ecuacions de Maxwell

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

... < ...

... (1/a^{2})·( d_{yz}^{2}[E_{y}·a^{3}xyz]+(-1)·d_{zy}^{2}[E_{z}·a^{3}xyz] ) , ...

... (1/a^{2})·( d_{zx}^{2}[E_{z}·a^{3}yzx]+(-1)·d_{xz}^{2}[E_{x}·a^{3}yzx] ) , ...

... (1/a^{2})·( d_{xy}^{2}[E_{x}·a^{3}zxy]+(-1)·d_{yx}^{2}[E_{y}·a^{3}zxy] ) ...

... >

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

... < ...

... (1/a)·( d_{x}[E_{y}·a^{3}xyz]+(-1)·d_{x}[E_{z}·a^{3}xyz] ) , ...

... (1/a)·( d_{y}[E_{z}·a^{3}yzx]+(-1)·d_{y}[E_{x}·a^{3}yzx] ) , ...

... (1/a)·( d_{z}[E_{x}·a^{3}zxy]+(-1)·d_{z}[E_{y}·a^{3}zxy] ) ...

... >

Principi:

E(x,y,z) = qk·(1/r^{2})·( < x,y,z >/r )

B(x,y,z) = (-1)·qk·(1/r^{2})·( < d_{t}[x],d_{t}[y],d_{t}[z] >/r )

Lley:

Sigui ( x = r & y = r & z = r ) ==> ( Maxwell-Ampere & Maxwell-Faraday )

Lley de Maxwell-Ampere en forma integral:

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

... qk+(1/3)·anti-potencial[ int[B(x,y,z)]d[t] ]

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

... d_{t}[q(t)]·k+(-1)·(1/3)·anti-potencial[ d_{t}[E(x,y,z,q(t))]+B(x,y,z,q(t)) ]

Lley de Maxwell-Faraday en forma integral:

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

... q·(k/r)+(2/3)·potencial[ int[B(x,y,z)]d[t] ]

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

... d_{t}[q(t)]·(k/r)+(-1)·(2/3)·potencial[ d_{t}[E(x,y,z,q(t))]+B(x,y,z,q(t)) ]

Lley:

Sigui ( x = r & y = r & z = r ) ==> ( Maxwell-Ampere & Maxwell-Faraday )

Lley de Maxwell-Ampere en forma diferencial:

rot[ E(x,y,z) ] = H(x,y,z)+(1/3)·int[B(x,y,z)]d[t]

rot[ B(x,y,z) ] = J(x,y,z,q(t))+(-1)·(1/3)·( d_{t}[E(x,y,z,q(t))]+B(x,y,z,q(t)) )

Lley de Maxwell-Faraday en forma diferencial:

anti-rot[ E(x,y,z) ] = P(x,y,z)+(2/3)·int[B(x,y,z)]d[t]

anti-rot[ B(x,y,z) ] = Q(x,y,z,q(t))+(-1)·(2/3)·( d_{t}[E(x,y,z,q(t))]+B(x,y,z,q(t)) )

Lley:

H(x,y,z) = qk·(1/r^{2})·...

... ( < 2axy+(-2)·azx+(1/3)·r,2ayz+(-2)·axy+(1/3)·r,2azx+(-2)·ayz+(1/3)·r >/r )

J(x,y,z) = (-1)·q·k·(1/r^{2})·...

... ( < (1/d_{t}[y])·d_{t}[ d_{t}[y]·ayx ]+(-1)·(1/d_{t}[z])·d_{t}[ d_{t}[z]·azx ], ...

... (1/d_{t}[z])·d_{t}[ d_{t}[z]·azy ]+(-1)·(1/d_{t}[x])·d_{t}[ d_{t}[x]·axy ], ...

... (1/d_{t}[x])·d_{t}[ d_{t}[x]·axz ]+(-1)·(1/d_{t}[y])·d_{t}[ d_{t}[y]·ayz ] >/r )+...

... (1/3)·d_{t}[q]·k·(1/r^{2})·( < r,r,r >/r )

Lley:

P(x,y,z) = qk·(1/r^{2})·...

... ( < a^{2}y^{2}z+(-1)·a^{2}yz^{2}+(2/3)·r, ...

... a^{2}z^{2}x+(-1)·a^{2}zx^{2}+(2/3)·r, ...

... a^{2}x^{2}y+(-1)·a^{2}xy^{2}+(2/3)·r >/r )

Q(x,y,z) = (-1)·q·k·(1/r^{2})·...

... ( < d_{t}[y]·a^{2}yz+(-1)·d_{t}[z]·a^{2}zy, ...

... d_{t}[z]·a^{2}zx+(-1)·d_{t}[x]·a^{2}xz, ...

... d_{t}[x]·a^{2}xy+(-1)·d_{t}[y]·a^{2}yx >/r )+...

... (2/3)·d_{t}[q]·k·(1/r^{2})·( < r,r,r >/r )

Principi:

E(x,y,z) = qk·(1/r^{2})·...

... ( f(br) )^{(-3)}·( < x·f(bx),y·f(by),z·f(bz) >/r )

B(x,y,z) = (-1)·qk·(1/r^{2})·...

... ( f(br) )^{(-3)}·( < d_{t}[x·f(bx)],d_{t}[y·f(by)],d_{t}[z·f(bz)] >/r )

Lley:

Sigui ( x = r & y = r & z = r ) ==> ( Maxwell-Ampere & Maxwell-Faraday )

Lley de Maxwell-Ampere en forma integral:

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

... qk+(1/3)·anti-potencial[ int[B(x,y,z)]d[t] ]

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

... d_{t}[q(t)]·k+(-1)·(1/3)·anti-potencial[ d_{t}[E(x,y,z,q(t))]+B(x,y,z,q(t)) ]

Lley de Maxwell-Faraday en forma integral:

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

... q·(k/r)·(1/f(br))+(2/3)·potencial[ int[B(x,y,z)]d[t] ]

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

... d_{t}[q(t)]·(k/r)·(1/f(br))+(-1)·(2/3)·potencial[ d_{t}[E(x,y,z,q(t))]+B(x,y,z,q(t)) ]

f(br) = 1

Principi: [ d'un planeta amb dia y nit ]

E(x,y,z) = qk·(1/r^{2})·...

... e^{(-3)·br}·( < x·e^{bx},y·e^{by},z·e^{bz} >/r )

B(x,y,z) = (-1)·qk·(1/r^{2})·...

... e^{(-3)·br}·( < d_{t}[x·e^{bx}],d_{t}[y·e^{by}],d_{t}[z·e^{bz}] >/r )

b = ( (2pi·i)/r )

Principi: [ de propulsió d'un coet ]

E(x,y,z) = qk·(1/r^{2})·...

... ( ln(br) )^{(-3)}·( < x·ln(bx),y·ln(by),z·ln(bz) >/r )

B(x,y,z) = (-1)·qk·(1/r^{2})·...

... ( ln(br) )^{(-3)}·( < d_{t}[x·ln(bx)],d_{t}[y·ln(by)],d_{t}[z·ln(bz)] >/r )

b = ( e/r )

Principi: [ d'ona de volum cosinosoidal ]

E(x,y,z) = qk·(1/r^{2})·...

... ( cos(br) )^{(-3)}·( < x·cos(bx),y·cos(by),z·cos(bz) >/r )

B(x,y,z) = (-1)·qk·(1/r^{2})·...

... ( cos(br) )^{(-3)}·( < d_{t}[x·cos(bx)],d_{t}[y·cos(by)],d_{t}[z·cos(bz)] >/r )

b = ( (2pi)/r )

Principi: [ d'ona de volum sinosoidal ]

E(x,y,z) = qk·(1/r^{2})·...

... ( sin(br) )^{(-3)}·( < x·sin(bx),y·sin(by),z·sin(bz) >/r )

B(x,y,z) = (-1)·qk·(1/r^{2})·...

... ( sin(br) )^{(-3)}·( < d_{t}[x·sin(bx)],d_{t}[y·sin(by)],d_{t}[z·sin(bz)] >/r )

b = ( pi/(2r) )

Ecuacións de variables estocástiques.

Teorema:

0 [< x [< oo

f(x) = (1/2)·e^{(1/2)·(-x)}

0 [< y [< oo

g(y) = (1/2)·e^{(1/2)·(-y)}

z = x+y

x = (z/2) & y = (z/2)

h(z) = p·(1/4)·e^{(1/4)·(-z)}·e^{(1/4)·(-z)} = (1/2)·e^{(1/2)·(-z)} & p = 2

Teorema:

0 [< x [< oo

f(x) = (1/3)·e^{(1/3)·(-x)}

0 [< y [< oo

g(y) = (2/3)·e^{(2/3)·(-y)}

z = x+y

x = (z/2) & y = (z/2)

h(z) = p·(2/9)·e^{(1/6)·(-z)}·e^{(2/6)·(-z)} = (1/2)·e^{(1/2)·(-z)} & p = (9/4)

Teorema:

( cos(x) )^{2}+( sin(x) )^{2} = 1

Demostració:

a^{2}+b^{2} = h^{2}

(a/h)^{2}+(b/h)^{2} = (h/h)^{2} = 1

Teorema:

( 1/sin(x) )^{2}·( 1+cos(x) )·( 1+(-1)·cos(x) ) = 1

( 1/cos(x) )^{2}·( 1+sin(x) )·( 1+(-1)·sin(x) ) = 1

Teorema:

( 1+cos(x) )·( 1+cos(x) )+( sin(x) )^{2} = 2

( 1+sin(x) )·( 1+sin(x) )+( cos(x) )^{2} = 2

Teorema:

( sin(x) )^{2}·( ( 1/(1+cos(x)) )+( 1/(1+(-1)·cos(x)) ) ) = 2

( cos(x) )^{2}·( ( 1/(1+sin(x)) )+( 1/(1+(-1)·sin(x)) ) ) = 2

Teorema:

( sin(x) )^{2}·( 1+( cot(x) )^{2} ) = 1

( cos(x) )^{2}·( 1+( tan(x) )^{2} ) = 1

Teorema:

( cos(x)+sin(x) )^{2}+(-1)·sin(2x) = 1

( cos(x)+(-1)·sin(x) )^{2}+sin(2x) = 1

Teorema:

( 1/( cos(x) )^{2} )·( cos(2x)+( sin(x) )^{2} ) = 1

( 1/( sin(x) )^{2} )·( (-1)·cos(2x)+( cos(x) )^{2} ) = 1