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Lettherebelight•§1.1•§1.2•§1.3•§1.4•§1.5•§1.6Diracdelta•§1.72Lettherebelight§1.1vrmq1.commutativeA+B=B+Aassociative(A+B)+C=A+(B+C)3Lettherebelight2.distributiveα(A+B)=αA+αBA−B=A+(−B)3.(dotproduct,scalarproduct)A·B=|A||B|cosθA·B=B·AA·(B+C)=A·B+A·CA·A=|A|2=A24Lettherebelight4.(crossproduct,vectorproduct)A×B=|A||B|sinθˆnA×B=−B×AA×(B+C)=A×B+A×CA×A=0A=Axˆex+Ayˆey+AzˆezA+B=(Ax+Bx)ˆex+(Ay+By)ˆey+(Az+Bz)ˆezA·B=AxBx+AyBy+AzBzA=A·A=A2x+A2y+A2z5LettherebelightA·B=AxBx+AyBy+AzBz=i,jδijAiBjδijKroneckerdelta,δij=ˆei·ˆej=1ifi=j0ifi=jA×B=ˆexˆeyˆezAxAyAzBxByBz=i,j,kεijkAiBjˆekεijkLevi-CivitasymbolLevi-Civitatensorεijk=ˆei·(ˆej׈ek)=⎧⎨⎩1ifijk=123,231,312−1ifijk=132,213,3210otherwise6LettherebelightLevi-Civita1.ˆei׈ej=kεijkˆek2.εijk=−εikj3.εijkεmnkk=δimδjn−δinδjm=δimδinδjmδjn4.n=jεijkεmjkj,k=δimδjj−δijδmj=3δim−δim=2δim5.m=iεijkεijki,j,k=2δii=67Lettherebelight1.(scalartripleproduct)A·(B×C)=B·(C×A)=C·(A×B)A·(B×C)=AxAyAzBxByBzCxCyCz2.(vectortripleproduct)A×(B×C)=(A·C)B−(A·B)CA×(B×C)=(A×B)×CnotassociativeA×(B×C)+B×(C×A)+C×(A×B)=08Lettherebelightr=xˆex+yˆey+zˆezr=|r|=√r·r=x2+y2+z2ˆer=r/rdl≡dr=dxˆex+dyˆey+dzˆezR≡r−r=(x−x)ˆex+(y−y)ˆey+(z−z)ˆezrfieldpointrsourcepointR=|R|=(x−x)2+(y−y)2+(z−z)29Lettherebelight§1.2∇T=∂T∂xˆex+∂T∂yˆey+∂T∂zˆezdT=∂T∂xdx+∂T∂ydy+∂T∂zdz=(∇T)·(dl)=|∇T||dl|cosθ∇TTT10Lettherebelight∇∇T=∂T∂xˆex+∂T∂yˆey+∂T∂zˆez=ˆex∂∂x+ˆey∂∂y+ˆez∂∂zTdel∇∇=ˆex∂∂x+ˆey∂∂y+ˆez∂∂z11LettherebelightAa=⇒∇T(gradient)A·B=⇒∇·v(divergence)A×B=⇒∇×v(curl)∇·v=ˆex∂∂x+ˆey∂∂y+ˆez∂∂z·(vxˆex+vyˆey+vzˆez)=∂vx∂x+∂vy∂y+∂vz∂z12Lettherebelight∇×v=ˆex∂∂x+ˆey∂∂y+ˆez∂∂z×(vxˆex+vyˆey+vzˆez)=ˆexˆeyˆez∂∂x∂∂y∂∂zvxvyvz=ˆex∂vz∂y−∂vy∂z+ˆey∂vx∂z−∂vz∂x+ˆez∂vy∂x−∂vx∂y∇r=∂x2+y2+z2∂xˆex+∂x2+y2+z2∂yˆey+∂x2+y2+z2∂zˆez∇r=xrˆex+yrˆey+zrˆez==rr=ˆer13Lettherebelight∇r=rr=ˆer,∇f(u)=dfdu∇u∇·A(u)=(∇u)·dA(u)du,∇×A(u)=(∇u)×dA(u)du∇r2=2r∇r=2r∇1r=−1r2∇r=−1r2ˆer=−rr3∇·r=ˆex∂∂x+ˆey∂∂y+ˆez∂∂z·(xˆex+yˆey+zˆez)=3∇×r=ˆexˆeyˆez∂∂x∂∂y∂∂zxyz=014Lettherebelight∇∇(f+g)=∇f+∇g∇·(A+B)=∇·A+∇·B∇×(A+B)=∇×A+∇×Bk∇(kf)=k∇f∇·(kA)=k∇·A∇×(kA)=k∇×A15Lettherebelightfg,A·B,fA,A×B∇(fg)=∇f(fg)+∇g(fg)∇1.∇fg∇f2.∇gf∇g=g∇ff+f∇gg∇C(fg)=gCf=fCg∇∇∇∇∇(fg)=g∇f+f∇g∇·(fA)=∇f·(fA)+∇A·(fA)∇1.∇fA∇f2.∇Af∇A=A·(∇ff)+f(∇A·A)∇C·(fA)=A·(Cf)=f(C·A)∇∇∇∇∇·(fA)=A·∇f+f∇·A16Lettherebelight∇×(fA)=∇f×(fA)+∇A×(fA)∇1.∇fA∇f2.∇Af∇A=−A×(∇ff)+f(∇A×A)∇C×(fA)=−A×(Cf)=f(C×A)∇∇∇∇∇×(fA)=−A×∇f+f∇×A17Lettherebelight∇·(A×B)=∇A·(A×B)+∇B·(A×B)∇1.∇AB∇A2.∇BA∇B=B·(∇A×A)−A·(∇B×B)∇C·(A×B)=B·(C×A)=−A·(C×B)∇∇∇∇∇·(A×B)=B·(∇×A)−A·(∇×B)18Lettherebelight∇×(A×B)=∇A×(A×B)+∇B×(A×B)∇1.∇AB∇A2.∇BA∇B=(B·∇A)A−B(∇A·A)+A(∇B·B)−(A·∇B)B∇C×(A×B)=(B·C)A−B(C·A)=A(C·B)−(A·C)B∇∇∇∇∇×(A×B)=(B·∇)A−B(∇·A)−(A·∇)B+A(∇·B)19Lettherebelight(B·∇)=(Bxˆex+Byˆey+Bzˆez)·ˆex∂∂x+ˆey∂∂y+ˆez∂∂z=Bx∂∂x+By∂∂y+Bz∂∂z=⇒(B·∇)A=Bx∂∂x+By∂∂y+Bz∂∂z(Axˆex+Ayˆey+Azˆez)=Bx∂Ax∂xˆex+By∂Ax∂yˆex+Bz∂Ax∂zˆex+Bx∂Ay∂xˆey+By∂Ay∂yˆey+Bz∂Ay∂zˆey+Bx∂Az∂xˆez+By∂Az∂yˆez+Bz∂Az∂zˆez20Lettherebelight∇(A·B)=∇A(A·B)+∇B(A·B)∇1.∇AB∇A2.∇BA∇B=B×(∇A×A)+(B·∇A)A+A×(∇B×B)+(A·∇B)B∇B×(C×A)=C(A·B)−(B·C)A⇓C(A·B)=B×(C×A)+(B·C)AC(A·B)=A×(C×B)+(A·C)B∇∇∇∇∇(A·B)=B×(∇×A)+(B·∇)A+A×(∇×B)+(A·∇)B21Lettherebelight