张量运算

1127x1.1a('Ã)='(aÃ)a£(kb)=k(a£b)a¢(kb)=k(a¢b)a¢(b£c);a£(b£c);a¢(b¢c);a£(b¢c)x1.2a¢(b£c)=¯¯¯¯¯¯a1a2a3b1b2b3c1c2c3¯¯¯¯¯¯=b¢(c£a)=c¢(a£b)=¡a¢(c£b)=¡b¢(a£c)=¡c¢(b£a)a;b;c()a¢(b£c)=0a;a;c()a¢(a£c)=01x1.3c£(a£b)=ffa;bcfa;bc£(a£b)=(b¢c)¢a¡(a¢c)¢b(a£b)£c=(a¢c)¢b¡(b¢c)¢aa£(ex£ey)=(a¢ey)¢ex¡(a¢ex)¢ey=ay¢ex¡ax¢eyex£(a£ey)=(ex¢ey)¢a¡(a¢ex)¢ey=¡ax¢eyc£(a£b)=(b¢c)¢a¡(a¢c)¢bf=c£(a£b)=c£dd=(a2b3¡a3b2)ex¡(a1b3¡a3b1)ey+(a1b2¡a2b1)ezf1=c2d3¡c3d2=c2(a1b2¡a2b1)+c3(a1b3¡a3b1)=a1(b2c2+b3c3)¡b1(a2c2+a3c3)+(a1b1c1¡b1a1c1)=a1(b¢c)¡b1(a¢c)f2=a2(b¢c)¡b2(a¢c)f3=a3(b¢c)¡b3(a¢c)f=(b¢c)¢a¡(a¢c)¢bx1.4ba;bca¢(b¢c)=(c¢a)¢b+c£(a£b)=(c¢a)¢b+(b£a)£ca¢(b¢c)=(a¢b)¢c+b£(a£c)=(a¢b)¢c+(c£a)£ba=a¢(ex¢ex)=(a¢ex)¢ex+(ex£a)£ex=ax¢ex+ex£(a£ex)2x1.5rrr1rrr=ex¢@@x+ey¢@@y+ez¢@@zr¢f=@fx@x+@fx@y+@fx@zr£f=(@fz@y¡@fy@z)¢ex+(@fx@z¡@fz@x)¢ey+(@fy@x¡@fx@y)¢ezr'=@'@x¢ex+@'@y¢ey+@'@z¢ezx1.6ra('Ã)='(aÃ))r('Ã)='(rÃ)+Ã(r')a£(kb)=k(a£b))r£(kb)=k(r£b)+rk£ba¢(kb)=k(a¢b))r¢(kb)=k(r¢b)+rk¢ba¢(a£c)=0)r¢(r£c)=0a£(ka)=0)r£(rk)=0r(f¢g);r£(f£g);r¢(f£g)r(f¢g)=r(f¢gc)+r(fc¢g)!(gc¢r)f+(f£r)£gc+(r¢fc)g+(g£r)£fc=(gc¢r)f+gc£(r£f)+(fc¢r)g+fc£(r£g)=(g¢r)f+(f¢r)g+g£(r£f)+f£(r£g)r£(f£g)=r£(f£gc)+r£(fc£g)!(r¢gc)f¡(r¢f)gc+(r¢g)fc¡(r¢fc)g=(gc¢r)f¡gc(r¢f)+fc(r¢g)¡(fc¢r)g=(g¢r)f¡(f¢r)g+f(r¢g)¡g(r¢f)3r¢(f£g)=r¢(fc£g)+r¢(f£gc)!¡fc¢(r£g)¡gc¢(f£r)=¡fc¢(r£g)+gc¢(r£f)=¡f¢(r£g)+g¢(r£f)(f¢r)g;(f¢r)'(f¢r)g(f¢r)g=(fc¢r)g=r(fc¢g)¡fc£(r£g)6=f(r¢g)(f¢r)'=f¢r'k(k¢r)r=k(k¢r)r=r(k¢r)¡k£(r£r)=r(k¢r)=k(f¢r)('g)=g[(f¢r)']+'[(f¢r)g](f¢r)g=r¢(fg)¡(r¢f)g(f£r)£g;(f£r)¢g;(f£r)'(f£r)£g=(fc£r)£g=r(fc¢g)¡fc(r£g)(f£r)¢g=f¢(r£g)(f£r)'=f£r'4x1.7rf(u)=rudf(u)dur¢A(u)=ru¢dA(u)dur£A(u)=ru£dA(u)du(a¢r)A(u)=(a¢r)udA(u)dur=ruddux1.8r¢(rr3)=rr¢ddr(rr3)=rr¢ddr(rerr3)=(rr¢er)ddr(1r2)=¡2r3r6=A(r)rr=er;r1r=¡err2r¢r=3;r¢er=2r;r¢(rr3)=4¼±(r)r£r=0;r£er=0;r£(rr3)=0r¢(rrn)=3¡nrn+4¼rn¡3±(r)r¢(rrn)=r¢(rr3¢1rn¡3)=r¢(rr3)¢1rn¡3+(rr3)¢r1rn¡3=4¼rn¡3±(r)+(rr3)¢[rr¢3¡nrn¡2]=4¼rn¡3±(r)+3¡nrn5x1.9rÃ=@Ã@rer+1r@Ã@µeµ+@Ã@zezr¢f=1r@@r(rfr)+1r@fµ@µ+@fz@zr£f=(1r@fz@µ¡@fµ@z)er+(@fr@z¡@fz@r)eµ+[1r@@r(rfµ)¡1r@fr@µ]ezr2Ã=1r@@r(r@Ã@r)+1r2@2Ã@µ2+@2Ã@z2x1.10rr=er;rz=ezr¢er=1r;r£er=0r¢eµ=0;r£eµ=1rezr¢ez=0;r£ez=0x1.11rÃ=@Ã@rer+1r@Ã@µeµ+1rsinµ@Ã@ÁeÁr¢f=1r2@@r(r2fr)+1rsinµ@@µ(sinµfµ)+1rsinµ@fÁ@Ár£f=1rsinµ[@@µ(sinµfÁ)¡@fµ@Á]er+1r[1sinµ@fr@Á¡@@r(rfÁ)]eµ+1r[@@r(rfµ)¡@fr@µ]eÁr2Ã=1r2@@r(r2@Ã@r)+1r2sinµ@@µ(sinµ@Ã@µ)+1r2sin2µ@2Ã@Á2x1.12rr=err¢er=2r;r£er=0r¢eµ=1rtanµ;r£eµ=1reÁr¢eÁ=0;r£eÁ=1rtanµer¡1reµ6IarraILB¢dl=2¼rB=¹0I)B=¹0I2¼reµr£B=r£(¹0I2¼reµ)=¹0I2¼[r£(1reµ)]=¹0I2¼[(r1r)£eµ+1r(r£eµ)]=¹0I2¼[rrddr(1r)£eµ+1rezr]=¹0I2¼[¡1r2(er£eµ)+1rezr]=0raILB¢dl=2¼rB=¹0ZZSJ¢dS)B=¹0Ir2¼a2eµr£B=r£(¹0Ir2¼a2eµ)=¹0I2¼a2[r£(reµ)]=¹0I2¼a2[(rr)£eµ+r(r£eµ)]=¹0I2¼a2[(er£eµ)+rezr]=¹0I¼a2ez=¹0Jx2.1(Einstein)7§(x1;x2;x3)§0(x01;x02;x03)x0i=3Xj=1®ijxj;i=1;2;3®ij=e0i¢ejPx0i=®ijxj;i=1;2;3(Einstein)(1)x0ix0i=xixi=const(1)®ij~®®=IIx2.2T3nTl¢¢¢m§§0®ijT0i¢¢¢j=®il¢¢¢¢¢¢®jm|{z}Tl¢¢¢mTnT0=TrT0i=®ijTj@@x0i=@xj@x0i@@xj=®ij@@xj8x2.3T0=ATA0T0ij=®il®jmTlmTijjiTii1Tij=TjiTii=05Tij=¡Tji3ab=Ã!T6=ba;Tij=aibjx2.4(Kronecker)±ij±ij±ij=1(i=j)±ij=0(i6=j)±ij=±ji±0ij=±ij±ijvj=viv¢Ã!I=Ã!I¢v=vx2.5{(levi{civita)ijk{ijkijk=+1(ijk=123;231;312)ijk=¡1(ijk=213;321;132)ijk=0(ijk=112;233;¢¢¢)ijk=¡jik0ijk=ijk=kij=jki(a£b)i=ijkajbk;(r£A)i=ijk@jAk9x2.6Aij=BijAij+Bij=CijCijÃ!C=kÃ!A,Cij=kAijÃ!C=Ã!AÃ!B,Cijlm=AijBlmÃ!AÃ!B6=Ã!BÃ!AÃ!CÃ!AÃ!B(Ã!AÃ!B)Ã!C=Ã!A(Ã!BÃ!C)Ã!A(Ã!B+Ã!C)=Ã!AÃ!B+Ã!AÃ!C(Ã!B+Ã!C)Ã!A=Ã!BÃ!A+Ã!CÃ!Ann¡2n2nÃ!A(j¢¢¢k)Bi¢¢¢l=XjXkAi¢¢¢j¢¢¢k¢¢¢l¢±jk=Ai¢¢¢j¢¢¢k¢¢¢l¢±jkÃ!AÃ!Bm+n¡2Ã!C=Ã!A¢Ã!BCim=AijBlm±jl=AilBlmÃ!AÃ!BÃ!C=Ã!A:Ã!BC=AijBlm±jl±imeieieje1e2E121eiejÃ!T=Tijeieja¢(bc)=(a¢b)c=a¢bc(ab)¢c=a(b¢c)=ab¢cab:cd=(a¢d)(b¢c)=a¢(b¢cd)=(ab¢c)¢d10x2.7r¢(fg)=(r¢f)g+(f¢r)gr¢Ã!T=@@x(ex¢Ã!T)+@@y(ey¢Ã!T)+@@z(ez¢Ã!T)r¢(EE)=(r¢E)E+(E¢r)Er¢(Ã!IE2)=@@x(ex¢Ã!IE2)+@@y(ey¢Ã!IE2)+@@z(ez¢Ã!IE2)=ex@E2@x+ey@E2@y+ez@E2@z=rE2r¢(Ã!IE2)=(r¢Ã!I)E2+(rE2)¢Ã!I=rE2IdS¢Ã!T=ZZZdVr¢Ã!TIdS¢(fg)=ZZZdVr¢(fg)11