0 y, b a,, a = U 0 b U0 b y y (3 95) b, 3 4,, x, 0 y b, b = 0 x ( ), 86

98 e - nx/ b y, sin n b y, b b = Cnsin n n = b ye - nx/ b (3 96) Cn, x = 0 (3 96 ),, U( y) - U0 Cn = b b U ( y) - 0 b y = Cnsin n b y U( y), Cn 3 4 ( b) = a + b = U 0 n = U0 b y + n = b y sin n yd y (3 97) b Cn sin n b ye- nx/ b (3 98) 3 4 (c), y = b = 0 = 0 y = 0 y = b, = 0 n ( 3 3 ) sin k y, y = b = 0 k = n b, y = 0, sin k y, y = b 3 4 (c) n = - y = - y ( sin k y) = - k cos k y = 0 k ( x, y) = = n ( n =,3,5, ) (3 99) b Cn sin n n =, 3, b ye - nx/ b ( 3 00),, 3 4, = (3 0), + + = 0 ( 3 0) z (,, z ) = f ( ) g() h( z) ( 3 0) d f d d f d + g, d g d + h d h d z = 0 ( 3 03) 87

99 (3 04)(3 03 ),, g d g d = - n d g d + n g = 0 ( 3 04) f d d d f d - n + h d h d z = 0 z, d h d z - k z h = 0 ( 3 05) d d d f d + k z - n f = 0 ( 3 06) (3 04 ) (3 05) ( 3 06), (3 04) : n n = 0, g() = A + B > 0, g() = Acos n+ Bsin n, ( ) g () = g( ), n, n n < 0 (3 05) : (3 06) k z k z k z = 0, h( z ) = A + Bz < 0, h( z ) = Acos kz z + Bsin kz z, k z = - k z > 0, h( z ) = Ach kz z + Bsh kz z d f d + d f d + ( kz) - n f = 0 ( 3 07) n, : ( ) n = 0 d f + d f d d + ( k z) f = 0 () f( ) = A0 J0 ( kz ) + B0 N0 ( kz ) J0 ( kz)n0 ( kz ) ( ) k z = 0 (3 ) n 88 = k z = 0 d f d + d f d - n f = 0 ( ) f() = A n n + Bn - n f ( ) = A + Bln

100 (4 ) n > 0, k z > 0 f ( ) = AJ n ( kz ) + BN n ( k z) J n ( kz)n n ( kz ) n n, , J n ( kz )N n ( kz ), J n ( kz)n n ( kz ) kz, ( n) i, 3 J0 ( kz ) ( i ) ( 0 ) 3 J 0 ( x) 8 i ( 0 ) i i i ( 0 )

101 , n, k z (,, z ) = f () g( ) h ( z ),,,, ( ) J n ( k )( 0, ) ( ) ni, J n ( ni ) = 0, n - J n ( k )[ 0, a], kia = ni 0 ( i j) a J 0 n ( ki ) J n ( kj ) d= a J n + ( kia) ( i = j) ( 3 08) [0, a] f ( k),, Ci ( 3 08), ( ) Ci = f( k) = CiJ n ( ki ) ( 3 09) i = a J n + ( ki a) a f ( k ) J 0 n ( ki ) d ( 3 0) J n ( k )( 0, )( ) ni, J n ( ni ) = 0, n > - J n ( k )[ 0, a], kia = ni a J n ( ki ) J n ( kj ) d= 0 0 ( i j) a - n ki a J n ( ki a) ( i = j) ( 3 ) [0, a] f ( k), Ci (3 ), Ci= a - f ( k) = n kia i = J n ( kia) Ci J n ( ki) ( 3 ) a 0 f( k) J n ( ki) d ( 3 3) N n ( k ) J n ( k ), N n ( k) , b, a, z = 0, = 0 = a, = 0 90

102 z = b, = f ( ) (,, z ),, = 0 n = 0, = a, = 0, f () J0 ( kz ) N0 ( kz ) z,, z = 0, z = b, h( z ), (, z) = f ( ) h( z) = ( A ch( kz z ) + A sh ( kz z ) ) ( B J0 ( kz) + B N0 ( kz ) ) z = 0 = 0, A = 0; = 0, B = 0 = a = 0, (, z) = Ash( kz z ) J0 ( kz ) ( 3 4) ( a, z) = Ash( kz z ) J0 ( kz a) = 0 kz a i i, (3 4) ( ( 0 ) kz i, z) =, z = b = f (), i = = ( 0 ) i a ( 3 5) Ai sh( kz i z ) J0 ( kz i) ( 3 6) f() = Ai sh ( kz ib) J0 ( kz i) i = 9

103 Ai, J0 ( kz m), 0 a d a 0 f ( ) J0 ( kz m f ( ) = U0, A m = ) d= A m sh( kz m b) a sh ( kz im ib) a 0 + = A m sh( kz m b) a 0 J 0 ( kz m ) d J0 ( kz i) J0 ( kz m ) d J ( kz m a) a f() J0 ( kz m ) d ( 3 7) sh ( kz m b) a 0 J ( kz m a) a U 0 J0 ( kz m) d= U0 a J ( kz m a) 0 kz m A m = (3 6), (, z) U0 kz m ash ( kz m b) J ( kz m a) U0 sh( k z m z ) J0 ( kz m) (, z ) = k z m ash( kz mb) J ( kz m a) m = ( 3 8) ( 3 9) ( 3 0) 3 7, (kz = 0 ), : = a, = 0 = = 0; = b = U0, 3 7 = 0 = = 0, g() n > 0, kz = 0, (, ) = ( A cos n+ B sin n) ( A+ B = 0 = 0 = = 0 = a = 0, 9 n = (, ) = n A = 0 - n ) ( 3 ) m ( m =,, 3,, ) m = ( Am / + Bm - m / ) sin m A m a m / + Bma - m / =

104 ( Bm, ) = m =, = b = U0, U0 = - A ma m / Am ( m / - a m / - m = A m ( b m / - a m / b - m / ) sin m m = An, U 0 sin n 0 m = n, (3 ), d= 0 m = U0( - cos n) n A n = 4 U 0 n b n A m ( b m / - a m / b - m / ) sin m ( 3 ) / ) sin m n sin d = An ( b n / - a n / b - n / ) / - a n / b - n = (, ) = n / - a n n / - a n n =, 3, 4 U0 b n / ( n =, 3, 5, ) / - n / / b - n sin n ( 3 3) / m = n 3 8 E0 = E0 ^x, a, z, 3 8, = 0, kz z = 0 3 8, E0 0 = - E0 x = - E0 cos ( 3 4) g() = g( n) n > 0 (, ) = n = 3 8 ( A ncos n+ Bnsin n) ( Cn n + Dn - n ) ( 3 5), E0, ( 3 5 ) (3 4),, (3 5)Bn = 0, Cn = 0, (, ) = - E0 cos+ A n - n cos n ( 3 6),, = 0,, (3 5) Dn, (x ), sin n n = 93

105 (, ) = n = Bn n cos n ( 3 7) An, Bn = a, (3 6) ( 3 7) =, 0 = r 0 - E0 acos+ A na - n cos n= Bn a n cos n ( 3 8) n = n = - E0 cos- n Ana - n - cos n= r nbna n - cos n ( 3 9) n = n =, ( 3 8 )(3 9 ), n = n, An = 0, Bn = 0 - E0 acos+ A a - cos= B acos ( 3 30) - E0 cos- A a - cos= r B cos ( 3 3) A =, B = - E0 ( r - ) r + E0 r + = - E0 cos+ r - r + a E0 a ( 3 3) cos(> a) ( 3 33) = - E 0 r + cos(< a) ( 3 34) E = -,,, E = - E0 r + cos= - E0 r + x E= - = E 0 ^x ( 3 35) r +,, r >, E< E0,, D D 94

106 3 4 3, = r r + r r r sin sin + r sin = 0 ( 3 36) ( r,, ) = f ( r) g( ) h() ( 3 37), fgh r sin, sin d r d f f d r d r + sin d g d sind g d + h (3 38), sin, f h d d r : d h = 0 ( 3 38) d d h d = - m d h d + m h = 0 ( 3 39) r d f d r + d gsind sind g d - m sin = 0 ( 3 40) d d r r d f d r - n ( n + ) f = 0 ( 3 4) d d sind g d + n ( n + ) sin- m sin g = 0 ( 3 4) (3 39 ) (3 4) ( 3 4),, = 0, m = 0, m = 0, (3 39 ) h() = A + B h() = h(+ ), h() = (3 4): ( ) n = 0, ( ) n > 0, n < 0 (3 4) f( r ) = A + Br - f ( r) = Ar n + Br - ( n + ) d sind sind g d + n( n + ) g - m sin g = 0 m = 0, : 95

107 ( ) m = 0, n > 0, ( ) m = 0, n = 0, g( ) = A P n ( cos ) + BQ n (cos) g( ) = A P0 ( cos ) + BQ0 (cos ) P n (cos ), Q n ( cos ), n ( ), P n (cos ) ( ), Q n ( cos ) (), [ -,] P n ( ), Q n ( ), Q n cos= (= 0,) Q n,, Q n 3 3 n P 0 (cos ) = P (cos ) = cos P (cos ) = (3cos - ) P 3 (cos ) = (5cos3-3cos ) 3 3 n P4 (cos ) = 8 (35cos4-30cos + 3),, ( r, ) = n = 0 A n r n P n ( cos ) + n = 0 Bn r - ( n + ) P n (cos) ( 3 43) m = 0, n = 0,,,, P n ( x) [ -,], P n ( x) P m ( x ) d x = - 0 ( n m ) n + ( n = m ) ( 3 44), [ -, ] f ( x), Cn ( 3 44 ), f ( x) = Cn P n ( x) ( x ) ( 3 45) n = 0 Cn = n + - f( x ) P n ( x ) d x ( 3 46) 3 9 E0,, E0, 3 0E0,, = 0, m = 0, ( r,)( 3 43), ( r, ) = n = 0 A n r n P n ( cos ) + n = 0 Bn r - ( n + ) P n (cos) 96

108 r = a, = 0 r,, Ana n P n (cos n = 0 ) + n = 0 = - E0 z = - E0 rcos Bna - ( n + ) P n (cos) = 0 An, Bn, ( 3 44) P m ( cos ), cos= - cos=, n = 0 - An an P n (cos ) P m ( cos ) d(cos) n = 0 + B na - ( n + ) P n (cos) P m (cos ) d(cos) - = A na n n + + B n a - ( n + ) n + = Bn = - An a n + ( 3 47), ( 3 43 )r, ( r, ) =, (3 47), n = 0 An r n P n (cos ) = - E0 rcos= - E0 rp (cos ) A = - E0 An = 0 ( n ) ( 3 48) B = - A a 3 = E0 a 3 ( 3 49) ( r,) = - E0 rcos+ E0 Er = - r = E 0 E= - r = - E 0 - a 3 cos r ( 3 50) + a3 r 3 cos ( 3 5) a 3 r 3 sin ( 3 5) = 0 E r = 3 0 E0 cos ( 3 53) (3 50), E0,, p = 4 0 E0 a 3 ( 3 54), 97

109 , E0,, 3, a U0 (), U0 (),,, = 0, m = 0, = 0,, Q n ( cos )(4 43 ), ( r, ) = n = 0 A n r n P n ( cos ) + n = 0 Bn r - ( n + ) P n (cos) r < a, r = 0,, Bn = 0, r = a (3 46),, 98 ( a, ( r, ) = ) = An = n + n = 0 n = 0 a n a n 0 = n + An r n P n (cos ) ( 3 55) An a n P n (cos) = U0 ( ) U 0 () P n ( cos ) d(cos ) - U0 ( ) P n (cos) sind U0 ( ) = Csin = C ( - cos ) = C [P 0 (cos ) - P (cos) ] ( 3 56)

110 (3 55), A0 = C, A = - ( r,) = C - C C a ( 3 57) r cos( r < a ) ( 3 58) a r > a, r,, An = 0, r = a,, Bn = n + ( a, ( r, ) = ) = n = 0 n = 0 Bn r - ( n + ) P n ( cos ) ( 3 59) Bna - ( n + ) P n ( cos ) = U0 ( ) = C [ P 0 (cos) - P (cos ) ] a n + C 0 [ P0 ( cos ) - P (cos ) ] P n (cos) sin d B0 = a C B = - ( r, ) = C a r - C a a r cos ( r > a) ( 3 60) = 0,,,,,,,,,, 3 5,,,,,,, = q 4 0 R = - / 0, q 4 0 R = q 0 4R = - / 0 99

111 ,, 4R Dirac Delta, 3 5, ( r - r ) = ( r - r )d V = V, f( r), 0 ( r r ) ( r = r ) ( r V ) 0 ( r V ) ( 3 6) ( 3 6) ( r - r ) = ( r- r) ( 3 63) f( r) ( r - r ) d V = f ( r ) ( r V ) V, f ( r)r = r f( r ) (3 6),, : ( r - r ) =( x - x )( y - y ) ( z - z ) ( 3 64) ( r - r ) = (- ) (- )( z - z ) ( 3 65) ( r - r ) = r sin ( r - r )(- )(- ) ( 3 66) 3 5,, 4R,, V 4R 4R d V = V 4 R = - 4 S 0 ( R 0, r r = ) ( R = 0, r = r ) ^RdS R R V ( rv ), S ^RdS R R V ( rv ), 00 d V = - = 4 ^R 4 V R d V ( 3 67)

112 S - V 4R d V = ^RdS R = 0 ( r V ) 0 ( r V ) (3 67) (3 6 ), (3 68 ) ( 3 6 ), 4R,,, ( 3 68) = - ( r - r ) ( 3 69) q 4 0 R = q 0 4R = - q( r - r ) 0 = - / 0 ( 3 70) = q ( r - r ) ( 3 7) = ( r - r ) ( 3 7), 3 6,,,,,,,, 3 6,,, V ( - ) d V = S n - n d S, =,, 4R 0

113 V 4R - 4R, r V,, V = - / 0 4R - ( r ) + ( r 4 0 R ) ( r - r d V = S 4R n - n 4R ) d V = S = - ( r - r ) 4R n - n 4R ( r ) ( r - r ) d V = V ( r )( r- r) d V = ( r) V ( r) = ( r ) V 4 0 R d V + 4 S R n - n R d S d S d S ( 3 73) (3 73), V S : V,, V,, ; V,, V V ; V, (3 73), V S, V, V n, ( r) = 4 S R n - n R V (3 73) d S ( 3 74) 3 q( 3 ) a o (), 4a S d S = q 4 0 d q 0 = q 4 0 d 3 3 (3 73), V a, o ( r ) = 0, R = a, 0 0 = ( r ) V 4 0 R d V + 4 S Rn - n R 0 = 4 S R ^n + d S R d S

114 = 0,, 0 = 4a ds + S 4a d S S = 4a d V + V 4a d S S = 4a d V + V 4a d S S C= 0 = 4a S d S 4a S d S = 0 = q 4 0 d ( 3 75) q,,,,, Sd S,, 0 0 n i ( 3 76) n i = i ( 3 76) (3 75 ), : ( ),, = 0 o, o o, (, o ) : ( ), q C = q 4 0 a, a, S, o S, 3 3(a), S P( r) ( r) = ( r ) 4 V R d V ( 3 77) R = ( r + r - rr cos ) / P, R, r, r rr cos, ( 3 77) lim = ( r ) d V r 4 V r r, r, r, 03

115 3 3 r = d V = 4 V q 4 r ( 3 78), r, r,, rr, E, r, E/ r, E r,,, V S ( 3 3 ( b) ) S, (3 73), S, (3 73), ( r ) ( r) = V 4 0 R d V + 4 S R n - n R d S ( 3 79) S SV, V,, ( r) = 4 S R n - n R d S ( 3 80),, ( r) V,, ( r) ( r) = ( r ) d V ( 3 8) V 4 0 R, ( 3 80) (3 8), S 3 6 (3 73), ( r ) ( r) = V 4 0 R d V + S 4R n - n 4R, 4R,, ( r ), S, n 04 d S

116 , = - / ( 3 8) G( ), G (3 8 ), ( 3 83 ), G = - ( r - r ) ( 3 83) G = - G/, G = - ( r - r ) ( G - G G) d V = V - V ( 86 ),, S G n - G n ( r) = G( r, r ) V d V + ( r )( r - r ) d V V d S = - G d V + ( r) V d V + S G n - G n d S ( 3 84) S V, ^n S,, ^n, ^n= - ^n,, G( r, r ) ( r) = V d V + S G n - G n d S ( 3 85), ( 3 83 ) G( r, r ) : ( ) S,, G S (3 83), ( 3 85), G S = 0 ( 3 86) G ( r) = V d V + G S n d S ( 3 87) ( ),, G S G n (3 83), ( 3 85), G ( r) = V S = 0 ( 3 88) d V - S G (3 ) n n + S n d S ( 3 89) = f ( 3 90) 05

117 ,, f, G3 G3 n + G 3 (3 83) G3, G3 (3 90), ( 3 9),, G3 G3 n + G 3 = f G3 G 3 n n - G 3 n + G3 = 0 (3 85) G3 ( r) = V S S d V - S = 0 ( 3 9) = f G3 fg3 d S ( 3 9) (3 87 ) (3 89) ( 3 9),, ;,,,,,,, ; ( ) ( ) (3 ) 06 G( r, r ) = 4R R = [ ( x - x ) + ( y - y ) + ( z - z ) ] / G( r, r ) = 4 R = [ ( x - x ) + ( y - y ) + ( z - z ) ] / R = [ ( x - x ) + ( y - y ) + ( z + z ) ] / R - R ( 3 93) ( 3 94)

118 , G( r, r ) = 4 R - a r R a, R, R, r3 4(a) G( r, r ) = 4 R - a r R a, R, R, r3 4( b) (4 ) ( 3 95) ( 3 96) G( r, r ) = ln r (3 97 ) (5 ) G( r, r ) = r ln r r = [ ( x - x ) + ( y - y ) ] / r = [ ( x - x ) + ( y + y ) ] / (6 ) (3 98 ) G( r, r ) = ln r r r a 3 4 ( 3 99), 3 5 ( a), 3 5 ( b) 3 5 3, G S = 0 ( 3 6 ), x = 0, 0 y b, G = 0 07

119 x = a, 0 y b, G = 0 y = 0, 0 x a, G = 0 y = b, 0 x a, G = 0 G = G x + G y (), = -( x - x ) ( y - y ) ( 3 00) 3 6 3,,, ( x, y ), 3 6,,, y = y, ( y < y ) ( y > y ), G : G + G = 0 ( 3 0) x y G x = 0, a = 0, G y = 0 = 0 ( G- G) y = y= 0, G - G y y = ( x - x ) y = y G x + G y = 0 ( 3 0) G x = 0, a = 0, G y = b = 0 ( G- G) y = y= 0, G - G y y,, G= m = G= m = y = y (3 03 )(3 04 ), (3 05) = ( x - x ) y = y Am sin m m x sh y ( 3 03) a a Bm sin m m x sh ( b - y) ( 3 04) a a A m sh m m = a y- B m sh m a ( b - y ) sin m x = 0 ( 3 05) a m = a m Am ch m a y+ B m ch m a ( b - Am sh m a y- B m sh m a ( b - (3 06), Am ch m a y+ B m ch m a ( b - (3 07) (3 08 ) 08 y ) = a a a 0 m ( x - y m ) sin a x = ( x - x ) ( 3 06) y ) = 0 ( 3 07) x m ) sin a xd x = m sin x( m =,,, ) ( 3 08) m a

120 Am = sin m m xsh a a ( b - m Bm = Am sh m a y y ) (3 09) (3 0 )( 3 03) (3 04), sin m m xsin a a G( r, r ) = x m = m sh m a b sh m a b ( 3 09) sh m ( 3 0) a ( b - y ) sh m m y sh a a ( b - sh m a ( b - y ) ( y < y ) m y) sh y( y > y a ) ( 3 ), 3 4,, ( 3 87 ) 3 3 a f (, ),, ( 3 96 ),, 3 4 ( b), ( r,, ) = S f(, ) G d S ( 3 ) n G( r, r ) = 4 R - a r R R = ( r + r - rr cos) / R = r + a - r a / r r cos rr G, G( r, r ) = G n 4 ( r + r - rr cos) / - G n = - G rr= a (3 4)(3 ), = 4 r r a + a - rr cos / ( 3 3) a - r 4a( r + a - racos) 3/ ( 3 4) a - r ( r) = S f (, ) 4a( r + a - racos) 3/ d S 0 0 = a 4 ( a - r ) sin d d f(, ) ( 3 5) ( r + a - racos) 3/ cos= coscos+ sinsincos(- ) 09

121 (3 5 ), f (, ),,,,,,, 9 3 3, R,, o, R, od,, 3 q, a (3 ), q, q, a (3 3 ), F, F q( q > 0 )(3 4), E = E0 ^x, x 3 5 q( 3 5 ),,, 0

122 q0, q, q q R , q, q 3 8 l x = y =, 3 7,, 3 9 ( x0, y0 ) l (3 8 ), , 3 9 a, U0, , l, b, a, 3, b, y = d y = b( - x ), U0, ( 3 0 ) z = 0 y 0 d, 0 U0, 3 3, ( x = a/ ) ( 3 ) U0,, 0 < x < a, z > ( ), E0 ^z,

123 3 3 5 E = E0 ^x, a ( ), 3 6 ( P = P0 ^z) 3 7, a, U0, - U0 ( 3 ), ( ) b(3 3), U0, U = 0, 3 9, 3 4, p, a, 3 a, b, a, d q, d > b > a,

124 3 a, z, z = 0 z = L,,, 3 3 a, L, 3 4 r = a r = a = z = 0 f ( x, y), ( z > 0) 3 6, = Acos, , = 0, = 0 r = a, 3 5 3

125 4, ( ) 80 ( Oersted),,, B( ),,, 4 B 4,,, Ia Ib,, Ia Ib/ R, R Ia Ib,, Ia Ib, 4, 4 () C C, I I,, I d l Id l, rr C, C, R = r - r, C C FC C 0, (4 ) = 0 4 C C I d l ( I d l ^R) R ( 4 ) 0 = H/ m ( 4 ),, C C FC C = 0 4 C C R = - R, FC C = - FC C I d l ( I d l ^R ) R ( 4 3) 4

126 4, ( 4 ) C C C C (4 ) FC C = C I d l 0 4 C I d l ^R R ( 4 4) ( 4 4 ) C r B( r), B( r) = 0 4 C I d l ^R R ( 4 5) (4 5 ) B, T ( [ ] ), T = Wb/ m (4 5 ) B, C r r B( r)r I, B( r) C C (4 5), I d l r d B( r) d B( r) = 0 I d l ^R 4 R I d l C B( r) B, (4 8) C = 0 I d l R 4 R 3 ( 4 6) d F = I d l B( r) ( 4 7) F = C Id lb ( 4 8), J, Jd V d B d B( r) = 0 J( r ) ^R 4 R d V ( 4 9), JS, JS d S d B B (4 ) ( 4 ) ( 4 5) d B( r) = 0 JS ( r ) ^R 4 R d S (4 0) B( r) = 0 J( r ) ^R 4 d V (4 ) V R B( r) = 0 JS ( r ) ^R 4 S R d S (4 ) (4 8), B F = JBd V (4 3) V 5

127 F = S JS Bd S (4 4) 4 I (4 ) z, z I d z P R R = + z ( 4 5 ) ^R = ^cos- ^zsin= R^- Id z P d B = 0 I d z( ^z^r ) 4 R z R^z ( 4 6 ) = 0 I d z 4( + z ) 3/ ^ ( 4 7 ) (4 7 ), I P B = 0 I ^ ( 4 8 ) 4 4 L I P I, I F = + L/ - L/ 0 I I L I ^z Bd z = - (4 9) ^ (4 9 ),, ( I I < 0) 4 JS^z y = 0, z JSd x( 4 3 ), (4 8), P x - x I I P y,

128 x, d B = - d B + d B ^x (4 0), I I P P ^ = - sin^x + cos^y (4 ) d B = - 0 JSsin B = - 0 ( x + y ) / d x^x = 0 JS sin ( x + y ) / d x^x (4 ) - 0 JS ^x ( y > 0) 0 JS ^x ( y < 0) (4 3) 4 3 J0 ^z, d 4 4, d y, J0 d y, (4 3 ),, B B = + d/ - - d/ 0 J0 d y ^x = - d/ - d/ 0 J0 d y ^x = y - - d/ 0 J0 d ^x y > d 0 J0 d ^x y < d d/ 0 J0 d y ^x + y 0 J0 dy ^x = - 0 J0 y^x - d y d (4 4) (4 8) ( 4 3 ) (4 4), 7

129 B, v q, S, l, 4 5,, = q Sl v ( 4 5 ) J= v ( 4 6 ), Fm = IlB = vsl B = qv B ( 4 7 ), Fm q v,, B, E, q 4 5 F = Fm + Fe = q( v B + E) (4 8) F ( 4 8) (Lorentz)89, d Fm = d qv B (4 9), V v Bd V = JBd V d Fm = S v Bd S = JS Bd S l v Bd l = Id l B (4 30), , I, y 4 6 (- y ), ^x B = B0 ^x, Fm = - e vb0 ^z (4 3) v, 8

130 , E, Fe = - e E Fm, ^y E 879,,,,, E 4 6 E, ( P P ) 4, 4, ;, ; 4 7 (4 7 ), S BdS = 0 (4 3) ( 4 3 ),,, BdS = S Bd V = 0 V, B = 0 (4 34 ), (4 33) (4 34) (4 34) (4 ), J^R R B = 0 4 J^R V R d V (4 35) = - J R = - R ( J) - J R (4 36) J = J( x, y, z ) (J x, y, z ),, J = 0, R = 0, (4 36 ), ( 4 35 ) 9

131 , 4 B L L S I, Bd l = 0 I (4 37) L, B L I, L P B ( 4 5 ), ( 4 37 ) Bd l= 0 I d l( d l ^R) L 4 L L R = - 0 I ( - d l d l )^R 4 L L R (4 38) 4 8, LP, P L d l P, L P = + d, d P P, L, P, L, d l, L L, LP d 4 8, ds = - d ld l, P 4 8 ( - d ld l )^R/ R, d l - d l, P, L - d l, P 0

132 (4 39 )( 4 38 ) L ( - d ld l )^R d= L R (4 39) Bd l = - 0 I L d= I (4 40) L 4 P, L : ( ) LL ( 4 9( a) ) P L,, = 0, L Bd l = 0 (4 4) 4 9 ( ) L L (4 9( b) ) L L SA, S I, LA A = (4 4) L, A (4 44 )( 4 40 ), L, A = - (4 43) = A - A = - - = - 4 (4 44) Bd l = 0 I (4 45) L Bd l = - 0 I (4 46) L, L L S I (4 0(a) ), 0 I, L

133 4 0 S (4 0 ( b) ), - 0 I i L S, B L i 0, Bd l = L 0Ii = 0 I (4 47) i (4 47) B S, S, B, J L S, (4 47) 0 Bd l = JdS (4 48) L S (4 48 ), (4 49 ) S, 0 BdS = JdS (4 49) S S B = 0 J (4 50) (4 50),,, ( 4 47 )( 4 48 ) ( 4 50 ) (4 50 ) (4 ), B = 0 4 J^R V R d V (4 5) ( 4 5 ), J x, y, z, (4 50) B, B, : L B ; B 4 4 3,

134 4 5 a I ( 4 ), B, ^, B = B ^ (4 5), L, L B 0 B= 0 I d= 0 I ( a) (4 53), L, J = I a, B = 0 J = 0 I a B= 0 I a ( a) (4 54) B= 0 I ( a) 0 I a ( a) ( 4 55 (a) ) ( 4 55 ( b) ) 4 6 R, R, R3 I (4 ), B 3

135 R B= 0 I R, B= 0 I R R R B= 0 I, B= 0 I R R3 - - R R 3 - R R3 B= 0, B= B= 0 I - 0 I R 0 I 0 I - ( - R ) ( R 3 - R ) 0 I R 3 - R ( R ) ( R R ) ( R R3 ) ( R3 ) - R, B= 0 I (4 56) (4 57) (4 58) (4 59) A, E = 0,, ( ),,,, B, B B, B, B A ( A = 0 ), B = A (4 60) (4 60 )A,, Wb/ m A, ( 4 5 ), d l^r R d l ^R R = - d l R l = d - R d l R d l d l, = 0, ( 4 5) R B = 0 I 4 d l L R = 0 I d l 4 L R (4 60 ) ( 4 6 ) 4 A = 0 Id l 4 L R (4 6) (4 6) (4 63)

136 i Ii J JS :, 0 Ii d li A = i 4 L i R ( 4 64a) A = 0 JS 4 S R d S ( 4 64b) A = 0 4 J V R d V (4 64c) (4 64 ),, ^z, ^z, B, (4 64) A, A, ( 4 64 ), A B, A?, ( 4 63 ), A = 0 Id l = 4 0 I L R 4 d l L R d l = d l R R + ( d l ) = - R R (4 65) d l (4 66) d l, d l= 0(4 66) (4 65 ), A = - 0 I 4 L R 0 I d l= - 4 S R d S (4 67) R = 0, (4 68 ) A = 0 (4 68) 4 3 A 4 3, ( 4 64c) A,, A, A A (4 60 ), (4 50) A = ( A) - A, A = 0, A = - 0 J (4 69) ( 4 69 )A,, A A = A x^x + Ay^y + A z^z, Ax, Ay A z Ax = - 0 Jx ( 4 70a) 5

137 Ay A z = - 0 Jy ( 4 70b) = - 0 Jz (4 70c) (4 64c)( 4 69 ), J = 0, ( 4 69 ), A = 0 (4 7), A L L S, Bd S = S ( A)dS = S L Ad l (4 7),,, f f, ( f ) = 0, (4 60 ), A( r) B, A = A( r) + f B, B = A = ( A + f ) = A (4 73) B,,, ( 4 60)A B ( 4 64) A A = 0,, B, A f (4 74 ) A = ( A + f ) = 0 (4 74) f = 0 (4 75) 4 7 L I (4 3 ) B,, B, (4 63 ) A(,, z ), Id zp P A = P R = [ ( z - A z^z = 4 0 I + L/ d z - L/ [ ( z - z ) + ] / ^z z ) + ] / (4 76) = 0 I 4 ln - z + L/ + [ ( z - L/ ) + ] / - ( z + L/ ) + [ ( z + L/ ) + ] / ^z (4 77) B = A = - A z ^ = 0 I - z + L/ 4 [ + ( z - L/ ) ] / +, L, 6 z + L/ [ + ( z + L/ ) ] / ^ (4 78)

138 B = 0 I ^ (4 79) LP(4 4) P A = 0 I d l 4 L r - r r - r = ( r + r - rr ) /, =, = r - r r + r r - rr r / = (4 80) r ( + ) / (4 8) r - rr P, rr, ( 4 8) r r - r = r = r r - ( + ) / = - + rr - r (4 83 )( 4 80 ) - rr r + r r r + 3 r (4 8) - rr r + r r + rr + (4 83) r A( r) = 4r 0 I d l+ 0 I L 4r 3 ( rr ) d l+ (4 84) L, Ld l= 0, ( 4 84 ) LS, 7

139 ( rr ) d l= - L ( rr ) d S (4 85) S ( rr ) = ( r ) r+ ( r ) r + r(r ) + r ( r) (4 86) r = 0, r= 0, ( r ) r = 0, ( r ) r= r, ( rr ) d l= L - S rds=ds r S P A( r) = 0 4r 3 S IdS r = 0 4r 3 ( mr) (4 87) m= S IdS, ( ), B = A = 0 4r 3 [3 ( m^r) ^r - m] (4 88) m= m^z, (4 88) B = 0 m [cos ^r + sin ^] (4 89) 4r 3 a,, z, I z, m =a I^z B, I, A A J0, J = 0, A, A, A 4 4,,,, 4 4,,, :,,,,, 8

140 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, M, M = lim V 0 m i V (4 90) i m V,,, N M A/ m M = Nm (4 9) 4 4,,,,, 4 8 A = 0 m^r (4 9) 4 R P ( x, y, z ) Md V, P ( x, y, z ) ( 4 5 ) d A = 0 M ^R 4 R d V (4 93) P, A = 0 4 R V M ^R d V= 0 4 M V R d V (4 94) 9

141 4 5 V, ( 49 ), (4 94) M R = R M - M R (4 95) (4 94 ) A = 0 4 R V V M R 0 (M) d V- 4 M V R d V (4 96) d V= - M ^n d S (4 97) S R SV, ^n S, P A = 0 4 M V R d V+ 0 4 M ^n d S (4 98) S R (4 98 )( 4 64b) (4 64c), M Jm, M^n JSm, Jm = M (4 99) JS m = M ^n ( 4 00) Jm JSm, P,,, P Bm = 0 4 JSm ^R S R d S+ 0 4 Jf JSf Jm, 30 B = 0 ( JSf + JS m ) 4 S R V Jm ^R R d V ( 4 0) JSm ^Rd S+ 0 ( Jf + Jm ) 4 ^Rd V ( 4 0) V R A = 0 ( JSf + JS m ) ( Jf + Jm ) d S+ 4 S R V R d V ( 4 03)

142 , b, l, M, M = M^z, M, (4 99 ) Jm = M = 0 (4 00) JSm JSm = M ^n = 0 = M ^n = M^z ^n = M ^ ^ M, P B, d z, d z Md z ^, d B d B = ^z 0 Mb d z [ ( z - z ) + b ] 3/ l B = 0 M d B = ^z 0 z z + b z - l ( z - l) + b ( 4 04) H, m, r H, Jf B B = 0 Jf,,,,, ( ) B = 0 ( Jf + Jm ) ( 4 05) Jm,,,, Jm = M (4 05 ), B = 0 ( Jf + M) = 0 Jf + 0 M B - M = Jf = H ( 4 06) 0 H = B - M ( 4 07) 0 H,, H M, A/ m B = 0 ( H + M) ( 4 08) (4 08 )B H,,, 3

143 (4 06 ),, S, ( H)dS = S JfdS ( 4 09) S, Hd l = If ( 4 0) L L S ( 4 0 ) H L, H, B L Bd l = 0 ( If + I m ) ( 4 ) L Im L, B = 0 ( H + M ), (4 0), L Md l = Im ( 4 ) M L B = 0, Jm m r,,,, M H, M H, M = m H ( 4 3) m,, B B = 0 ( H + M) = 0 ( + m ) H = H ( 4 4) = 0 ( + m ) = 0 r ( 4 5) r = = + m ( 4 6) 0, r m, m > 0, m < 0, m = 0 r, = 0 ( ), (4 3 ) r,, ;,, B H 3

144 Bx By B z = H x + Hy + 3 H z = H x + Hy + 3 Hz = 3 H x + 3 Hy + 33 Hz ( 4 7),, B H (4 4 ), B H : B = H ( 4 8),,,,, 4 0, a, If ( 4 7 ), ( ) H, L, L Hd l = H = If H = I f ^ (> 0 ) ( 4 9) ( ) B, B, B = H = I f ^ ( 0 < < a) ( 4 0) B = 0 H = 0 If ^ (> a ) ( 4 ) (3 ) MM = B - H, 0 M = H - H = 0 - H = (4 ) If ^ (0 < < a) ( 4 ) M = 0 (> 0 ) ( 4 3) 33

145 Jm = M = < a, Im = L Md l = L ^ ^ ^z = 0 ( 4 4) z 0 M 0-0 If ^ d ^ = - If ( 4 5) 0 (4 5),, Im (= 0), If, JS m = M ^n = - 0 If 0 a ^ ^= - I Sm = ajsm = If ( 4 6) 0 a^z - 0 If = - Im ( 4 7) 0 4 5,, 4 5 B 4 8 (a),, h0, S,, Bd S = ( B n - B n )S = 0 ( 4 8) S ^n( B - B ) = 0 ( 4 9) 4 5 H 4 8 ( b), L, h0,, l, L 34

146 4 8 ^n, ^t, ^t^n = ^n, Hd l = ( H t - H t )l = ( JS f^n )l ( 4 30) L ( H - H )^t = JSf( ^t ^n) ( 4 3) JSf, JSf = 0,, H t = H t ( 4 3), ( ),,, ( 4 9 ) ( 4 3 ) ( 4 8 ( b) ) tan tan = ( 4 33) M Md l = Im ( 4 34) L (4 34) 4 8 (b ) ( M - M ) ^t = JS m (^t ^n) ( 4 35) JS m A A, A, A, 35

147 A = A ( 4 36) 4 6,,, L Hd l = 0 ( 4 37) H = 0 ( 4 38) m, H = - m ( 4 39) m, A ( 4 39 ) E = -,,,, m = H, H, (4 34 ) B H ( 4 08 ), [ 0 ( H + M) ] = 0 ( 4 40) H = - M ( 4 4) m, m = - M ( 4 4) A/ m m H = m ( 4 43) H = ( - m ) = - m = m m = - m ( 4 44) = 0, m m = 0 ( 4 45), ( 4 4 ), M,, B, 0 ( H n + M n ) = 0 ( H n + M n ) ( 4 46) H n - H n = M n - M n ( 4 47), S m, S m = M n - M n = ( M - M ) ^n ( 4 48) 36

148 ,,,, m = 4 m V R d V+ 4 S m S R d S ( 4 49) m, (4 44 ),, H t = H t, m S = m S ( 4 50), B n = B n,, m n S m = n S ( 4 5), 4 3 A, Jm JS m, m S m,, A Jm JS m, m S m ;, m,,, 4 a (4 9 ), M = M 0 ^z, z B H : M = M0 ^z, JSm Jm = M = 0 ( 4 5) = M0 ^z ^r = M0 sin ^ ( 4 53),, (4 9( a) ), B( r) = 0 4 d S JSm ^R S R (4 55)(4 54 ) d S= 0 4 M0 sin ^ R S R 3 d S ( 4 54) d S= a sin d d ^ ( 4 55) B( r) = 0 M0 a ^ R R 3 sin d d ( 4 56) z P, z, 4 9 (b ), 37

149 4 9 4 R = r = z^z, r= a^r ( 4 57) z^z - a^r= ( z - acos ) ^z - asin ^ ( 4 58) R = ( z + a - z acos ) / ( 4 59) ^ (4 58 ), ^ R = ^ ( z - acos ) ^z - asin ^ ^ = ( z - acos ) ( cos ^x + sin ^y) + asin ^z ( 4 60) (4 59) (4 60 )( 4 56), cos d = sin d = 0 ( 4 6) 0 0 B = 0 M0 a 4 3 sin 3 d d ^z 0 ( z + a - azcos ) 3/ = 0 0 M 0 a 3 ^z ( z > a) 3 z M0 ^z (0 < z < a) z < 0, z B = - 0 M0 a 3 3 z 3 ^z ( z < - a ) 3 0 M0 ^z ( - a < z < 0 ) ( 4 6) ( 4 63) z B, M, z B ^z (z < 0 z 3 < 0 ) H = B - M, z H 0 38

150 H = M0 a 3 3 z 3 ^z ( z > a) 3 M0 ^z - M0 ^z = - 3 M 0 ^z ( z < a) (4 64)H M : M, (4 0) m = - M = 0 ( 4 65) Sm = M0 ^z^r= M0 cos ( 4 66), z m = 4 = 0 0 M0 a 3 M0 cos a sin ( z + a - a zcos ) / d d 3 z ( z > a) M0 z 3 (0 < z < a ) H = - m = M0 a 3 - ( 4 67) 3 z 3 ^z ( z > a) 3 M 0 ^z (0 < z < a) z < 0 z H H = - m = z B B = 0 ( H + M) = - M 0 a 3 3 z 3 ^z ( z < - a) - 3 M 0 ^z ( - a < z < 0) ( 4 64) M0 a z 3 ^z ( z > a) 0 -, ( 4 68) ( 4 69) M0 3 ^z + M0 ^z = ( 4 70) 3 0 M0 ^z ( z < a) 4 7, A m,,, 4, 39

151 d I( 4 ), B,, I I, I, I (4 ( a) ), 4 4, I, I I ( 4 ( b) ) II I, H t = H t B n = B n, I= - I ( 4 7) + I= I ( 4 7) I, II,,, 4 3 ( a) ( b) = 0, = 9 0 = 9 0, = 0 B B 40

152 ( ), = 0 ;, ;( 4 7) (4 7) I= I, I 0 ( 4 73) H, B = lim H = lim I = lim 0 I + 0 = 0 I ( 4 74) ( ), = 0,, (4 7 )( 4 7 ) I= - I, I I ( 4 75),,, 4 3 J = J0 ^z ( a) ( 4 76) A, BJ ^z, A ^z, A, z (< a) A, (> a ) A,, A z = A z = - 0 J0(< a) A z = A z = 0 (> a) < a, ( 4 77), B ( 4 77) ( 4 78) A z = J0 3 + C ( 4 79) A z = J0 + C B = A = - A z ^ = 3 0 J0 - C ^ = 0, B, C, ( 4 80) B > a, ( 4 78), = - A z ^ = 3 0 J0 ^ (< a) ( 4 8) B A z = C ( 4 8) B = A = - A z ^ = - = a, H, C ^ 4 ( 4 83)

153 A z > a B = a C = - = 0 J0 3 = A z = a 4 4 (a), ( b)j B ( 4 84) 3 0 J0 a3 ( 4 85) a 3 ^ (> a) ( 4 86) J z B 4 4 a, M = M 0 ^z,,, o, 4 5, M = M0 ^z, m = - M = 0,,, m r m r r 3 4 3, ( 4 87 ) m ( r,) = An r n P n (cos n = 0 + sin sin m = 0 ( 4 87) ) + n = 0 Bn r - ( n + ) P n (cos ) ( 4 88) r < a, m, r = 0 m, ( 4 88 ) Bn, m( r, ) = n = 0 An r n P n (cos) ( r < a) ( 4 89) r > a, m, r m, (4 88) An, 4

154 m r = a m r r = a = Bn r - ( n + ) P n (cos) ( r > a) ( 4 90) n = 0 - m r m r = a = m r = a ( 4 9) r = a = - ( M - M )^r = - Sm ( 4 9) M = M0 ^z, M = 0, Sm = M 0 cos= M 0 P ( cos ) (4 9 )( 4 9 ) ( 4 89) ( 4 90 ), An = Bn = 0 ( n ) m= 4 3 a3 M H = - m = - A = 3 M 0, B = 3 M 0 a 3 m = 3 M 0 rcos( r a) ( 4 93) m = 3 M a 3 0 r cos= m^r 4r ( r a) ( 4 94) H = - m = - 3 M 0 ( cos ^r - sin ^) = - = 3( m^r) 4 r 3 ^r - B = 0 ( H + M), B B = 0-3 M 0 a3 - cos ^r - sin ^ r 3 3 M 0 ^z ( r < a) ( 4 95) r 3 m r 3 ( r > a) ( 4 96) 3 M 0 ^z + M0 ^z = 3 0 M0 ^z ( r < a) ( 4 97) = 0 3( m^r) 4 r 3 ^r - (4 95 )( 4 97 ),, H m r 3 ( r > a) ( 4 98) M = M 0 ^z, B M (4 96)(4 98 ), m 4 b ( 4 ) I, a P B B 4 a I, B 4 3 K I, 43

155 B = 0 K I tan d K d K, B I I, I I ( sec / - ), 4 5 d, a ( 4 ),, J= J0 ^z, B 4 6 a ( 4 3 ), J = J0 ^z, b, d, P R = 6mm, R = 7mm ( 4 4 ), I = 00A, 4 8 R, D, I ( 4 5 ), 4 9 z B 44 Jz = r + 4 r ( r a)

156 4 0 J = J0 r^z ( r a) A B, B A = - 0 J J A = 0 4 V R d V 4 a q,, 4 3 J J, J = J J = - J B 4 4 a,, M = ( A z + B) ^z, A, B 4 5,, a, = 0, = 9 0 ( 4 6 ) ,

157 5 5 5,, ei, : ei = - d dt = - d B n dt ds ( 5 ) S ei ( 5 ) ( 5 ) E, E ei ei = - ei = C E dl ( 5 ) d dt = C E dl = - d dt S B ds ( 5 3) ( 5 3) C, C 5 ei C B, C ( ), N, N, ei = - d = - dt d dt N ( 5 4) = 5, t, ( 5 3),, 46

158 ei = C E dl = - S ( 5 5), S E n ds = - S E = - B n t ds ( 5 5) B n t ds ( 5 6) B t ( 5 7) ( 5 7),,,, B, B= 0, B A B= A, ( 5 7) E = - E + t ( A) A = 0 ( 5 8) t E+ A t,, E + A = - t E = - A - ( 5 9) t, A, A = 0, t E = - ( 5 0) ( 5 9) :, - ;, - A t 5 3, B C v t Ca d dt = lim t 0 t = Cb ( 5 ), lim t 0 B( t + t S b B( t+ t) t+ t Cb S b Ca S a C Ca t) ds S - B( t) ds ( 5 ) a, B( t) t Cb V S, t+ t S ( S= S a + S b+ S ), 47

159 5 S B( t + t) ds = S B( t + t) ds b - S a B( t + t) ds S + B( t + t) ds = 0 ( 5 ) S a S a t 0, B( t+ t), B( t + t) = B( t) + B t t + ( 5 3) ( 5 3) ( 5 ) S a B( t + S B( t + S a t) ds S = B( t) ds + t a t) ds = S B( t) ds + t S ds= dlv t, ( 5 5) S B( t + t) ds = t Ca B( t) ( dlv ) + ( 5 4) ( 5 6) ( 5 ) S b B( t + t) ds S - B( t) ds= a t S a + B( t) ds + ( 5 4) t B( t) S ds + ( 5 5) t ( t) Ca, ( 5 7) ( 5 ), C Ca d dt = S = S B( t) ( dlv ) + t B( t) ds + t Ca ( B( t)v ) dl ( 5 6) t ( 5 7) B ds + t C ( Bv ) dl Cb, B ds + t S ( Bv ) ds ( 5 8) ( 5 8) B S 48

160 ,, C C E dl = - S B ds t + S [ Bv ] ds ( 5 9), ( 5 9), S E ds = S ( vb) - B ds ( 5 0) t ( 5 0) C S,, E = - B + ( vb) ( 5 ) t ( 5 ), v = 0, ( 5 ) ( 5 7), B = 0, B, ( 5 ) t E = ( vb) ( 5 ), 5 h w ( 5 3),, B= B 0sin( t) z, z, ei ei = C E dl = S E ds = - S B = B 0 cos( t) z t ei = S - B0 cos( t) z n ds = - B ds t B 0 Scos cos( t) 49

161 S = hw 5 x ( 5 4),, B= B 0z, ei = C E dl S = ( vb) ds = C ( vb) dl = B 0 Ssin( t) S = hw 5, J f , S, t, S V,, S ( ) V, S J f ds = - q t = - VdV ( 5 3) t V q V, V V ( 5 3), ( V ), ( 5 4), V J f + V dv = 0 ( 5 4) t J f + V = 0 ( 5 5) t ( 5 3) ( 5 5),, E J f V, S J f ds = 0 ( 5 6a) J f = 0 ( 5 6b) 5 3,, P 50 = 4 dv ( 5 7) 0 V R

162 R,, J P A = V 0 J 4R R, A dv ( 5 8), J, = ( t), J = J ( t),, J P? P t A t? ( 5 7) ( 5 8)?, J, ( 5 7) ( 5 8) P A, R,, t P A R J, t- R/ c ( c ), t R/ c P A, J A, ( Maxwell) J d, ( ) 6,,, P A J,?,, ( R/ c, ),, J d, L, f, Lf c, L30cm, f = 50H z, L f Lf = 3050 = 500( cm / s) c 3cm, L. 3cm, f = 9375MHz, Lf = cm / sc,, A ( 5 7) ( 5 8), J f= 0 5 3, I ( 5 6), I,,, l H dl = I

163 a I Ba = b I Bb = - 0I ^ a 0I ^ b B = Ba + Bb = 0I - a = S B ds = R 0I R a a + w = 0hI ln ( R a + w ) R b ( R b + w) Ra ei = - d = - dt h a d a ^ b R + w b - R b 0h di ( Ra ln + w ) R b dt ( Rb + w ) Ra h d b b di / dt> 0, ei< 0, 5 6 R b, a ei = - 0h di dt ln ( R a + w ) R a = 0h di dt ln R a ( R a + w ) 5 4,,,,, ( 5 7), ( ),, J f = 0 ( 5 9) S J f ds = 0 ( 5 30), ( 5 30) k = 0 ( 5 3) k 5 7 ( 5 3), ( k ),,, 5

164 ,,, E = 0 ( 5 3) E = - ( 5 33) l E dl = 0 ( 5 34) k U k = 0 ( 5 35) ( 5 35), N, U I ( =,,, N ), E = 0 ( E = - S N P in = U I = ) H = J f ( J f = 0) P in = - S ( EH ) ds = - V ( EH ) dv = V ( H ) dv ( 5 36) ( 5 36) S ( H ) = H ( ) - ( H ) ( 5 36) = - J f = - ( J f) P i n = V - ( J f) dv = - S J f ds S U,, P in = - S J f N ds = = N - S U J f ds = = U I 5 5, l a l b( 5 9), l a I a, l b ba ( l a ), ba 53

165 a I a Mba, M ba l a ba = Mba I a ( 5 37) l b,, I = LI ( 5 38) 5 9 L, L ( 5 37) ( 5 38), t l b l a l a I a l b S b ba ba S = Bba dsb ( 5 39) b Bba S b I a, Bba = Aba,, I a ba = l b ba S = ( Aba ) dsb = Aba l dl b ( 5 40) b b Aba ( 5 8), 0 4 l a I adl a R Mba = dl b = l b l a dl a dlb 0I a 4 0 dla dl b 4 l l b a R R = M bai a ( 5 4) ( 5 4), ( 5 4),, ( 5 4) b a a b S = Ba b dsa = M a bi a ( 5 43) a M a b= M ba,, M, H ( ) W b/ A, a I a ei = l b E dl = -, b d dt = - M di a dt ( 5 44) ( 5 44) H A / s, V, H 54

166 ( 5 0),, L = 0 4 l l dl dl R ( 5 45) dl dl l R, dl dl R= 0, ( 5 45),,, ( 5 45),,, dl dl I, ei = l E dl = - d dt = - L di dt ( 5 46) , di / dt> 0, ei< 0, H, A / s, V, H 5 5 0, N ( 5 ), B N = N R R L = rb = B = 0N I 0N I r 0N I r adr = = 0N a I ln R R 0N ai ln R R 5 65 I, ( 55