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1、Lettherebelight1:SSSInthebeginningGodcreatedtheheavensandtheearth.NowtheearthwaslessandemptydarknesswasoverthesurfaceofthedeepandtheSpiritofGodwashoveringoverthewaters.AndGodsaid“Lettherebelight”andtherewaslight.MEUJbV($13Ykkk111000k1EnXM1Lettherebelight1:111:1.11.21.31.41.5IX1.6Diracdelta1.7u|AnEnX
2、M2Lettherebelight1:1.1!$QkqkXvrIkvkXmutativeA+B=B+A(associative(A+B)+C=A+(B+C)EnXM3Lettherebelight1:2.Idistributive(A+B)=A+BAB=A+(B)3.:(dotproductscalarproduct)AB=|A|B|cosAB=BAA(B+C)=AB+ACAA=|A|2=A2EnXM4Lettherebelight1:4.(crossproductvectorproduct)AB=|A|B|sinnAB=BAA(B+C)=AB+ACAA=0!“A=Axex+Ayey+Azez
3、A+B=(Ax+Bx)ex+(Ay+By)ey+(Az+Bz)ezAB=AxBx+AyBy+AzBzA=pAA=qA2x+A2y+A2zEnXM5Lettherebelight1:AB=AxBx+AyBy+AzBz=XijijAiBjijKroneckerdeltaij=eiej=1ifi=j0ifi6=jAB=exeyezAxAyAzBxByBz=XijkijkAiBjekijkLevi-CivitasymbolLevi-Civitatensorijk=ei(ejek)=1ifijk=1232313121ifijk=1322133210otherwiseEnXM6Lettherebeligh
4、t1:Levi-Civita51.LmXeiej=Xkijkek2.eIpKXijk=ikj3.EeIijkmnk|zk=imjninjm=iminjmjn4.“-n=jijkmjk|zjk=imjjijmj=3imim=2im5.n“-m=iijkijk|zijk=2ii=6EnXM7Lettherebelight1:n!n1.nI(scalartripleproduct)A(BC)=B(CA)=C(AB)A(BC)=AxAyAzBxByBzCxCyCz2.n(vectortripleproduct)A(BC)=(AC)B(AB)CA(BC)6=(AB)CnotassociativeA(BC
5、)+B(CA)+C(AB)=0EnXM8Lettherebelight1:o!mr=xex+yey+zezr=|r|=rr=px2+y2+z2er=rrdldr=dxex+dyey+dzezmRrr0=(xx0)ex+(yy0)ey+(zz0)ezr|:fieldpoint:r0:sourcepointR=|R|=p(xx0)2+(yy0)2+(zz0)2EnXM9Lettherebelight1:1.2!FFT=Txex+Tyey+TzezdT=Txdx+Tydy+Tzdz=(T)(dl)=|T|dl|cosAFTTCz=TCzEnXM10Lettherebelight1:!FT=Txex+
6、Tyey+Tzez=exx+eyy+ezzTdelQq=exx+eyy+ezzEnXM11Lettherebelight1:aAa=TF(gradient)AB=v(divergence)AB=v(curl)n!v=exx+eyy+ezz(vxex+vyey+vzez)=vxx+vyy+vzzEnXM12Lettherebelight1:o!v=exx+eyy+ezz(vxex+vyey+vzez)=exeyezxyzvxvyvz=exvzyvyz+eyvxzvzx+ezvyxvxy!Kr=px2+y2+z2xex+px2+y2+z2yey+px2+y2+z2zezr=xrex+yrey+zr
7、ez=rr=erEnXM13Lettherebelight1:r=rr=erf(u)=dfduuA(u)=(u)dA(u)duA(u)=(u)dA(u)dur2=2rr=2r1r=1r2r=1r2er=rr3r=exx+eyy+ezz(xex+yey+zez)=3r=exeyezxyzxyz=0EnXM14Lettherebelight1:8!F!Qq5(f+g)=f+g(A+B)=A+B(A+B)=A+BXJk(kf)=kf(kA)=kA(kA)=kAEnXM15Lettherebelight1:“fgABfAAB(fg)=f(fg)+g(fg)k|A51.fgPf2.gfPg=gff+fg
8、g2|$A5C(fg)=gCf=fCgm(fg)=gf+fg(fA)=f(fA)+A(fA)k|A51.fAPf2.AfPA=A(ff)+f(AA)2|$A5C(fA)=A(Cf)=f(CA)m(fA)=Af+fAEnXM16Lettherebelight1:(fA)=f(fA)+A(fA)k|A51.fA=f2.Af=A=A(ff)+f(AA)2|$A5C(fA)=A(Cf)=f(CA)m(fA)=Af+fAEnXM17Lettherebelight1:(AB)=A(AB)+B(AB)k|A51.ABPA2.BAPB=B(AA)A(BB)2|$A5C(AB)=B(CA)=A(CB)m(AB)
9、=B(A)A(B)EnXM18Lettherebelight1:(AB)=A(AB)+B(AB)k|A51.ABPA2.BAPB=(BA)AB(AA)+A(BB)(AB)B2|$A5C(AB)=(BC)AB(CA)=A(CB)(AC)Bm(AB)=(B)AB(A)(A)B+A(B)EnXM19Lettherebelight1:(B)=(Bxex+Byey+Bzez)exx+eyy+ezz=Bxx+Byy+Bzz=Il(B)A=Bxx+Byy+Bzz(Axex+Ayey+Azez)=BxAxxex+ByAxyex+BzAxzex+BxAyxey+ByAyyey+BzAyzey+BxAzxez+B
10、yAzyez+BzAzzezEnXM20Lettherebelight1:(AB)=A(AB)+B(AB)k|A51.ABPA2.BAPB=B(AA)+(BA)A+A(BB)+(AB)B2|$A5B(CA)=C(AB)(BC)AC(AB)=B(CA)+(BC)AC(AB)=A(CB)+(AC)Bm(AB)=B(A)+(B)A+A(B)+(A)BEnXM21Lettherebelight1:!UF(f)F(f)uIF(A)(A)(A)u(f)=exx+eyy+ezzexfx+eyfy+ezfz=2fx2+2fy2+2fz22fIfLaplacianIEnXM22Lettherebelight1:ILaplacian22=x2+y2+z22f=2fx2+2fy2+2fz2ILaplacianEI2A=2x2+2y2+2z2(Axex+Ayey+Azez)=2Axx2+2Axy2+2Axz2ex+2Ayx2+2Ayy2+2Ayz2ey+2Azx2+2Azy2+2Azz2ezLaplacianE2A=(2Ax)ex+(2Ay)ey+(2Az)ezexeyezEnXM23Lettherebelight1:F(f)=()f=05(Cg)(Cf)=(CC)gf=0(g)(f)=(gg)(ff)=(gf)gf6=0F(A)6=2A(A)=()A=0|B
