微分几何答案-前五章

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1、‡‡‡©©©AAAÛÛÛSSSKKK)))‰‰‰SSSKKK˜˜˜1.T´E3˜‡(V)C†.béuE3,‡mf8U,T(U)=U¿…T±U¥?¿ü:ål.y²T´˜‡ÜÓC†.yyy²²²:²·?U,†½n2.11‡y²ÄƒÓ.(LbT´C2.)2.a(t)´ëYŒ‡•þŠ¼ê,y²:(1)ja(t)j´~ê…=ha(t);a0(t)i=0;(2)bja(t)j6=0.Ka(t)••ØC…=a(t)^a0(t)=0.yyy²²²:(1)ja(t)j=~ê,ja(t)j2=ha(t);a(t)i=~ê,ha(t);a(

2、t)i0=2ha(t);a0(t)i=0,ha(t);a0(t)i=0.(2)dja(t)j6=0,a(t)••ØC,e(t):=a(t)´ü~•þ,e0(t)=0.ja(t)jdhe(t);e(t)i=1,khe(t);e0(t)i=0,=:e(t)?e0(t).Ïd,e0(t)=0,00a(t)a0(t)100e(t)==e(t),0=e(t)^e(t)=^(+()a(t))=a(t)^a(t).ja(t)jja(t)jja(t)j3.y5Ÿ1.1Ú5Ÿ1.2.555ŸŸŸ1.1vi(1i4)´R3o‡•þ.(1)v1^(v2^v3)=hv1;

3、v3iv2hv1;v2iv3;(2)Lagrangeðª:hv1^v2;v3^v4i=hv1;v3ihv2;v4ihv1;v4ihv2;v3i;(3)(v1;v2;v3)=(v2;v3;v1)=(v3;v1;v2).yyy²²²:vi=(xi;yi;zi)(1i4).(1)d½Â,OŽkijkv^(v^v)=(x1;y1;z1)^x2y2z2123x3y3z3ijk=x1y1z1y2z3y3z2x2z3+x3z2x2y3x3y2=(x2y1y3x3y1y2+x2z1z3x3z1z2;x1x2y3+x1x3y2+y2z1z3y3z1z2

4、;x1x2z3+x1x3z2y1y2z3+y1y3z2)=((x1x3+y1y3+z1z3)x2(x1x2+y1y2+z1z2)x3;(x1x3+y1y3+z1z3)y2(x1x2+y1y2+z1z2)y3;(x1x3+y1y3+z1z3)z2(x1x2+y1y2+z1z2)z3)=hv1;v3iv2hv1;v2iv3:12(2)d½Â,OŽkijkijkhv1^v2;v3^v4i=hx1y1z1;x3y3z3ix2y2z2x4y4z4=h(y1z2y2z1;x1z2+x2z3;x1y2x2y1);(y3z4y4z3;x3z4+x4z3;

5、x3y4x4y3)i=y1y3z2z4y1y4z2z3y2y3z1z4+y2y4z1z4+x1x3z2z4x1x4z2z3x2x3z1z4+x2x4z1z3+x1x3y2y4x1x4y2y3x2x3y1y4+x2x4y1y3=(x1x3+y1y3+z1z3)(x2x4+y2y4+z2z4)(x1x4+y1y4+z1z4)(x2x3+y2y3+z2z4)=hv1;v3ihv2;v4ihv1;v4ihv2;v3i:(3)d½Â,OŽkijk(v;v;v)=hv;v^vi=h(x2;y2;z2);x3y3z3i231231x1y1z1=h(x2;y

6、2;z2);(y3z1y1z3;x3z1+x1z3;x3y1x1y3)i=x2y3z1x2y1z3x3y2z1+x1y2z3+x3y1z2x1y3z2=x1(y2z3y3z2)+y1(x2z3+x3z2)+z1(x2y3x3y2)=hv1;v2^v3i=(v1;v2;v3):A^þãª,k(v2;v3;v1)=(v3;v1;v2):555ŸŸŸ1.2f=f(x;y;z)´R3(½ö§˜‡«•)þ˜‡¼ê,F(x;y;z)=(P(x;y;z);Q(x;y;z);R(x;y;z))´R3(½ö§˜‡«•)þ˜‡•þ

7、.K(1)r^(r

8、f)=rot(gradf)=0,(2)hr;r^Fi=div(rotF)=0.yyy²²²:(1)d½Â,OŽijk@f@f@f@@@r^(rf)=r^(;;)=@x@y@z@x@y@z@f@f@f@x@y@z@2f@2f@2f@2f@2f@2f=(;+;)=0:@y@z@z@y@x@z@z@x@x@y@y@x(2)Šâ½Â,OŽijk@@@@@@hr;r^Fi=h(;;);i@x@y@z@x@y@zPQR@@@@R@Q@R@P@Q@P=h(;;);(;+;)i@x@y@z@y@z@x@z@x@y@2R@2Q@2R@2P@2Q@2P=++

9、=0:@x@y@x@z@y@x@y@z