Axy= Bxydy=Μ0 Jx(Dy)2 Axz= Bxzdz=Μ0 Jx(Dz)2

Supplement2 toeucidean 4dimensionalelectromagnetism

HereI am writingasmallsupplement toeuclidean 4dimensionalelectromagnetismthat is mostabout themagneticalvectorpotential.

Axy=∫Bxydy=µ0∬jx(dy)2 Axz=∫Bxzdz=µ0∬jx(dz)2

Axct=∫Bxctcdt=µ0∬jx(cdt)2

Ax=Axy+Axz-Axct=µ0∬jx((dy)2+(dz)2-(cdt)2)

Ayx=∫Byxdx=µ0∬jy(dx)2 Ayz=∫Byzdz=µ0∬jy(dz)2

Ayct=∫Byctcdt=µ0∬jy(cdt)2

Ay=Ayx+Ayz-Ayct=µ0∬jy((dx)2+(dz)2-(cdt)2)

Azx=∫Bzxdx=µ0∬jz(dx)2 Azy=∫Bzydy=µ0∬jz(dy)2

Azct=∫Bzctcdt=µ0∬jz(cdt)2

Az=Azx+Azy-Azct=µ0∬jz((dx)2+(dy)2-(cdt)2)

Usx/c=∫(Esx/c)dx=µ0∬(ρ0vt)(dx)2

Usy/c=∫(Esy/c)dy=µ0∬(ρ0vt)(dy)2

Usz/c=∫(Esz/c)dz=µ0∬(ρ0vt)(dz)2

Us/c=Usx/c+Usy/c+Usz/c=µ0∬(ρ0vt)((dx)2+(dy)2+(dz)2)

A42=Ax2+Ay2+Az2+(Us/c)2 A4=(-Ax;-Ay;-Az;(Us/c))

WhereAxisthemagneticalvectorpotential fromcurrentsflowing inx-direction

Axyisthemagneticalvectorpotential fromcurrentsflowing inx-direction in they-direction ,Axzisthemagneticalvectorpotential fromcurrentsflowinginx-direction in thez-direction , Axctisthemagneticalvectorpotential fromcurrentsflowing inx-direction in thetime dimension , Ayisthemagneticalvectorpotential fromcurrentsflowing iny-direction , Ayxisthemagneticalvectorpotential fromcurrentsflowing iny-direction in thex-direction , Ayzisthemagneticalvectorpotential fromcurrentsflowing iny-direction in thez-direction , Ayct is the magneticalvector potential from currentsflowing in y-direction in the time dimension , Azisthemagneticalvectorpotential from currentsflowinginz-direction , Azxisthemagneticalvectorpotential fromcurrentsflowing inz-direction in thex-direction , Azyisthemagneticalvectorpotential fromcurrentsflowing inz-direction in they-direction , Azct is the magneticalvector potential from currentsflowing in z-direction in the time dimension , Us/cistheelectrostatic potential/c , Usx/cistheelectrostatic potential/c inx-direction , Usy/cistheelectrostatic potential/c iny-direction , Usz/cistheelectrostatic potential/c inz-direction , A4isthe 4dimensionalelectromagneticalvectorpotential.

Φxy=∬Bxydydx=∫Axydx Φxz=∬Bxzdzdx=∫Axzdx

Φyx=∬Byxdxdy=∫Ayxdy Φyz=∬Byzdzdy=∫Ayzdy

Φzx=∬Bzxdxdz=∫Azxdz Φzy=∬Bzydydz=∫Azydz

WhereΦxyisthe magnetic flux fromcurrentsflowing inx-direction in thexy-plane ,Φxzisthemagnetic flux fromcurrentsflowing inx-direction in thexz-plane , Φyxisthemagnetic flux fromcurrentsflowing iny-direction in thexy-plane , Φyzisthemagnetic flux fromcurrentsflowing iny-direction in theyz-plane , Φzxisthemagnetic flux fromcurrentsflowing inz-direction in thexz-plane , Φzyisthemagnetic flux fromcurrentsflowing inz-direction in theyz-plane.Bxyisthemagnetic flux density fromcurrentsflowing inx-direction in they-direction ,Bxzisthemagnetic flux density fromcurrentsflowing inx-direction in thez-direction , Byxisthemagnetic flux density fromcurrentsflowing iny-direction in thex-direction , Byzisthemagnetic flux density fromcurrentsflowing iny-direction in thez-direction , Bzxisthemagnetic flux density fromcurrentsflowing inz-direction in thex-direction , Bzyisthemagnetic flux density fromcurrentsflowing inz-direction in they-direction , Esx/cistheelectrostaticfield/c in thex-direction , Esy/cistheelectrostaticfield/c in they-direction ,Esz/cistheelectrostaticfield/c in thez-direction.

Ux=∫Exdx=∫(d(Usxcdt)/(cdT))-dΦyx/dT-dΦzx/dT=∫(d(Usxcdt)/(cdT))-∫(d(Ayxdy)/dT)-∫(d(Azxdz)/dT)=vtUsx/c+∫(dUsx/(cdT))cdt-vyAyx-∫(dAyx/dT)dy-vzAzx-∫(dAzx/dT)dz=vtµ0∬(ρ0vt)(dx)2+µ0∫(d(∬(ρ0vt)(dx)2)/dT)cdt-vyµ0∬jy(dx)2-µ0∫(d(∬jy(dx)2)/dT)dy-vzµ0∬jz(dx)2-µ0∫(d(∬jz(dx)2)/dT)dz

Uy=∫Eydy=∫(d(Usycdt)/(cdT))-dΦxy/dT-dΦzy/dT=∫(d(Usycdt)/(cdT))-∫(d(Axydx)/dT)-∫(d(Azydz)/dT)=vtUsy/c+∫(dUsy/(cdT))cdt-vxAxy-∫(dAxy/dT)dx-vzAzy-∫(dAzy/dT)dz=vtµ0∬(ρ0vt)(dy)2+µ0∫(d(∬(ρ0vt)(dy)2)/dT)cdt-vxµ0∬jx(dy)2-µ0∫(d(∬jx(dy)2)/dT)dx-vzµ0∬jz(dy)2-µ0∫(d(∬jz(dy)2)/dT)dz

Uz=∫Ezdz=∫(d(Uszcdt)/(cdT))-dΦxz/dT-dΦyz/dT=∫(d(Uszcdt)/(cdT))-∫(d(Axzdx)/dT)-∫(d(Ayzdy)/dT)=vtUsz/c+∫(dUsz/(cdT))cdt-vxAxz-∫(dAxz/dT)dx-vyAyz-∫(dAyz/dT)dy= =vtµ0∬(ρ0vt)(dz)2+µ0∫(d(∬(ρ0vt)(dz)2)/dT)cdt-vxµ0∬jx(dz)2-µ0∫(d(∬jx(dz)2)/dT)dx-vyµ0∬jy(dz)2-µ0∫(d(∬jy(dz)2)/dT)dy

Uct=∫Ectcdt=∫(d(Axctdx)/dT)+∫(d(Ayctdy)/dT)+∫(d(Azctdz)/dT)=vxAxct+∫(dAxct/dT)dx+vyAyct+∫(dAyct/dT)dy+vzAzct+∫(dAzct/dT)dz=vxµ0∬jx(cdt)2+µ0∫(d(∬jx(cdt)2)/dT)dx+vyµ0∬jy(cdt)2+µ0∫(d(∬jy(cdt)2)/dT)dy+vzµ0∬jz(cdt)2+µ0∫(d(∬jz(cdt)2)/dT)dz

U=Ux+Uy+Uz+Uct=∫Exdx+∫Eydy+∫Ezdz+∫Ectcdt=∫(d(Uscdt)/(cdT))-∫(d(Axdx)/dT)-∫(d(Aydy)/dT)-∫(d(Azdz)/dT)=vtUs/c+∫(dUs/(cdT))cdt-vxAx-∫(dAx/dT)dx-vyAy-∫(dAy/dT)dy-vzAz-∫(dAz/dT)dz=vtµ0∬(ρ0vt)((dx)2+(dy)2+(dz)2)+µ0∫(d(∬(ρ0vt)((dx)2+(dy)2+(dz)2))/dT)cdt-vxµ0∬jx((dy)2+(dz)2-(cdt)2-µ0∫(d(∬jx((dy)2+(dz)2-(cdt)2))/dT)dx-vyµ0∬jy((dx)2+(dz)2-(cdt)2-µ0∫(d(∬jy((dx)2+(dz)2-(cdt)2))/dT)dy-vzµ0∬jz((dx)2+(dy)2-(cdt)2-µ0∫(d(∬jz((dx)2+(dy)2-(cdt)2))/dT)dz

WhereUistheelectricpotential ,Uxistheelectric potential inx-direction,Uyistheelectric potential iny-direction ,Uzistheelectric potential inz-directionandUctistheelectric potential in thetime dimension.

vx2+vy2+vz2+vt2=c2 c=(vx;vy;vz;vt)

vxisthex-komponent of the velocity ,vyis they-komponent of the velocity ,vzisthez-komponent of the velocityandvtis the time velocity , cis thestandard light speed (4velocity), ρ0isthe charge densityandjxisthex-komponent of the currentdensity , jyisthey-komponent of the currentdensity , jzis thez-komponent of the currentdensity ,µ0isthemagneticalconstant

jx2+jy2+jz2+(ρ0vt)2=(ρ0c)2 ρ0c=(jx;jy;jz;(ρ0vt))

(ds4)2=(cdT)2=(dx)2+(dy)2+(dz)2+(cdt)2

ds4=cdT=(dx;dy;dz;cdt)

Whereds4is thesmallestpossible 4distanceanddTisthesmallestpossibleowntime interval

E2=Ex2+Ey2+Ez2+Ect2 E=(Ex;Ey;Ez;Ect)

WhereEistheelectricfieldandExis thex-komponent of the electricfield ,Eyisthey-komponent of the electricfield ,Ezisthez-komponent of the electricfieldandEctistheelectricfield komponent in the time dimension.

Thissupplementshouldbereadtogetherwhitotherpartsofeuclidean 4dimensionalelectromagnetism (for instancetheformula for Uis corrected inthissupplement( thatformula vasa little bitwrong in ”euclidean 4dimensionalelectromagnetism” howevertheformulas for UxUyUzandUctare correct in”euclidean 4 dimensionalelectromagnetism”)) andismostlyabout thevectorpotential andhowitfitstogetherwhittheelectric potential ( thatisascalar) I hopethatthis wouldhelpyoutoconstruct time (zeropoint) energy converters.