陈维恒微分几何答案

rXYX C(0,r)M(t C XMCC 0,0)CM MCMNM=( ( ), ( )x ty t ( )x tM COCMCM Csinrtrt ( )y tCNCCcosrrt M (sin ,cosrtrt rrt) 2 ( )(3,6 ,6 )r ttt 2 * 2 1 ( )(1,2 ,2 ) 12 rttt t *( )rt 123 ( ,)c cCc 2 * 123 2 1 cos( ),)( )(22) 12 rt CrtCcc tc ta t a 222 123 1ccc 132 2 ,0 2 ccac ( )r t 22 (,0, 22 ) 4 clPll cP c( )rr tP 0 tt cl 1/ / ll 1011 ( )|,lcrr tttttt 012 1, )i l/ /ll i ( i dist ldist l, ) i l 010 ( )|, ii crr tttttt2 i l2,3,4,i 12 / / / / / / / n llll n ll 0 ( r tt 201 12 )() / / nn nn r tt l tt P 0 ( )r t 0201 0 12 ()() ( ) nn nn r ttr tt r t tt 0 /( )lr t/lcP ( )r t 0 t( )x 0 0t 0 t ( )yf x( )xzg 0 ( )0 x t 0 ( )0 x t 0 t 0 ( )t( )xx t 0 ( )t 0 ( )t( )xx t( )xth 0 ( )t ( )( ( )( )yy ty h xf x( )( ( )( )zz tz h xg x 0 t( )yf x( )zg x 222 1xyz 22 xyx 0z 222 2 1 1x 22 2222 11 () 24 xyz yz xyxxy 1 sin 2 yt 11 cos 22 xt 11 cos 22 zt02t 11111 ( )(,s )ttcos ,sinco 22222 r tt 2.2 1.,0a 1,.( )(,)r tacht asht at0tb 2 x yach a ,0,x. 3,( )( cos , ln(sectan )sin )r tat attat0,t. 1 222 ( )(, ),|( )|(1)2.r tasht acht ar tash tch tacht 00 |( )|22. bb sr tdtachtdtashb 2, ( )( ,),( )(1, ttt xtyachr tt achr tsh aaa). 2 00 |( )|1 xx tx sr tdtshdtash aa. 3 1 ( )(sin , (cos ) cos r tat at t 2222 2 00 1 |( )|sin(cos2)lncos . cos tt sr tdtatatdtat t 2. dr ds 1( )( cos , sin ,),0.r tat at bt a 2 33 ( )(cos,sin,cos2 ).r tttt 1( )(sin , cos , )r tat at b 22 11 (sin , cos , ). |( )| drdr dtdr at at b dsdtdsdt ab r t 2 22 ( )( 3cossin ,3sincos , 2sin2 )r tttttt 22 ( )1 ( 3cossin ,3sincos , 2sin2 ). |5sin cos | |( )| drr t ttttt dstt r t 3.c 22 22 1, ,0. xyz xacha b aba ( ,c0,0)a ( , , )x y z. ,zt, tt xachybsh aa ( )(, ) tt r tachbsht aa c ( )(,1) tbt r tshch a aa 22 22 00 |( )| zz abtz sr tdtchdtabsh aaa t e4. ,.( )rr t 2 (0)(1,0, 5),( )( , ,) t rr tt 2 ( )( , ,) t r ttt e 32 11 ( )(,) 32 t r tttec,c. ,0t(0)(0,0,1)(1,0, 5)rc (1,0, 6).c 32 11 ( )(1,6). 32 t r ttte 2.3Frenet 1. 1=,2ln ,.0 a rat ata t 2 323 = 3,3 ,3.rttttt 3=sin,1 cos,.ra ttatbta0 t4 33 = cos,sin,cos2 .rtt 1 22 22 ( )( ,),( )0, aaaa r tart tttt3 2 422 432 222 ( )( ),. aaa r trt ttt 2 2 2 3 |( )( )|2 . 1 |( )| r trtt a t r t 2 22 ( )(33 ,6 ,33 ),( )6 ,6,6 ,r ttttrttt 22 ( )( )81, 2 ,1 .r trttt t 2 2 3 |( )( )|1 . 31 |( )| r trt t r t 3( )(1 cos, sin , ),( )sin , cos ,0 ,r tatat b rtat at 2 ( )( )cos ,sin ,cos1 .r trtabt abt at 224 3 3 2 222 4sin |( )( )| 2 . |( )| 4sin 2 t a ba r trt r tt ba 4 22 ( )( 3cossin ,3sincos , 2sin2 ),r tttttt 2323 ( )6cos sin3cos,6sin cos3sin, 4cos2 ,rtttttttt 233222 ( )( )12sincos, 12sincos, 9sincos.r trttttttt 3 |( )( )|3 . 25|sin cos | |( )| r trt tt r t 2. 1 22 ( )=cos , sin ,0.r tat at btab 2,( )=cos , sin , t r tat bt e0t,0ab. 1( )sin , cos ,( )=cos ,sin ,0 ,r tat at brtatat 2 ( )( )sin ,cos ,.r trtabtabt a =0Xr,( )( ) =0Xrr trt .sincos0bt xbt yazabt 2( )sin , cos ,( )=cos ,sin , tt r tat bt ertatbt e ( )( )cossin,sincos,. tt r trtbettaettab =0Xr,( )( ) =0Xrr trt ,=0t,0,1 ,.rarrba ab ,2 xy z ab . 3., sin 1ln1 z xshxyy zexx 0,0,0Frenet. ,xx syy szz s, . s0s 0,0,0,x sy sz s 222 sin1 (1)ln12 13 z xshxyy zexx xyz 1 , 2,s cos4 5 1 z xchx xyy y x zezx x ,0s 1 000 3 xyz. 333 0(0)(, 333 r). 3 , 4 , 5,s 22 2 2 2 0 sincos* 1 1 zz xxyyzz xshx xchx xyy yy y xx ze ze zx x x ,0s 121 1 00,0,0, 999 9 xyzr 2 . 9 6 |0 | 9 r, 066622 0,000, 66322 |0 | r r ,0 . 4. 222 22 +=9 =3 xyz xz 2,2,1 ,xx syy szz s, . s0s 2,2,1,x sy sz s 222 22 222 91 32 13 xyz xz xyz 1 , 2,s 04 05 xxyyzz xxzz ,0s 12 0,0,0 33 xyz 2. 3 12 2 00, 33 3 r. 3 , 4 , 5, 222 22 0 0 0 xxyyzz xxxyyyzzz xxxzzz ,0s 11 00,0,0, 33 xyz 11 00, 33 r 022 |0 |,00, 32 |0 | r r r 2 , 2 2 222 0 =00 =. 366 2 222 2210 366 xyz0,49xyz. 5. 2 2 1 1 , ,0 ,0 ( )0,0,0 ,0 0, ,0 t t ett r tt t et , ,0t00t0t 2 1 3 2 0,( ),1,0 t tr te t 2 1 3 2 0,( )0,1, t tr te t 0,(0)0,1,0 ,(0)0,1,0 ,(0)0,1,0 .trrr ,( )0.t r t. 2 1 64 46 0,( ),0,0 t trte tt 2 1 64 46 0,( )0,0, t trte tt 0,(0)0,0,0 ,(0)0,0,0 ,(0)0,0,0 .trrr 0.0t , 2 2 1 3 1 6 ( )12 ( ),1,0 4 |( )| 1 t t r t te t r t e t 0t 2 ( )( ) ( )=sgn 460,0, 1 | ( )( )| r trt tt r trt 2 2 2 1 3 1 6 sgn 46 2 ( )( )( )1,0 4 1 t t t ttte t e t , 2 2 1 3 1 6 12 ( )0,1, 4 1 t t te t e t 0t 2 ( )sgn 461,0,0tt 2 2 2 1 3 1 6 sgn 46 1 ( )0,1 4 1 t t t te t e t 0,0,1,0 ,0,0,1 ,1,0,0t 0,0,1,0 ,1,0,0 ,0,0, 1t 2.4Frenet 1.31. 1=,2ln ,.0 a rat ata t 2 323 = 3,3 ,3.rttttt 3=sin,1 cos,.ra ttatbta0 t4 33 = cos,sin,cos2 .rtt 1 22 22 ( )( ,),( )0, aaaa r tart tttt3 2 34 2 2 2 2 2 26 ( )= 0, ( ),( ),( ) 2 1 |( )( )| aa rt tt r t rt rt t a t r trt 2 22 ( )(33 ,6 ,33 ),( )6 ,6,6 ,r ttttrttt 2 2 2 ( )=6,0,6 ( ),( ),( ) 1 31 |( )( )| rt r t rt rt t r trt 3( )(1 cos, sin , ),( )sin , cos ,0 ,r tatat b rtat at 2 22 2 ( )=cos ,sin ,0 ( ),( ),( ) cos1 |( )( )| rtatat r t rt rt b bat r trt 4 22 ( )( 3cossin ,3sincos , 2sin2 ),r tttttt 3232 2 ( )=6sin21cossin ,6cos21sincos ,8sin2 ( ),( ),( ) 8 25sin2 |( )( )| rtttttttt r t rt rt t r trt 2323 ( )6cos sin3cos,6sin cos3sin, 4cos2 ,rtttttttt 2.33.0,0,0 sin 1ln1 z xshxyy zexx 3331 12 (0)(,),0,. 3339 99 rr ,* 222 33 2 3 3 24 0 3cos3sin 1212 3 11 zzz xxxyyyzzz cos 1 xchx xshx xxchx xyy yy yyy y x xxxxx xxx ze ze zze zx xx ,0s 83 03, 818181 r 3 2 0 ,0 ,0 1 0 2 |0 | rrr r 3.,( )rr s 1 ( )( )rss ds. 1 r, 2 1 r, 2 1 2r 32 1 2 35 22 ,rrr rr 2 32 , | |,. | r rr rr rrr 4. 2 2 11 + d ds . 11111dddd r dsdsdsds 2 2 11 + d ds ,s 212111 0 dddd dsdsdsds 2 , 1111 0 dddd dsdsdsds 1 0 d ds , 11 0 dd dsds , 111 0 dd r dsds 0 111d rr ds 0 r,. 0 |rrc,.c 1 0 d ds , 5.( ) s, ( )( )( ),( )( )( ), ( )( )( )sssssssss ( ) s,( )( ) ( )( ) ( )( ) ( )sa ssb ssc ss s , ( )( ) ( )( ) ( )( ) sabcsbc sabcsac sabcsab ,0, ( )( ) ( )( ) ( ) abc ssss 6.1, 2, 3,. 1,c( )( )0r scr s( )( )0r scs. ,( )( )0r scr s,( )0r sc. ,.( )0r sc0 2c,.0rc ,0rc. 01,.0rc 3c,0rc0rcr, 0 d rcrc ds ,|rcc,. 7. 123 ( );( ),( ),( )r ssss( )r s, 3 1 i ijj j d ds ,1,3i0 ijji . 0( )( ) j i ijjiijji d d ss dsds . 8. 33 22 1+1 ( )=, 11 332 sss r ss,s 111 ( ), |( )| 1 222 ss r sr s s. 2 112 ( ),0 ,| ( )|. 4 14 1 4 1 r sr s ss s 33 22 2 2 , , 112 ( )1,1,0 ,. 88 4 1 | | r r r r sss s r 111 ( )( ), 222 ss sr s ( )11 ( ),0 22 | ( )| r sss s r s 111 ( )( )( ),. 222 ss sss 9.( ) s( )rr s 2 1, 2 1 d ds ,Frenet. 2 , 32 32 2 3 2 2 | 1. | , , . | 1 . | d ds 2 1 . | 1 2 1 . 1 10.rr t; ( ),( ), ( )r tttt 33 ,| |sgn 2222 ,|,|rrrrrr 23 |r, 222 ,|,|rrrrr 2 23 |r, | |rr | | |sgn|rr,.| 33 ,| | ,sgn. 2.5 1., ,0 . ,.( )rr ss, . .a st( ) saconst. ,( )0sa( )0sa0,( ) s. 22 010 0,0 2 0 d ds ,const. c,c,cc. ,scaa. ,0ca,ac,( )r s. 2.cc , 0 ( )( ) s t ss ds( ) dt s ds . c dds dtdt d c dt d c dt 2 2 2 1 ddd cc dtdtdt , 22 cos 1sin 1Ac tBc t. 2 |1 22 222222 cos1sin12cos 1sin 11c t Ac t BABc tc t 2 2 1 1 0 A B AB 1 2 Ae Be , 22 cos 1, sin 1,0c tc t d dt t, 22 2 1 sin 1,cos 1,0 1 c tc ta c 2 |1 222 12 2 | c 2 12 sin 1cos 11 1 1 ac t ec t ea c 0t 2 1 t c , 2 2 | | 0,| 1 c eaa c 2 2 1 t c 2 3 2 1 t c , 1 0e a ,a 1 e 2 e 3 2 1 c ae c 22 2 1 sin 1,cos 1, 1 c tc t c c ,s 22 200 1 sin1( ), cos1( ), 1 ss rcs ds dscs ds ds cs c 3.( )3sin ,2cos , 3sinr tttttt 1 2cos,2sin, 22 uu r uu . ( )13cos , 2sin , 3cosr tttt, ( )3sin , 2cos ,sinrtttt ( )3cos ,2sin ,cosrtttt, ( )( )2 3cos2, 4sin , 2cos2 3r trtttt, ,8r rr. ,r rr 32 |11 , 44 | rr rrr . 1 ( )sin,cos, 1 22 uu ru, 1 11 ( )cos,sin,0 2222 uu ru, 1 11 ( )sin,cos,0 4242 uu ru, 11 11 ( )( )sin,cos, 22 22 2 uu ruru 1 , ,8r rr. 111 11 11 32 111 , |11 , 44 | rrr rr rrr . 11 ,. r t. 1( ) r u 4. 1: ,crcht sht t , 2: , 22 uu ee cru1 ,. 3 E 1 c 2 c ,s, 22 tttt eeee chtht tt 11 0 22 11 0,0, 1,10 2222 001 tt ee t 11 0 22 11 0 22 001 1 c 2 ,c. 2.6 1., .,Bertrand. ,. 12 ,c c 1 c 1111 ( );( ),( ),( )r ssss 1( ) s 2 c 1 c s 21( ) ( )r ss( )r s s 221111111 |1r 1 1 0(s)( ) sc 21 |( )( )| | ( )| | |r sr ssc 1 c 2 c 12 1212 d ds 11222121221 |0/ /,r 12 ( )( )ssc 1 c 2 c 0. .1st c rk:( r s) s ( )r s 1( ) ( )( )r sr ss s 1 1r 1 1 2 22 1 1 1 1 | r r 11 2 22 1 cos,cos 1 00 0 1 ctg 0 1ctg 0 ctg 1( ) ( )( )r sr ss 1 rr 1/ / 1 2 22 1 1 1 1 22 sgn 1, s 111 22 sgn |r 1/ /1( ) r s( )r s( )r s 1 c 2 c 1 c 2 c 2 c 1 c 1 c 2 c 1 c 1( ) rr s s 1c c 2211 ( )( )( )r sr scss c 1( ) s c1 2211 :( )( )( )( )cr sr ssss 1 r 2 r 1212 r 1 21 1r 1121 10rsc 1 c 1( ) rr s 1( ) s 1 c 1 ) s 2 c 2 c 21 ( )( )( )( ) (r sr sss 1 rs 2 ( )/ / 1 22111 :( )( )( )( )( )( )cr sr sssss 1211111 1-+-+r 22 c 1 c 11 ( )( )( )( )/ /( )ssssrs 112 0r 222 1111111 0, ,1-,-,+,0 22 111 1-0,0 1 22 1 1 ,arctan 1 1 1 tan( ) s ds 21111 11 11 ( )( )( )tan( )( ) ( )( ) r sr sss dss ss = ( )r r ss 1 1 ( )( )( ) ( ) r sr ss s s 1 1( ) ( )( )( ) ( )( ) s rsss ss ( )r s( )0s 1 1 ( )( ) ( ) rss s ) 1( ) r s 1 0( )/ / (rss 0s 0r( )rr s0s 2 2 2 0 00 111 lim00|0 s rc s | ( )r s0c 2 2 0 ( )000 2 s r srso s 0 1 00r0c 2 2 2 0 00 111 lim| ( )000| s r srcc s 2 2 2 0 2 0 0 11 lim|000 | 2 s s sc s c 2 222 0 2 0 00 111 lim 2 s sscc s 2 22 0 20 2222 0 00 4 lim0 11 2 s s sscc ( )rr ss ( )rr ss 1123 ( )( )( ) ( )( ) ( )( ) ( )r sr sssssss 222 123 ( )( )( )( )R ssss 2 2 1 3 0 ( )( ) lim0 s r ssr sRs s r ss 2 3 11 ( ) ( )( )( ) 6 s r ssr sssss 23 2 ( )( ) ( )( ) 26 ss ssss 33 3 ( ) ( ) ( )( ) 6 ss sssos 2 2 112 ( )2( )1( ) ( )r ssr ssssss 2332 312 123 ( )( )( ) ( )( )( ) ( )( )( )( ) 333 sss sssssosss 22 s 2 312 12 ( )( )( ) 2( )0,1( ) ( )0,( )( )( ) ( ) 333 sss sssssss 123 2 1( ) ( )0,( ),( ) ( )( ) ( ) s sss sss 12 2 1( ) ( )( )( )( ) ( )( ) ( )( ) ( ) s r sr sssss sss 2.7 1. r 1cos , sin,02rat btt tsin , cos,cos ,sinrat btratb 33 22222222 sincos r xyx yab xyatbt 2,racht bsht ,rasht bchtracht bsht 3 22222 r ab a sh tb ch t 3 2 ,rt t ,1,2 ,0,2rtr 3 22 2 14 r t 4sin,1 cosra ttat 1 cos, sin,sin , cosratatrat at 1 222cos r at 5, t rt ach a 1 1,0, tt rshrch aaa , 2 1 r t ch aa 6cos , ln sectansin,0 2 raaa 2 sin sin ,cos,cos ,sin coscos a raaraaa tan r a 2.,=,. . 22 3 222 cos ,sin sincos ,cossin cos2sin ,sin2cos 2 r r r r 3. 2 1 1 r s s ,.s 2 0 2 0 2 0 22 00,00,00 1 0arctan 1 0cos arctanln |1| 0sin arctan11 ln |1|, 11 s s s xy sdss s x sxs dsss y sys dss r ssss 4.1. 1=cos , sin,02rat btt. 2222 33 22 11 ,c sincos | 1cossin , r ty xbta atbt r tt r ttab ab os ,sint 2.,racht bsht 2222 2222 33 11 , | 1 , r ty xbcht a b ch ta sh t r abba r ttch tsh t ab sht 3. 2 ,rt t 2 32 11 ,2 1+4 | 11 4 ,3 2 r ty x t r r tttt ,1t 4.sin,1 cosra ttat 2 11 ,sin , 22cos | 1 sin,1 cos r ty xat a at r r tta ttat 1 cost 5, t rt ach a . 2 11 , | 1 12 ,2 2 r t ty xsh at r sh a att r tttshach aa 1 6cos , ln sectansin,0 2 raaa. 1 cos,sin cos | 1 sec , ln sectan r a aa r raa 5. 1 222 xya. 0 2 cos , sin |, ( )cos, sin, ( )( )sin,cos ( )= ( )+( )cossin, sincos t r tat at s srtdtatt a ssss r saasr s aaaa sss r sr scssacsacs aaa s a 2 x yach a . 0 2 , | ( )= ( )+, | t t r tt ach a t srtdtash a tt cashcash rtt aa r tr tcasht tt a chchrt aa 3sin ,1 cosr tttt. 0 2 1 cos ,sin |4cos 2 ( )= ( )+4cossinsin,3coscos 22 | t rttt t srtdt rttt r tr tcttctc rt 2 t 3.1 222 222 222 222 222 222 1. 1 ,coscos , cossin , sin, 22 +=1 ,seccos , secsin , tan, 22 += xyz abc rabc xyz abc rabc xyz abc 22 22 2 22 22 1 ,tancos , tansin , sec,0, 2 +=2 1 ,cos ,sin , 2 =2 ,2 rabc xy z ab r u vauv buvu xy z ab r u va uvb uvuv 222 22 222222 2.+=1N= 0 0 1 , = 0 01=, ,0 , 221 1,. 111 2 3 xyzSu v uvuv xyz uvuvuv 22 222222 . 3, 2 1,1. 1 1211 , , 1,0 NP/ /NP u vu v txtu ytv zt uv xyztuvt u vtu tvt 22 222222 221 2P =, 11 uvuv uvuvuv , 1 2 , 2 , 2222 , , 2 , 2 xy y 2222 , 2222 u y 2 2 v v1 1 N= 0 0 1 , = 0 010 0 1 , = 0S 2 2 222 2 2 2 2 2 2 u u xyxy 222 yzy2 2 2 2 2 2 2 =2z z , , 2 u 22 222222 2222 3, 1,* 11 221 2, 11 * , u vu v xy uv zz uvu xyz uvuvuv uv uv uvuv , 1 v 222 222 3. +=111 +=1 1 111 , 222 xyzxzxzyy abcacacbb xzy u acb xzy u acb yaacc vxuvuzuv buuu 22 22 1 2 1111 ,0, , 2222 2 ,+2 2 , 22 u u acac ruuvubu uuuu xyxyxy zz ababab xy c ab xy cz ab uavaubvb vz ucxy uu r,0,1 22 uauba b v u u 3 1 23 4 121324 121324 4.14, OP,1OPOP1PPP P, OPOPPPP P iiii t ix y z ttrt rt r 1324 121312241324 1PPP P = 1PPPP + PPP PPPP P0 t rrt y b c v v 1 1c1 u u u2 u u2 u 2 2 +2 2 2 2 2 z z2 2 c z z2 2 uavauvau ucxy y, , 2 , , r 1 1 u u 1 1 u u u u 11111 1 1111c c1 u, ,u, ,u, ,u, , uu22u22 , , 22 2 , , u22 xyxy + + + + + + abababab xyxyxy y y ab bab b xy, ,uc, uauba b 0,0 ua 0,0,0 222u ,0, 22 , , ,0, 3 4. 4. OPOP r r 3 3 1 1 1 1 1 1 z z, ,u u,z z, , ,z z, u u2 2 2 y y b b2 2 , 1 1 1 1 1 1 1 1 1 1 u u 2 2 2 ,2 y y 2 2 ,2 2 2 22 22 2 2 12 111 2 22 22 5.=,cos , sin , cossin 2cos0,2 cos , 0,0,0 4 2 cos,2 cos sin , 4 r r u vuv uv bvxaya uvauva uauvuuav c c r vbv a r vavavv bv ab 2 22 2 , 4 b ab 3.2 2 2 1. 0 0 0 0 0 , 0 rcr dr drr r r drrc 2. ,cos ,sin , sin ,cos ,0 ,cos ,sin , cos ,sin , cossin cossin uv uv r u vf vu f vu g v rf vu f vurfvu fvu g v rrf v g vu f v g vuf v fv xf vuyf vuzg v f v g vuf v g vu 0,0, 0,0, ,0,0,/ /, 0 0 uv f v fv f v fv g vZ g v ZZh u v r u vx u vy u vz u vr u vh u vrr xyz xyzh uuu xyz xyzh vvv S 22 22 2222 2222 2 2 0 , . . =cos ,sin , xy z zh uu xy z zh vv xyxy uv zz uv xyzf st xyf z Srf zfzzS 3. , ,X= + =0, =,X 0 0 0, , uv r u vavl urvlurl u u vavl uvlu va rx y f x y 0 0 0 , u z v v z Z ZZ Z Z , 22 2 2 0 0 v z v v xyzy 22 2 S S 2 u v v uv zz uvuv y y2 2z z z 3. 3 zgz z z f f 1,0,0,1, 0, 10 010, , , xxyy xyxy x yx rfrf rr rr r r f 0 y ff x yxfyf xyf x y x f x yF y 222 222 000 000 4.1, ,., , , xyz a b cabcc abc c bb a xyz xyzf ab 123 222 111 222 222 22 3 1, ,. ,0,0, 0,0,1,2,3 1 1 i c c c bb a fcc bb affcfc fbfb fa cc bb afi xyz abc xyz abc xy ab 2 33 000 000 222 000 222222 000 1 ,1,2,3, 1 0, i iii ij ijijij ijijijij z c xyz xyzni abc xyz n n aabbcc xxyyzz ij aabbcc 5.cos , sin ,2 ,. 1, 2:. t uv uv rvu vu vcut ve cr r crr 22 2 0 0 . yz z 00 ab 0000 b . . 1 22 2 2222 22 2 3 1, 1, 1 1 1 c c z yzy 22 2 xy y ab 3 1 c 2 bcb 22 2 3 1, 1, c y y bcb 2 2 33 2 1 1 ij j i ijij z z x x0 ni 33 c c ij j j c 0 0 j j 0 0 nj j 2 2 x x0 x0 0 0 a a 1 1 1 1 1 12 2sin , cos ,cos ,sin ,1 22 cos,( ),cos, ( ) 22 | ,( ) t uvuv uv uv uv uv u drdudv rtrrre r dtdtdt rvu vu vruu rtrtrtr t rtr trtr t rtr trtr t rtr t=, ( ) = 4 v rtr t 3.3 1.: (1).( cos , sin , ( ). (2).( cos , sin , ( ), ruv uvv ruv uvuava 22 2222 222222 (1).(cos ,sin ,0) (sin , cos ,( ) cossin1,0,( ) 2( ) u v uuvv rvv ruv uvv ErvvFr rGruV IEduFdudvGdvduuv dv 22 222 22222 (2).(cos ,sin ,( ) (sin , cos , ) 1( ),( ), 2(1( )2( )( u v uuvv rvvu ruv uv a EruFr rauGrua 22 )IEduFdudvGdvu duau dudvuadv 2. 22 2222222 22 , auavuva r uvauva uva , . 2222 222222222222 2222 222222222222 22 22 22222222 2 2222 2 ()44 , ()()() 42 ()4 , ()()() 44 ,0, ()() 4 () u v uv auvaauva u r uvauvauva auva uvaa v r uvauvauva aa ErFGr uvauva a I uva 22 ()dudv 3., 22 20PduQdudvRdv . .: 20ERFQGP 22 2 ( )( ua 2 2) uadv 2 ( )( 2) u2 u2( )(2 22 avuvavuva 2 v v 22 , 2 2 avuvavuvav v va uvaa uv 222222 2 va uva uva u 2222 2 , , 2 2 2 2222 4 4 2 () ) 22222 4 ()( 222 u 2222 , () 22222 , u 2222 , 2 a I 2 )44 4auva a44 4) ) 222222222222 ) ) , , 2222222222222222 )auva)44 4) )()()()( 22222222222222222222 )()()()()()()( 22222222222222 222222 2 ( 22222) 2 ( 22 2 ( 222 ()() , 222222 2 ()() )2 ( ( )()() 22222222222222222 )()()()()( 22222222222 r 2 2222 , 22222222) ) I 2 2 ( 3. 2 2 ( )( ( ( ( ( 2 2( ) ( )( )( ) ( ( ( ( 2 2( ) ( ) ) 2 2 2 2 222 , vava 2222 vauvav2 22 . 2 2 2 2 2 2 2 22 1212 1212 121212121212 1212 1212 2020 2 , ,( ()0 2 ()0 dudu PduQdudvRdvPQR dvdv d ud ud u d uQR d vd vPd v d vP d r d rd r d rEd ud uF d ud vd vd uGd vd v d u d ud ud u EFG d v d vd vd v RQ EFG PP ER20FQGP ) 4. 22222222 0 0 2 0 222222222 ( coscos , cossin , sin ) cos,0,cos cos coscos raaa EaFGaIada d drra d drr aaa d 2 222 2 0 0 22 0 000 0 1 coscos tan cos 1 0,tan,tanln sectan 2cos ( ,tanln sectan) c rc 5. 2222 ()Iduuadv (1). 12 :0:CuvCuv0 (2) 22 123 :,:,: 22 aa Cuv Cuv Cv1. (3).,uav uav1v 2v2 222 2 2 2 2222222222 2 2 2 os2 2 2 da d d 22 a da d 2 2 2 22222222 d 2 2 2 2 22222222222 22 2 0 0 0 cos 1 coscos 0 tanln sectan)ln sectan) 2 2 0 0 0 0 0 0 0 1 0 , 0 ,tanln sectanln sec 0 tan 0 0 tanln sectanln sectantanln sectanln sec,tanln sectanln sec 0 I (1) 1 C C1:0:0 0 0 (2)(2) 1 1 a CuCu 1 1: : a (3)(3)u 222 0 2 2 r a a2 2 n , sn ,n , , 222 co 2 2 r cosco 2 2 2 2 2 2 1 cos1 cos os s2 2 222222 12 22 2222222 22 222222 ()1,0, (1).0,0 0, , () cos ()() () ()( 2 IduuadvEFGua uvdudvdudv uvuv C C drrdu uuadv v drr duuadvuuav dv vuadv v dvuadvvua2 2 22 2 2222 2 2 ) (1) (0,0),cos 1 (1)(1) 1 :arccos,. 1 v adv va a advav a a 1 123 (2).,(0,0),(,1),(,1) 22 aa C C COAB 2 11 2242 00 7 ()44 26 7 , 6 a du OAuadvvvdva dv OBaABa 11122233 ,Cd uad vCd uad vCv0 13 222224 2 113 2 1212 222222 1122 2 cos 3 ()() 1 4 22 arccos,arccos 33 cos1 0 d ud uv A d uuad vd uv v ABA d ud ua d vd v O d ua d vd ua d v O 123 (3).,(0,0),(,1),( ,1)C C COAaBa 1 22222 0 22 2ln( 33 a u a DD AEGF dudvua dudvua dudva221) 1 1),(,1)(,1) 2 ),( (a(1), ) 1 424 1 0 7 7 0 2 a dvdv 7 4444 424 dv v444 4 424 0 0 1 1 222222222222 Cd uad vCd uad v 2222222222222 2 C2 2 13 3 2222222 1111 22 1 ( ( 1 1 2 arccos, 3 , cos 12 0 0 d ud uud u 13 3 d 2 2 2 )2 222 1 1 2 d ud u 1212 O O O d u u2 2 1 d vd v2 1 () 222 ABABA 2 arccos, a O O 1 1 (3).,(3). 1 1,1 1 ),(,( a ,(,( ( ( , ,ABaABa, , () ) 2222 ()() 2222 () ) 22 2 1 ad vad v 1 1 3.5 1.(cos ,sin , ),02 ( cos , sin ,),02 , tt rachachtt aa rvu vu auuv 22222 *2222 2222* : () , ,: tt Ichdta chd aa Ivadudv tt a chva uvashuII aa t uvash a ,. 2.:( (cos ), (sinsin ), () . ra conuv auv b uv 2 22222 :(2 coscos,2 cossin,2 () 22222 2 cos,(cossin 22 (4) ( ( )cos ,( )sin , ( ), ( )( )( ) uvuvuvuvuv raab uvuv uvarvu vubu Ivb dudv rf vu f vu g v Ifv dufvgv dv 22222 2222 222222 ,2 cos, 22 ( )4,( )( )1 ( )4, ( )2 ln4,. (4cos ,4sin ,2 ln4 :,2 cos. 22 uvuv uuvva fvvbfvgv f vvbg vbvvbII rvbuvbubvvb uvuv uva ) 3.: ) ) 2222 ,2 cossin,2sin 22222 , ( ), v 2i2i ,( ), (, dvdv 2222 ( )( )( ) 2222 2 vuvuuvuvu 2i i i ( )( ) 2 2 2222 2 2 (coscosr(cos(coscos r ),os ,( )sin , (,( ) i( 2222222 22 ,2 cos, 2222 , v