electromagnetisme

div[E(x,y,z)] = d_{xyz}^{3}[ anti-potencial[E(x,y,z)] ]

anti-div[E(x,y,z)] = d_{xyz}^{3}[ potencial[E(x,y,z)] ]

anti-potencial[E(x,y,z)] = ...

... int-int[E_{x}(x,y,z)]d[y]d[z]+int-int[E_{y}(x,y,z)]d[z]d[x]+int-int[E_{z}(x,y,z)]d[x]d[y]

potencial[E(x,y,z)] = ...

... int[E_{x}(x,y,z)]d[x]+int[E_{y}(x,y,z)]d[y]+int[E_{z}(x,y,z)]d[z]

div[E(x,y,z)] = ...

... d_{x}[ E_{x}(x,y,z) ]+d_{y}[ E_{y}(x,y,z) ]+d_{z}[ E_{z}(x,y,z) ]

anti-div[E(x,y,z)] = ...

... d_{yz}^{2}[ E_{x}(x,y,z) ]+d_{zx}^{2}[ E_{y}(x,y,z) ]+d_{xy}^{2}[ E_{z}(x,y,z) ]

anti-potencial[ rot[E(x,y,z)] ] = 0

potencial[ anti-rot[E(x,y,z)] ] = 0

rot[E(x,y,z)] = ...

... < ...

... x·( d_{y}[E_{y}(x,y,z)]+(-1)·d_{z}[E_{z}(x,y,z)] ), ... 

... y·( d_{z}[E_{z}(x,y,z)]+(-1)·d_{x}[E_{x}(x,y,z)] ), ...

... z·( d_{x}[E_{x}(x,y,z)]+(-1)·d_{y}[E_{y}(x,y,z)] ) ...

... >

anti-rot[E(x,y,z)] = ...

... < ...

... yz·( d_{yy}^{2}[E_{y}(x,y,z)]+(-1)·d_{zz}^{2}[E_{z}(x,y,z)] ), ... 

... zx·( d_{zz}^{2}[E_{z}(x,y,z)]+(-1)·d_{xx}^{2}[E_{x}(x,y,z)] ), ...

... xy·( d_{xx}^{2}[E_{x}(x,y,z)]+(-1)·d_{yy}^{2}[E_{y}(x,y,z)] ) ...

... >

rot[E(x,y,z)]+E(x,y,z) = anti-potencial-vector[E(x,y,z)]

anti-rot[E(x,y,z)]+E(x,y,z) = potencial-vector[E(x,y,z)]

anti-potencial[ anti-potencial-vector[E(x,y,z)] ] = anti-potencial[E(x,y,z)]

potencial[ potencial-vector[E(x,y,z)] ] = potencial[E(x,y,z)]

Camp eléctric de un cub rectangular rúbic central:

E_{x}(x,y,z) = ax+byz

E_{y}(x,y,z) = ay+bzx

E_{z}(x,y,z) = az+bxy

div[E(x,y,z)] = 3a

anti-div[E(x,y,z)] = 3b

anti-potencial[E(x,y,z)] = b·( (1/4)·(yz)^{2}+(1/4)·(zx)^{2}+(1/4)·(xy)^{2} )+3a·xyz

potencial[E(x,y,z)] = a·( (1/2)·x^{2}+(1/2)·y^{2}+(1/2)·z^{2} )+3b·xyz

rot[E(x,y,z)] = ...

... a·< x·( 1+(-1) ), y·( 1+(-1) ), z·( 1+(-1) ) > = < 0,0,0 >

anti-rot[E(x,y,z)] = ...

... a·< yz·( 0+(-0) ), zx·( 0+(-0) ), xy·( 0+(-0) ) > = < 0^{2},0^{2},0^{2} >

Camp eléctric de ecuador-meridià rúbic central:

E_{x}(x,y,z) = ae^{isx}+be^{isy+isz}

E_{y}(x,y,z) = ae^{isy}+be^{isz+isx}

E_{z}(x,y,z) = ae^{isz}+be^{isx+isy}

div[E(x,y,z)] = a·(is)·( e^{isx}+e^{isy}+e^{isz} )

anti-div[E(x,y,z)] = b·(is)^{2}·( e^{isy+isz}+e^{isz+isx}+e^{isx+isy} )

anti-potencial[E(x,y,z)] = ...

... a·( e^{isx}yz+e^{isy}zx+e^{isz}xy )+...

... ( b/(is)^{2} )·( e^{isy+isz}+e^{isz+isx}+e^{isx+isy} )

potencial[E(x,y,z)] = ...

... b·( e^{isy+isz}x+e^{isz+isx}y+e^{isx+isy}z )+...

... ( a/(is) )·( e^{isx}+e^{isy}+e^{isz} )

rot[E(x,y,z)] = ...

... a·(is)·< x·( e^{isy}+(-1)·e^{isz} ), y·( e^{isz}+(-1)·e^{isx} ), z·( e^{isx}+(-1)·e^{isy} ) >

anti-rot[E(x,y,z)] = ...

... a·(is)^{2}·< ...

... yz·( e^{isy}+(-1)·e^{isz} ), zx·( e^{isz}+(-1)·e^{isx} ), xy·( e^{isx}+(-1)·e^{isy} ) ...

... >