曲面的第一基本形式

x2.22.2.1.P;QS:r=r(u;v),r(u;v);r(u+¢u;v+¢v),Taylor,¡¡!PQ=ru¢u+rv¢v+o(j¢uj+j¢vj);¡¡!PQj¡¡!PQj2=r2u¢u2+2ru¢rv¢u¢v+rv¢v2+o(j¢uj2+j¢vj2);PQ,,du=¢u;dv=¢v,j¡¡!PQj2(I)I=r2udu2+2ru¢rvdudv+rvdv2;E=r2u;F=ru¢rv;G=r2v;I=Edu2+2Fdudv+Gdv2:,,.E;F;G,.EG¡F20;du;dv,E0;G0.1r(u;v)=½(u)+vb.ru=½0(u);rv=b,E=r2u=j½0(u)j2;F=ru¢rv=½0(u)¢b;G=r2v=b2;,I=j½0(u)j2du2+2½0(u)¢bdudv+b2dv2:652xy.xyr(x;y)=fx;y;0g,I=dx2+dy2:3x2+y2+z2=a2.,r(µ;Á)=facosµcosÁ;acosµsinÁ;asinµg;I=a2(dµ2+cos2µdÁ2):,r(u;v)=½2a2uu2+v2+a2;2a2vu2+v2+a2;a(u2+v2¡a2)u2+v2+a2¾I=4a4(u2+v2+a2)2(du2+dv2)=4¡1+1a2(u2+v2)¢2(du2+dv2):4r(u;v)=ff(v)cosu;f(v)sinu;g(v)gI=[f0(v)]2du2+([f0(v)]2+[g0(v)]2)dv2:2.2.22.1.8:u=u(¹u;¹v)v=v(¹u;¹v)S:r=r(u;v),(u;v)(¹u;¹v)I=Edu2+2Fdudv+Gdv2;¹I=¹Ed¹u2+2¹Fd¹ud¹v+¹Gd¹v2;66:¹E=r¹u¢r¹u=µru@u@¹u+rv@v@¹u¶¢µru@u@¹u+rv@v@¹u¶=Eµ@u@¹u¶2+2Fµ@u@¹u@v@¹v¶+Gµ@v@¹u¶2;¹F=r¹u¢r¹v=Eµ@u@¹u@u@¹v¶+Fµ@u@¹u@v@¹v+@v@¹u@u@¹v¶+Gµ@v@¹u@v@¹v¶;¹G=r¹v¢r¹v=Eµ@u@¹v¶2+2Fµ@u@¹v@v@¹v¶+Gµ@v@¹v¶2:J=@u@¹u@u@¹v@v@¹u@v@¹v#Jacobi,¹E¹F¹F¹G#=JtEFFG#J;du=@u@¹ud¹u+@u@¹vd¹v;dv=@v@¹ud¹u+@v@¹vd¹v;[du;dv]=[d¹u;d¹v]Jt;¹I=[d¹u;d¹v]¹E¹F¹F¹G#d¹ud¹v#=[d¹u;d¹v]JtEFFG#Jd¹ud¹v#=[du;dv]EFFG#dudv#=I:1(¹u;¹v),¹E;¹F;¹G67(u;v)E;F;G,2.1¹E¹G¡¹F2=¯¯¯¯@(u;v)@(¹u;¹v)¯¯¯¯2(EG¡F2):2.2R3.f:f(P)=P¢T+P0R3,S:r=r(u;v)fS¤:r¤(u;v)=f±r(u;v).r¤u=ru¢T;r¤v=rv¢T;E¤;F¤;G¤S¤,T,E¤=r¤u¢r¤u=(ru¢T)¢(ru¢T)=ru¢ru=E;F¤=F;G¤=G,SS¤.2.2.31.C:r(t)=r(u(t);v(t));t2[a;b]S:r=r(u;v),,s=Zbajr0(t)jdt=ZbasEµdudt¶2+2Fµdudtdvdt¶+Gµdvdt¶2dt=ZbapI5xy,x=x(t);y=y(t).xyI=dx2+dy2,s=Zt2t1pI=Zt2t1sµdxdt¶2+µdydt¶2dt=Zt2t1qx02+y02dt:().686r(µ;Á)=fcosµsinÁ;sinµsinÁ;cosÁg,8:µ=logcot(¼=4¡t=2)Á=¼=2¡t;0·t·¼=2;(,\).I=sin2Ádµ2+dÁ2;dµdt=1sin(¼=2¡t);dÁdt=¡1;,s=Z¼=20sEµdµdt¶2+2FµdµdtdÁdt¶+GµdÁdt¶2dt=Z¼=20p2dt=¼=p2:2..CC¤S:r=r(u;v)P0(u0;v0),u=u(t);v=v(t);u=u¤(t¤);v=v¤(t¤);P0(u0;v0)µrududt+rvdvdt¶¯¯¯¯P0;µrudu¤dt¤+rvdv¤dt¤¶¯¯¯¯P0;,CC¤P0(u0;v0)ÁcosÁ=¡rududt+rvdvdt¢¢¡rudu¤dt¤+rvdv¤dt¤¢¯¯rududt+rvdvdt¯¯¯¯rudu¤dt¤+rvdv¤dt¤¯¯¯¯¯¯¯P069dtdt¤cosÁ=Edudu¤+F(dudv¤+dvdu¤)+Gdvdv¤pEdu2+2Fdudv+Gdv2pE(du¤)2+2Fdu¤dv¤+G(dv¤)2¯¯¯¯¯P0;,()du:dvdu¤:dv¤,.2;,,,¼.2.3dr=rudu+rvdv±r=ru±u+rv±vEdu±u+F(du±v+dv±u)+Gdv±v=0.2.4S:r=r(u;v)u-v-µcosµ=FpEG;()F=0.7ds2=du2+(u2+a2)dv2;u+v=0;u¡v=0(,).u+v=0,u=u1;v=v1,du1=¡dv1;u¡v=0,u=u2;v=v2,du2=dv2.µ,cosµ=Edu1du2+F(du1dv2+du2dv1)+Gdv1dv2pEdu21+Fdu1dv1+Gdv21pEdu22+Fdu2dv2+Gdv22:ds2=du2+(u2+a2)dv2E=1;F=0;G=u2+a2:cosµ=u2+a2¡1u2+a2+1:70u+v=0u¡v=0u=0;v=0cosµ=u2+a2¡1u2+a2+1¯¯¯¯u=0v=0=a2¡1a2+1:3.()S:r=r(u;v);(u;v)2D()D.(,C.Go®man)u-v-D8:.,8:,DPP1P2P3(),PP1P2P3r(u;v);r(u+¢u;v);r(u;v+¢v);r(u+¢u;v+¢v);Taylor,¡¡!PP1=r(u+¢u;v)¡r(u;v)=[ru(u;v)+²1]¢u;¡¡!PP2=r(u;v+¢v)¡r(u;v)=[rv(u;v)+²2]¢v;lim¢u!0²1=lim¢v!0²2=0.,¢u;¢v¡¡!PP1¼ru(u;v)¢u;¡¡!PP2¼rv(u;v)¢v;SPP1P2P3¼SPP1P2P3=j¡¡!PP1£¡¡!PP2j=jru£rvj¢u¢v=pEG¡F2¢u¢v:71¢¾=pEG¡F2¢u¢vd¾=pEG¡F2dudvS.¾=lim¢u!0¢v!0XpEG¡F2¢u¢v=ZZDpEG¡F2dudv:3E;F;G,¾.,¾.8:u=u(¹u;¹v)v=v(¹u;¹v),¹E¹G¡¹F2=·@(u;v)@(¹u;¹v)¸2(EG¡F2);¾.4R,¾,,xy;¾,u=x;v=y,,I=dx2+dy2,¾=ZZRdxdy:.8ds2=du2+(u2+a2)dv2,u=§av;v=1.u=§av;v=1A(u=0;v=720);B(u=a;v=1);C(u=¡a;v=1).ABCS=ZZDpEG¡F2dudv=ZZDpu2+a2dudv=2Za0pu2+a2duZ1audv=2Za0(1¡ua)pu2+a2du=·upu2+a2+a2ln(u+pu2+a2)¡23a(u2+a2)3=2¸a0=23¡p23+ln(p2¡1)#a2:9r(µ;Á)=f(b+asinÁ)cosµ;(b+asinÁ)sinµ;acosÁg,a;b(ba),0·µ·2¼;0·Á·2¼.I=(b+asinÁ)2dµ2+a2dÁ2;S=ZZ0·µ·2¼0·Á·2¼pEG¡F2dµdÁ=Z2¼0Z2¼0a(b+asinÁ)dÁdµ=4¼2ab:2.2.4,,..2.2.573,,..2.5S:r=r(u;v)a(u;v)b(u;v),(¹u;¹v),,r¹ur¹va;b.a;beda=a1ru+a2rv;b=b1ru+b2rv;ed(¤)b2du¡b1dv;¡a2du+a1dv;,,¸¹,,¸(b2du¡b1dv)=d¹u;¹(¡a2du+a1dv)=d¹v;¹u;¹vu;v,¹u=¹u(u;v);¹v=¹v(u;v).a;b,@(¹u;¹v)@(u;v)=¸¹¯¯¯¯¯a1a2b1b2¯¯¯¯¯6=0;¹u;¹vS.,(*)du=¹a1d¹u+¸b1d¹v¸¹¯¯¯¯¯a1a2b1b2¯¯¯¯¯;dv=¸b2d¹v+¹a2d¹u¸¹¯¯¯¯¯a1a2b1b2¯¯¯¯¯;74r¹u=rudud¹u+rvdvd¹u=1¸¯¯¯¯¯a1a2b1b2¯¯¯¯¯(a1ru+a2rv)ka;r¹v=rudud¹v+rvdvd¹v=1¹¯¯¯¯¯a1a2b1b2¯¯¯¯¯(b1ru+b2rv)kb:2.6.,(u;v),a=ru;b=¡FEru+rv;,a;bS.,S(¹u;¹v),r¹uka;r¹vkb,(¹u;¹v).,Adu+Bdv=0;C±u+D±v=0;,E+F(dvdu+±v±u)+Gdvdu±v±u=0;(¤¤)E¡F(AB+CD)+GABCD=0:,Adu+Bdv=0;.(**)E+F(¡AB+±v±u)+G(¡AB)±v±u=0;75±v±u=¡BE¡AFBF¡AG:10z=axy.r(u;v)=fu;v;auvg;u-r(u)=fu;v0;auv0g;,ru=f1;0;av0g,u-.u-.,z=axy,z=axy..u-±u:±v=1:0,,u-Edu+Fdv=0;z=axyE=1+a2v2;F=a2uv;G=1+a2u2,¡2udu=2a2v1+a2v2dv;¡lnu2=ln(1+a2v2)+lnC1;u2(1+a2v2)=C1;u-.u-v2(1+a2u2)=C2;C1;C2.76