微分几何习题课第四章课后答案

‡©AÛSK‘€òŸ[e-mail]4²ŒÆêƆOŽÅ‰ÆÆApril24,2014Jj1m2­‚ÛÜnØ3­¡ÛÜnØSK˜2.a(t)•þŠ¼ê§y²µ1jaj=~ê…=ha(t);a0(t)i=0¶2a(t)•ØC£…??š¤…=a(t)^a0(t)=0y²µ£1¤duddtjaj2=ddtha;ai=2ha;a0ilájaj=~ê…=ha(t);a0(t)i=0£2¤ea(t)•ØC§Kkü ~•þb¦µa(t)=f(t)bla0(t)=f0(t)b§ka(t)^a0(t)=f(t)f0(t)b^b=0SK˜‡L5§ea(t)^a0(t)=0§dua(t)6=0§-b(t)=a(t)=ja(t)jKkjb(t)j=1§ld£1¤§hb(t);b0(t)i=0§lhb0(t);a(t)i=0)b0(t)?a(t)2db(t)½Âa(t)=f(t)b(t)§ùpf(t)=ja(t)j6=0§lka0(t)=f0(t)b(t)+f(t)b0(t)dKa(t)^a0(t)=f(t)b0(t)^a(t)=0§b0(t)ka(t)a(t)??š§b0(t)=0§=b(t)~•þ§la(t)•±ØCSK˜4.fO;e1;e2;e3gIe§f1;2;3g˜‡˜†§y²µ1fO;e(1);e(2);e(3)gIe¶2fO;e1;e2;e3g†fO;e(1);e(2);e(3)g½•ƒÓ…=ó˜†y²µ£1¤dufO;e1;e2;e3gIe§½=hei;eji=1i=j0i6=j´˜‡˜†§½=f1;2;3gÙgV§lkhe(i);e(j)i=1i=j0i6=jlfO;e(1);e(2);e(3)gIeSK˜£2¤TfO;e1;e2;e3g†fO;e(1);e(2);e(3)gC†§le(i)=3Xj=1tjiej'üXê§tji=1j=(i)0j6=(i)é?¿˜†§Ù_Sê^()L«§|^1ª½Â§kdetT=X(1)()t(1)1t(2)2t(3)3=(1)()l…=()óꣽ=ó˜†¤ž§detT=1¶…=()Ûꣽ=Û˜†¤ž§detT=1üIe½•ƒÓ…=ó˜†SK2.­‚r(t)=(x(t);y(t))§y²Ù­Çµ(t)=x0(t)y00(t)x00(t)y0(t)[(x0(t))2+(y0(t))2]32y²µs­‚r(t)lëê§Kdsdt=jr0(t)j=p(x0(t))2+(y0(t))2t=drdtdtds=(x0(t);y0(t))p(x0(t))2+(y0(t))2;n=(y0(t);x0(t))p(x0(t))2+(y0(t))2dFrenetúª§n=dtds=dtdtdtds=[x0y00x00y0]
(y0(t);x0(t))((x0(t))2+(y0(t))2)2'üàB(t)=x0(t)y00(t)x00(t)y0(t)[(x0(t))2+(y0(t))2]32SK1.¦e­‚­Çµ1y=ax2¶2x2a2+y2b2=1;(a;b0)¶3r(t)=(acosht;bsinht)¶4r(t)=(t;acoshta)(a0))µ£1¤y=ax2Œw¤­‚r(x)=(x;ax2)§ldSK2á=µ(x)=2a(1+4a2x2)32£2¤T­‚ý§Ùëꐧr(t)=(acost;bsint);t2[0;2]§ldSK2á=µ(t)=asint
(bsint)(acost)
bcost(a2sin2t+b2cos2t)32=ab(a2sin2t+b2cos2t)32=a4b4(a4y2+b4x2)32SK£3¤†dSK2(ص(t)=asinht
bsinhtacosht
bcosht(a2sinh2t+b2cosh2t)32=ab(a2sinh2t+b2cosh2t)32£4¤†dSK2(ص(t)=1acoshta(1+sinh2ta)32=1acoshta(cosh2ta)32=1acosh2taSK3.­‚C34‹I(r;)eL«r=f()§y²­‚­ÇLˆª()=f2()+2dfd
2f()d2fd2hf2()+dfd
2i32y²µ|^4‹Iچ‹I'X§T­‚3†‹IeL«x()=f()cos;y()=f()sinlkµx0()=f0()cosf()sin;y0()=f0()sin+f()cosx00()=f00()cos2f0()sinf()cosy00()=f00()sin+2f0()cosf()sin|^SK2¿ò±þˆª‘\á=()=x0()y00()x00()y0()[(x0())2+(y0())2]32=f2()+2dfd
2f()d2fd2hf2()+dfd
2i32SK5.y²µE3¥K­‚r(t)­ÇÚLÇ©O(t)=jr0(t)^r00(t)jjr0(t)j3;(t)=(r0;r00;r000)jr0^r00j2y²µslëê§Kdsdt=jr0(t)j§lt(t)=r0(t)jr0(t)j)r0(t)=jr0(t)jt(t)=hr0(t);r0(t)i1=2t(t)dFrenetúª§kr00(t)=hr00(t);r0(t)ijr0(t)jt(t)+jr0(t)jdtdsdsdt=hr00(t);r0(t)ijr0(t)jt(t)+jr0(t)j2n(t)lr0(t)^r00(t)=jr0(t)j3t(t)^n(t)=jr0(t)j3b(t)SKüB(t)=jr0(t)^r00(t)jjr0(t)j3Ӟ·‚b(t)=r0(t)^r00(t)jr0(t)^r00(t)ju´n(t)=b(t)^t(t)=(r0(t)^r00(t))^r0(t)jr0(t)^r00(t)j
jr0(t)j=hr0(t);r0(t)ir00(t)hr0(t);r00(t)ir0(t)jr0(t)^r00(t)j
jr0(t)jéb(t)=r0(t)^r00(t)jr0(t)^r00(t)j¦§¿|^Frenetúªnjr0(t)j=dbdsdsdt=b0(t)=r0(t)^r000(t)jr0(t)^r00(t)j+1jr0(t)^r00(t)j0r0^r00SKü†n‰SȧB(t)=(r0;r00;r000)jr0^r00j25P2.1K·‚3˜„ëêeOŽ­Ç!LÇÚFrenetIeúªµ8:t(t)=r0(t)jr0(t)jn(t)=hr0(t);r0(t)ir00(t)hr0(t);r00(t)ir0(t)jr0(t)^r00(t)j
jr0(t)jb(t)=r0(t)^r00(t)jr0(t)^r00(t)j±9(t)=jr0(t)^r00(t)jjr0(t)j3;(t)=(r0;r00;r000)jr0^r00j2SK4.¦e­‚­ÇÚLǵ1r(t)=(acosht;asinht;bt)(a0)¶2r(t)=(3tt2;3t2;3t+t2)¶3r(t)=(a(1sint);a(1cost);bt)(a0)¶4r(t)=(at;p2alnt;at)(a0))µ£1¤dur0(t)=(asinht;acosht;b)r00(t)=(acosht;asinht;0)r000(t)=(asinht;acosht;0)ljr0(t)j=pa2sinh2t+a2cosh2t+b2r0(t)^r00(t)=(absinht;abcosht;a2)jr0(t)^r00(t)j=apb2sinh2t+b2cosh2t+a2SK(r0;r00;r000)=asinhtacoshtbacoshtasinht0asinhtacosht0=a2bl(t)=jr0(t)^r00(t)jjr0(t)j3=apb2sinh2t+b2cosh2t+a2(a2sinh2t+a2cosh2t+b2)3=2(t)=(r0;r00;r000)jr0^r00j2=bb2sinh2t+b2cosh2t+a2£2¤dur0(t)=(32t;6t;3+2t)r00(t)=(2;6;2)r000(t)=(0;0;0)SKljr0(t)j=p18+44t2r0(t)^r00(t)=(18;12;18)jr0(t)^r00(t)j=6p22(r0;r00;r000)=32t6t3+2t262000=0l(t)=jr0(t)^r00(t)jjr0(t)j3=6p22(18+44t2)3=2(t)=(r0;r00;r000)jr0^r00j2=0SK£3¤dur0(t)=(acost;asint;b)r00(t)=(asint;acost;0)r000(t)=(acost;asint;0)ljr0(t)j=pa2+b2r0(t)^r00(t)=(abcost;absint;a2)jr0(t)^r00(t)j=apa2+b2(r0;r00;r000)=acostasintbasintacost0acostasint0=a2bSKl(t)=jr0(t)^r00(t)jjr0(t)j3=aa2+b2(t)=(r0;r00;r000)jr0^r00j2=ba2+b2£4¤dur0(t)=(a;p2at;at2)r00(t)=(0;p2at2;2at3)r000(t)=(0;2p2at3;6at4)ljr0(t)j=a1+1t2SKr0(t)^r00(t)=(p2a2t4;2a2t3;p2a2t2)jr0(t)^r00(t)j=p2a2t21+1t2(r0;r00;r000)=ap2atat20p2at22at302p2at36at4=2p2a3t6l(t)=jr0(t)^r00(t)jjr0(t)j3=p2t2a(t2+1)2(t)=(r0;r00;r000)jr0^r00j2=p2t2a(t2+1)2SK6.y²µ­‚r(s)=(1+s)323;(1s)323;sp2!;s2(1;1)±slë꧿¦§­Ç§LÇÚFrenetIey²µdur0(s)=p1+s2;p1s2;1p2jr0(s)j=1§lslëêdd·‚t(s)=p1+s2;p1s2;1p2_t(s)=14p1+s;14p1s;0SKd_t(s)=(s)n(s)á§(s)=j_t(s)j=14r21s2;n(s)=r1s2;r1+s2;0!b(s)=t(s)^n(s)=p1+s2;p1s2;1p2_b(s)=14p1+s;14p1s;0d_b(s)=(s)n(s)á§(s)=p24p1s2SK7.­‚r(t)=8:(e1t2;t;0);t0(0;0;0);t=0(0;t;e1t2);t01y²µr(t)´˜^K­‚§…3t=0ž­Ç=0¶2¦r(t)(t6=0)žFrenetIe§¿?Øt!0žFrenetIe4y²µ£1¤dur(t)=(x(t);y(t);z(t))§ùpx(t)=(e1t2t00t0;y(t)=tz(t)=(0t0e1t2t0‡yr(t)K­