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科学认识、运用客观世界的基本特性(18)(修改补充版)
(接(17))
22. 各种时空多线矢的矢算
狭义相对论纠正经典物理学“绝对时间”的错误观念,得到4维时空的1线矢。
但是,尚未解决4维时空矢量各种矢算,及其必然产生的各种高次、线,多线矢的表达。
因而,至今也尚不能正确地认识、区分,各种时空多线矢,不能全面、正确地进行各种相应的矢算,以致造成许多国际流行的严重错误。
必须创新4维时空的矢算,予以解决。
其实,4维时空矢算的基本原则对于任何维的矢量都是一样的。只因维数的不同而有不同的结果。
27.1.多线矢的加减法
对于,矢量的加减法,都是各种相同矢量的各维分量相加减;而矢量的乘法就因维数的不同而有显著的差别,
对于正交系:
27.2.4维时空多线矢的叉乘法
对应各不同维各分量组成相应高维分量,作为叉乘积的相应分量:
A[1线矢]叉乘B[1线矢]=AB[2线矢]
={(A0Bj-AjB0)[0j基矢]+(AkBl-AlBk)[kl基矢]
,jkl=123循环求和},
共有c(4,2)=6维,不能表达为1线矢。其模长:
AB={(A0Bj-AjB0)^2+(AkBl-(AlBk)^2
,jkl=123循环求和}^(1/2),
AB[2线矢]叉乘C[1线矢]=(ABC)[3线矢]
={(A0Bj-AjB0)Ck[0jk基矢]+(AkBl-AlBk)C0[0kl基矢]
+(A0Bj-AjB0)Cl[0jl基矢]+(AkBl-AlBk)Cj[jkl基矢]
,jkl=123循环求和}
={(A0Bj-AjB0)Ck[l*基矢]+(AkBl-AlBk)C0[j*基矢]
+(A0Bj-AjB0)Cl[-K*基矢]+(AkBl-(AlBk)Cj[0*基矢]
,jkl=123循环求和}
=ABC[1*线矢], 共有c(4,3)=4维,其模长:
ABC={((A0Bj-AjB0)Ck)^2+((AkBl-AlBk)C0)^2
+((A0Bj-AjB0)Cl)^2+((AkBl-AlBk)Cj)^2
,jkl=123循环求和}^(1/2),
ABC[3线矢]叉乘D[1线矢]=(ABCD)[标量]
={A0BjCkDl,jkl=123循环求和},
AB[2线矢]叉乘CD[2线矢]=(AB,CD)[22线矢]
={(A0Bj-AjB0)(C0Dk-CkD0)[0j,0k基矢]
+(A0Bj-AjB0)(C0Dl-ClD0)[0j,0l基矢]
+(A0Bj-AjB0)(CkDl-ClDk)[0j,kl基矢]
+(AkBl-AlBk)(ClDj-CjDl)[kl,lj基矢]
+(AkBl-AlBk)(CjDk-CkDj)[kl,jk基矢]
,jkl=123循环求和},共有c(6,2)=15维,
(AB,CD)[22线矢]叉乘E[1线矢]=(AB,CD)E[22,1线矢]
={(A0Bj-AjB0)(C0Dk-CkD0)El[0j,0k,l基矢]
+(A0Bj-AjB0)(C0Dl-ClD0)Ek[0j,0l,k基矢]
+(AkBl-AlBk)(ClDj-CjDl)E0[kl,lj,0基矢]
+(AkBl-AlBk)(CjDk-CkDj)E0[kl,jk,0基矢]
,jkl=123循环求和},共有4c(3,2)=12维,
(AB,CD)[22线矢]叉乘EF[2线矢]=(AB,CD,EF)[222线矢]
={(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0)[01,02,03基矢]
+(A0Bj-AjB0)(C0Dk-CkD0)(EkFl-ElFk)[0j,0k,kl基矢]
+(A0Bj-AjB0)(C0Dk-CkD0)(ElFj-EjFl)[0j,0k,lj基矢]
+(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)[0j,0k,jk基矢]
+(AkBl-AlBk)(ClDj-CjDl)(E0Fj-EjF0)[kl,lj,0j基矢]
+(AkBl-AlBk)(ClDj-CjDl)(E0Fk-EkF0)[kl,lj,0k基矢]
+(AkBl-AlBk)(ClDj-CjDl)(E0Fl-ElF0)[kl,lj,0l基矢]
+(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1) [23,31,12基矢]
,jkl=123循环求和},共有c(6,3)=20维,
(AB,CD,EF)[222线矢]叉乘G[1线矢]
=(AB,CD,EF)G[222,1线矢]
={(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)Gl
+(AkBl-AlBk)(ClDj-CjDl)EjFk-EkFj))G0
,jkl=123循环求和}, 共有3c(3,2)=6维,
(AB,CD,EF)[222线矢]叉乘GH[2线矢]
=(AB,CD,EF,GH)[2222线矢]
={(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0)
(GjHk-GkHj)[01,02,03, jk基矢]
+(A0B1-A1B0)(C0D2-C2D0)(EjFk-EkFj)
(GkHl-GlHk)[01,02,jk,kl基矢]
+(A0B2-A2B0)(C0D3-C3D0)(E0F3-E3F0)
(GjHk-GkHj)[02,03,jk,kl基矢]
+(A0Bj-AjB0) (C2D3-C3D2)(E3F1-E1F3)
(G1H2-G2H1)[0j,23,31,12基矢]
+(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1)
(GjHk-GkHj)[23,31,12, jk基矢]
,jkl=123循环求和}
=(AB,CD,EF)[222*线矢] , 共有c(6,4)=15维,
(AB,CD,EF,GH)[2222线矢]叉乘I[1线矢]
={(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0) (GjHk-GkHj)Il
[01,02,03, jk,l基矢]
+(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0) (GjHk-GkHj)Il
[01,02,03, 12,3基矢]
,jkl=123循环求和}, 共有3c(3,2)=6维,
(AB,CD,EF,GH)[2222线矢]叉乘I[1线矢]已不能再叉乘任何[1线矢]。
(AB,CD,EF,GH)[2222线矢]叉乘IJ[2线矢]
=(AB,CD,EF,GH,IJ)[22222线矢]
={(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0) (GjHk-GkHj)
(IkJl-IlJk) [01,02,03, jk,kl基矢]
+(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1) (GjHk-GkHj)
(IkJl-IlJk) [23,31,12, jk,kl基矢]
,jkl=123循环求和}
=(AB,CD,EF,GH,IJ)[2*线矢], 共有c(6,5)=6维,
(AB,CD,EF,GH,IJ)[22222线矢]已不能再叉乘任何[1线矢]。
(AB,CD,EF,GH,IJ)[22222线矢] 叉乘KL[2线矢]
={(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0) (G2H3-G3H2)
(I3J1-I1J3) (K1L2- K2L1) [标量]},
27.3.4维时空多线矢的点乘法
消去对应各维分量中相同维的分量,作为点乘积的相应分量:
A[1线矢]点乘B[1线矢]= A.B[标量]
={(AaBa),a=0,1,2,3}[标量],
各类相同矢量的点乘积都是相应的[标量]
AB[2线矢]点乘C[1线矢] =AB.C[1线矢]
={(A0Bj-AjB0)Cj[0基矢]-(A0Bj-AjB0)C0[j基矢]
+(AkBl-AlBk)Ck[l基矢]-(AkBl-AlBk)Cl[k基矢]}
,jkl=123循环求和}, 共有4维,
ABC[3线矢]点乘D[1线矢]=ABC.D[标量]
={(A0Bj-AjB0)CkDl+(AkBl-AlBk)C0Dj
+(A0Bj-AjB0)ClDk+(AkBl-AlBk)CjD0
,jkl=123循环求和}[标量]
ABC[3线矢]点乘DE[2线矢]=ABC.DE[1线矢]
={(A0Bj-AjB0)(D0Ej-DjE0)(Ck[k基矢]+Cl[l基矢])
+(AkBl-AlBk)(DkEl-DkEl)(C0[0基矢]+Cj[j基矢])
,jkl=123循环求和},共有4维,
(AB,CD)[22线矢]点乘E[1线矢]=(AB,CD)E[22.1线矢]
={(A0Bj-AjB0)(C0Dk-CkD0)
((E0+Ej)[0k基矢]-(E0+Ek))[0j基矢]
+(A0Bj-AjB0)(C0Dk-CkD0)
((E0+Ej)[0l基矢]-(E0+El))[0j基矢]
+(A0Bj-AjB0)(CkDl-ClDk)(-Ek+El)[0j基矢]
+(A0Bj-AjB0)(CkDl-ClDk)(E0-Ej)[kl基矢]
+(AkBl-AlBk)(ClDj-CjDl)(-El+Ej)[kl基矢]
+(AkBl-AlBk)(ClDj-CjDl)(Ek-El)[lj基矢]
+(AkBl-AlBk)(CjDk-CkDj)(-Ej+Ek)[kl基矢]
+(AkBl-AlBk)(CjDk-CkDj)(Ek-El)[jk基矢]
,jkl=123循环求和}, 共6维,
(AB,CD)[22线矢]点乘EF[2线矢]=(AB,CD)EF[22.2线矢]
={(A0Bj-AjB0)(C0Dk-CkD0)(-E0Fk+EkF0)[0j基矢]
+(A0Bj-AjB0)(C0Dk-CkD0)(E0Fj-EjF0)[0k基矢]
+(A0Bj-AjB0)(C0Dl-ClD0)(-E0Fl+ElF0)[0j基矢]
+(A0Bj-AjB0)(C0Dl-ClD0)(E0Fj-EjF0)[0l基矢]
+(A0Bj-AjB0)(CkDl-ClDk)(-EkFl+ElFk)[0j基矢]
+(A0Bj-AjB0)(CkDl-ClDk)(E0Fj-EjF0)[kl基矢]
+(AkBl-AlBk)(ClDj-CjDl)(-ElFj+EjFl)[kl基矢]
+(AkBl-AlBk)(ClDj-CjDl)(EkFl-ElFk)[lj基矢]
+(AkBl-AlBk)(CjDk-CkDj)(-EjFk+EkFj)[kl基矢]
+(AkBl-AlBk)(CjDk-CkDj)(EkFl-ElFk) [jk基矢]
,jkl=123循环求和}, 共有C(4,2)=6维,
(AB,CD,EF)[222线矢]点乘G[1线矢]=(AB,CD,EF)G[222.1线矢]
={(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0)
(G1[02,03基矢]-G2[01, 03基矢]+G3[01,02基矢])
+(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0)G0
([02,03基矢]-[01,03基矢]+ [01,02基矢])
+(A0Bj-AjB0)(C0Dk-CkD0)(EkFl-ElFk)
(Gj[0k,kl基矢]-Gk[0j,kl基矢]
-G0[0j,kl基矢]-Gl[0j,0k基矢])
-(A0Bj-AjB0)(C0Dk-CkD0)(EkFl-ElFk)Gj[0k,kl基矢]
+(A0Bj-AjB0)(C0Dk-CkD0)(EkFl-ElFk)Gl[0j,0k基矢]
+(A0Bj-AjB0)(C0Dk-CkD0)(ElFj-EjFl)Gl[0j,0k基矢]
-(A0Bj-AjB0)(C0Dk-CkD0)(ElFj-EjFl)Gj[0j,0k基矢]
+(A0Bj-AjB0)(C0Dk-CkD0)(ElFj-EjFl)G0[0j,lj基矢]
-(A0Bj-AjB0)(C0Dk-CkD0)(ElFj-EjFl)Gk[0j,lj基矢]
+(A0Bj-AjB0)(C0Dk-CkD0)(ElFj-EjFl)Gj[0k,lj基矢]
-(A0Bj-AjB0)(C0Dk-CkD0)(ElFj-EjFl)G0[0k,lj基矢]
+(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)G0[0k,jk基矢]
-(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)Gj[0k,jk基矢]
+(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)Gj[0j,0k基矢]
-(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)Gk[0j,0k基矢]
+(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)Gk[0j,jk基矢]
-(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)G0[0j,jk基矢]
+(AkBl-AlBk)(ClDj-CjDl)(E0Fj-EjF0)G0[kl,lj基矢]
-(AkBl-AlBk)(ClDj-CjDl)(E0Fj-EjF0)Gj[kl,lj基矢]
+(AkBl-AlBk)(ClDj-CjDl)(E0Fj-EjF0)Gk[lj,0j基矢]
-(AkBl-AlBk)(ClDj-CjDl)(E0Fj-EjF0)Gl[lj,0j基矢]
-(AkBl-AlBk)(ClDj-CjDl)(E0Fj-EjF0)Gl[kl,0j基矢]
+(AkBl-AlBk)(ClDj-CjDl)(E0Fj-EjF0)Gl[kl,0j基矢]
+(AkBl-AlBk)(ClDj-CjDl)(E0Fk-EkF0)G0[kl,lj基矢]
-(AkBl-AlBk)(ClDj-CjDl)(E0Fk-EkF0)Gk[kl,lj基矢]
+(AkBl-AlBk)(ClDj-CjDl)(E0Fk-EkF0)Gk[lj,0k基矢]
-(AkBl-AlBk)(ClDj-CjDl)(E0Fk-EkF0)Gl[lj,0k基矢]
+(AkBl-AlBk)(ClDj-CjDl)(E0Fk-EkF0)Gj[kl,0k基矢]
-(AkBl-AlBk)(ClDj-CjDl)(E0Fk-EkF0)Gl[kl,0k基矢]
+(AkBl-AlBk)(ClDj-CjDl)(E0Fl-ElF0)Gk[lj,0l基矢]
-(AkBl-AlBk)(ClDj-CjDl)(E0Fl-ElF0)Gl[lj,0l基矢]
+(AkBl-AlBk)(ClDj-CjDl)(E0Fl-ElF0)Gj[kl,0l基矢]
-(AkBl-AlBk)(ClDj-CjDl)(E0Fl-ElF0)Gl[kl,0l基矢]
-(AkBl-AlBk)(ClDj-CjDl)(E0Fl-ElF0)Gl[kl,lj基矢]
+(AkBl-AlBk)(ClDj-CjDl)(E0Fl-ElF0)G0[kl,lj基矢]
+(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1)G2 [31,12基矢]
-(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1)G3 [31,12基矢]
-(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1)G3 [23, 12基矢]
+(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1) G1[23, 12基矢]
+(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1)G1 [23,31基矢]
-(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1)G2 [23,31基矢]
,jkl=123循环求和}, 共9维,
(AB,CD,EF)[222线矢]点乘GH[2线矢] =(AB,CD,EF)GH(4)[222.2线矢]
={(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0)
((G0H1-G1H0)[02,03基矢]-(G0H2-G2H0)[01, 03基矢]
+(G0H3-G3H0) [01,02基矢])
+(A0Bj-AjB0)(C0Dk-CkD0)(EkFl-ElFk)
((G0Hj-GjH0)[0k,kl基矢]-(G0Hk-GkH0)[0j,kl基矢]
+(GkHk-GlHk)[0j,0k基矢])
+(A0Bj-AjB0)(C0Dk-CkD0)(ElFj-EjFl)
((G0Hj-GjH0)[0k,lj基矢]-(G0Hk-GkH0)[0j,lj基矢]
+(GlHj-GjHl)[0j,0k基矢])
+(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)
((G0Hj-GjH0)[0k,jk基矢]-(G0Hk-GkH0)[0j,jk基矢]
+(GjHk-GkHj)[0j,0k基矢])
+(AkBl-AlBk)(ClDj-CjDl)(E0Fj-EjF0)
((GkHl-GlHk)[lj,0j基矢]-(GlHj-GjHl)[kl,0j基矢]
+(G0Hj-GjH0)[kl,lj基矢])
+(AkBl-AlBk)(ClDj-CjDl)(E0Fk-EkF0)
((GkHl-GlHk)[lj,0k基矢]-(GlHj-GjHl)[kl,0k基矢]
+(G0Hk-GkH0)[kl,lj基矢])
+(AkBl-AlBk)(ClDj-CjDl)(E0Fl-ElF0)
((GkHl-GlHk)[lj,0l基矢]-(GlHj-GjHl)[kl,0l基矢]
+(G0Hl-GlH0)[kl,lj基矢])
+(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1)
((G2H3-G3H2)[31,12基矢]-(G3H1-G1H3)[23, 12基矢]
+(G1H2-G2H1)[23,31基矢])
,jkl=123循环求和}, 共9维,
(AB,CD)E[22,1线矢]点乘F[1线矢]=(AB,CD)E. F[22,1.1线矢]
={(A0Bj-AjB0)(C0Dk-CkD0)ElFl[0j,0k基矢]
+(A0Bj-AjB0)(C0Dk-CkD0)ElF0([0k,l基矢]-[0j,l基矢])
+(A0Bj-AjB0)(C0Dk-CkD0)El(Fk[0j,l基矢]-Fj[0k,l基矢])
+(A0Bj-AjB0)(C0Dl-ClD0)EkFk[0j,0l基矢]
+(A0Bj-AjB0)(C0Dl-ClD0)EkF0([0l,k基矢]-[0j,k基矢])
+(A0Bj-AjB0)(C0Dl-ClD0)Ek(Fl[0j,k基矢]-Fj[0l,k基矢])
+(AkBl-AlBk)(ClDj-CjDl)E0F0[kl,lj基矢]
-(AkBl-AlBk)(ClDj-CjDl)E0Fj[kl,0基矢]
+(AkBl-AlBk)(ClDj-CjDl)E0Fk[lj,0基矢]
+(AkBl-AlBk)(ClDj-CjDl)E0Fl([lj,0基矢]-[kl,0基矢])
+(AkBl-AlBk)(CjDk-CkDj)E0F0[kl,jk基矢]
-(AkBl-AlBk)(CjDk-CkDj)E0Fl[jk,0基矢]
-(AkBl-AlBk)(CjDk-CkDj)E0Fj[kl,0基矢]
+(AkBl-AlBk)(CjDk-CkDj)E0 Fk([jk,0基矢]+[kl, 0基矢])
,jkl=123循环求和},
出现2类矢量,[kl,lj基矢]9维,[kl,0基矢]6维,共15维。
(AB,CD)E[22,1线矢]点乘FG[2线矢]
=(AB,CD)E. FG [22,1.2线矢]
={(A0Bj-AjB0)(C0Dk-CkD0)El
((F0Gj-FjG0)[0k,l基矢]-(F0Gk-FkG0)[0j,l基矢]
+(FlGj-FjGl)[0j,0k,j基矢]-(FkGl-FlGk)[0j,0k,k基矢])
+(A0Bj-AjB0)(C0Dl-ClD0)Ek
((F0Gj-FjG0)[0l,k基矢]-(F0Gl-FlG0)[0j,k基矢]
+(FjGk-FkGj)[0j,0l,j基矢]-(FkGl-FlGk)[0j,0l,l基矢])
+(AkBl-AlBk)(ClDj-CjDl)E0
((FkGl-FlGk)[lj,0基矢]-(FlGj-FjGl)[kl,0基矢]
-(F0Gj-FjG0)[kl,lj,j基矢]-(F0Gk-FkG0)[kl,lj,k基矢])
+(AkBl-AlBk)(CjDk-CkDj)E0[kl,jk,0基矢]
((FkGl-FlGk)[jk,0基矢]-(FjGk-FkGj)[kl,0基矢]
-(F0Gj-FjG0)[kl,jk,j基矢]-(F0Gk-FkG0)[kl,jk,k基矢])
,jkl=123循环求和},
出现2类矢量,[kl,lj,k基矢]9维,[kl,0基矢]6维,共15维。
类似的,还有其它多线矢的点乘,各多线矢和各点乘多线矢的自乘。
(未完待续)
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