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1x1.1..................................................................................1x1.2.......................................................................................2711x3.1............................................................................11x3.2.................................................................15x3.3....................................................................................18x3.4..................................................................................19x3.5...............................................................................21232531IIIx1.11.1¾xx=3x2+4xy¡8y2;¾xy=¡12x2¡6xy¡2y2;¾yy=2x2+xy+3y2;¾xz=¾yz=¾zz=0x+3y+z+1=0(1,-1,1)T§n=§(1;3;1)p11A(1,-1,1)n0=(¡1;1;¡1)n¢n0=1p110AnA_¡!¾=¯¯¯¯¯¯¯¡97=207=240000¯¯¯¯¯¯¯T=¾ijniej=12p11(3;31;0)Tn=T¢n=4811T¿=pjTj2¡T2n=p1454221.2xTx=xap0¡32¹Ua;Ty=yap0;Tz=zap0;aUp0F=ITds=Z2¼0dµZ¼0Ta2sin'd'Fy=Fz=0Fx=Z2¼0dµZ¼0µ¡32¹Ua¶a2sin'd'=¡6¼¹aUx6¼¹aU11.2x1.21.9ijklmn=¯¯¯¯¯¯¯±il±im±in±jl±jm±jn±kl±km±kn¯¯¯¯¯¯¯(1-1)ijklmk=±il±jm¡±im±jli;j;k1-1i;j;ki;j;kl;m;ni;j;kl;m;n1;2;31-111i=jl=mijklmkk6=i;j;l;mijklmk=¯¯¯¯¯¯¯±il±im0±jl±jm0001¯¯¯¯¯¯¯=±il±jm¡±im±jl¯¯¯¯¯¯¯a11a12a13a21a22a23a31a32a33¯¯¯¯¯¯¯=ijka1ia2ja3k¯¯¯¯¯¯¯±i1±i2±i3±j1±j2±j3±k1±k2±k3¯¯¯¯¯¯¯=lmn±il±jm±kn=ijkijklmn=¯¯¯¯¯¯¯±i1±i2±i3±j1±j2±j3±k1±k2±k3¯¯¯¯¯¯¯¢¯¯¯¯¯¯¯±l1±l2±l3±m1±m2±m3±n1±n2±n3¯¯¯¯¯¯¯=¯¯¯¯¯¯¯±i1±i2±i3±j1±j2±j3±k1±k2±k3¯¯¯¯¯¯¯¢¯¯¯¯¯¯¯±l1±m1±n1±l2±m2±n2±l3±m3±n3¯¯¯¯¯¯¯=¯¯¯¯¯¯¯±is±sl±is±sm±is±sn±js±sl±js±sm±js±sn±ks±sl±ks±sm±ks±sn¯¯¯¯¯¯¯=¯¯¯¯¯¯¯±il±im±in±jl±jm±jn±kl±km±kn¯¯¯¯¯¯¯ijklmk=¯¯¯¯¯¯¯±il±im±ik±jl±jm±jk±kl±km±kk¯¯¯¯¯¯¯1.10(a)(A£B)£(C£D)=[A¢(B£D)]C¡[A¢(B£C)]D=(aiei£bjej)£(ckek£dlel)=(aibjijmem)£(ckdlklnen)=aibjckdlijmklnmntet=aibjckdlijm(±kt±lm¡±km±lt)et=aibjckdlijlek¡aibjckdlijkel=[A¢(B£D)]C¡[A¢(B£C)]D(b)[A¢(B£C)]D=[D¢(B£C)]A+[A¢(D£C)]B+[A¢(B£D)]C1(a)[A¢(B£C)]D¡[A¢(B£D)]C=¡(A£B)£(C£D)2(A£B)£(C£D)=(aiei£bjej)£(ckek£dlel)=aibjijmem£ckdlklnen=aibjckdlijmklnnpmep=aibjckdl(±in±jp¡±ip±jn)klnep=aibjckdlkliej¡aibjckdlkljei=¡[A¢(D£C)]B¡[D¢(B£C)]A[A¢(B£C)]D=[D¢(B£C)]A+[A¢(D£C)]B+[A¢(B£D)]C2-=aibjckdl(ljkei+ilkej+ijlek¡ijkel)=aibjckdlem(ljk±im+ilk±jm+ijl±km¡ijk±lm)=aibjckdlem(njk±ln±im+iln±kn±jm+inl±jn±km¡njk±in±lm)=aibjckdlem[(njk(±ln±im¡±in±lm)+iln(±kn±jm¡±jn±km)]=aibjckdlem(njkplipnm+ilnpkjpnm)=aibjckdlem(pjknlinpm+ilnpkjpnm)(p,n)=01.11r¢(rnr)=(n+3)rnr£(rnr)=0r2rn=n(n+1)rn¡2r[f(r)]=f0(r)rrr=xiei;r=jrjr¢(rnr)=@(rnxi)@xi=nrn¡1xixir+3rn=(n+3)rnr£(rnr)=@@xiei£(rnxjej)=@rnxj@xiijkek=rn±ijijkek+xjnrn¡1@r@xiijkek=xjnrn¡1xirijkek=nrn¡2(xiei£xjej)=nrn¡2r£r=0r2rn=@2rn@xi@xi=@@xi³nrn¡1xir´=n(n¡2)rn¡3xixir+3nrn¡2=n(n+1)rn¡2r[f(r)]=@@xi[f(r)]ei=f0(r)@r@xiei=f0(r)xirei=f0(r)rr@r@xi=@pxjxj@xi=12pxjxj@xlxl@xi=1rxl@xl@xi=1rxl±li=xir1.1231.2(a)r¢('a)=r'¢a+'r¢a=@('ai)@xi=@'@xiai+'@ai@xi=r'¢a+'r¢a(b)r£('a)=r'£a+'r£a=ijk@('ak)@xjei=ijk@'@xjakei+ijk'@ak@xjei=r'£a+'r£a(c)r¢(a£b)=(r£a)¢b¡(r£b)¢ar¢(a£b)=@@xiei¢(ajej£bkek)=@@xiei¢(ajbkjklel)=@aj@xibkjkl±il+@bk@xiajjkl±il=@aj@xibkijk+@bk@xiajijk=(@@xiei£ajej)¢bkek+(@@xiei£bjej)¢akek=(r£a)¢b+(r£b)¢a(d)r£(r£a)=r(r¢a)¡r2a=ijk@@xjµklm@am@xl¶ei=(±il±jm¡±im±jl)@2am@xj@xlei=@@xiµ@aj@xj¶ei¡@2(aiei)@xj@xj=r(r¢a)¡r2a(e)r£(a£b)=a(r¢b)¡(a¢r)b+(b¢r)a¡b(r¢a)r£(a£b)=@@xiei£(ajej£bkek)=@@xiei£(ajbkjklel)=@@xiajbkjklilmem=@@xiajbk(±jm±ki¡±ij±km)em=@@xiajbiej¡@@xiaibkek=ajej@bi@xi+bi@@xiajej¡bkek@ai@xi¡ai@bk@xiek=a(r¢b)¡(a¢r)b+(b¢r)a¡b(r¢a)(f)r(a¢b)=a£(r£b)+(a¢r)b+b£(r£a)+(b¢r)aa£(r£b)+(a¢r)b=ijkajklm@bm@xlei+aj@bi@xjei=(±il±jm¡±im±jl)aj@bm@xlei+aj@bi@xjei=aj@bj@xiei)=aj@bj@xiei+bj@aj@xiei=@(ajbj)@xiei=r(a¢b):0B@0121202011CA4x+3y+z=1.x+3y+z=1n=1p11(i+3j+k),T=1p110B@0121202011CA0B@1311CA=1p11(5i+7j+3k):¾=(T¢n)n=(1p11(i+3j+k)¢1p11(5i+7j+3k))n=2911i+3j+kp11:¿=T¡¾=1p11(5i+7j+3k)¡2911p11(i+3j+k)=111p11(26i¡20j+4k)52.1u=2xy2;v=2x2y:dxu=dyvdx2xy2=dy2x2y=)y2¡x2=constant2.2jvj=p2y2+x2+2xyy2+2xy=:uv2ydy+2xdy+2ydx=0=)dyy+dxx+y=0=)uv=¡x+yy=)(u=¡(x+y)f(x;y)v=yf(x;y)=)jvj=jfjp2y2+x2+2xy=)jfj=1)f(x;y)=§1(u=¡(x+y)v=y(u=(x+y)v=¡y2.4vi=xi1+t;txi=»it=0xi=»idxivi=ds()dxixi=(1+t)=dsdxixi=ds=(1+t)xi=ciexps1+ttxi=»ix»1=y»2=z»3(2-1)7dxidt=vi()dxidt=xi1+txi=di(1+t)t=0xi=»idi=»ix»1=y»2=z»3(2-2)(2-1),(2-2)txi=»it=0xi=»i[]2.5vijvjtvijvjtvi=fi(x;y;z)jvjdxivi=ds=)dxifijvj=ds=)dxifi=jvjdsxf1=yf2=zf3dxidt=vi=)dxidt=fijvj=)dxifi=jvjdtxf1=yf2=zf31u=x+tv=¡y+tw=0(»;´;³)Lagrange(1)Lagrange(2)Lagrange(3)LagrangeEuler(4)t=0(-1,1,0)(-1,1,0)t=0(1)8:dxdt=u=x+tdydt=v=¡y+tdzdt=w=0=)8:x=c1et¡t¡1y=c2e¡t+t¡1z=c3t=0(x;y;z)=(»;´;³)8:x=(»+1)et¡t¡1y=(´+1)e¡t+t¡1z=³(2)8:uL=dxdt=(»+1)et¡1vL=dydt=¡(´+1)e¡t+1wL=dzdt=0(3)¡!aL=@2¡!x@t2=¡(»+1)et;(´+1)e¡t;0¢8¡!aE=@¡!u@t+ui@¡!u@i=(x+t+1;y¡t+1;0)Euler@¡!u@t(4)8:dxds=u=x+tdyds=v=¡y+tdzds=w=0=)8:x=c1es¡ty=c2e¡s+tz=c3t=0(xy=c1c2z=0(-1,1,0)(xy=¡1z=08:x=(»+1)et¡t¡1y=(´+1)e¡t+t¡1z=³t=0(»;´;³)=(¡1;1;0)8:x=¡t¡1y=2e¡t+t¡1z=0t=s(-1,1,0)(»;´;³)=(se¡s¡1;(2¡s)es¡1;0)s8:x=set¡s¡t¡1y=(2¡s)es¡t+t¡1z=0=)t=08:x=se¡s¡1y=(2¡s)es¡1z=09x3.13.1VV1V2,PV1V2,SV.Pu,PF,:ddtZZZVFdV=ZZZV@F@tdV+ISFV¢ndS+ZZP(F1¡F2)u¢ºdSnºSP.():,FV1V2,V1;V2,:ddtZZZV1FdV=ZZZV1@