micro-economia: geometría una caisha de costat quadrat

F(x,z) = 16·( p( 2x^{2}+4xz )+(-1)mx^{2}z )


d_{x}[F(x,z)] = p( 4x+4z )+(-1)(2m)xz
d_{z}[F(x,z)] = p( 4z )+(-1)mx^{2}


x = 0
z = 0


p=(m^{2}/16)


d_{x}[F(x,z)] = (m^{2}/4)( x+z )+(-1)(2m)xz
d_{z}[F(x,z)] = (m^{2}/4)( z )+(-1)mx^{2}


x=(m/4)
z=(m/4)

constructor matemàtic borrós dos daus

en dos daus:


p=q


1=1
2=2
3=3
4=4
5=5
6=6


p( p=q ) = (6/36) = (1/6)


p+q=2


1+1=2


P( p+q=2 ) = (1/36)


p+q=12


6+6=12


P( p+q=12 ) = (1/36)


p+q=3


1+2=3
2+1=3


P( p+q=3 ) = (2/36) = (1/18)


p+q=11


5+6=11
6+5=11


P( p+q=11 ) = (2/36) = (1/18)


p+q=4


1+3=4
2+2=4
3+1=4


P( p+q=4 ) = (3/36) = (1/12)


p+q=10


4+6=10
5+5=10
6+4=10


P( p+q=10 ) = (3/36) = (1/12)


p+q=5


1+4=5
2+3=5
3+2=5
4+1=5


P( p+q=5 ) = (4/36) = (1/9)


p+q=9


3+6=9
4+5=9
5+4=9
6+3=9


P( p+q=9 ) = (4/36) = (1/9)


p+q=6


1+5=6
2+4=6
3+3=6
4+2=6
5+1=6


P( p+q=6 ) = (5/36)


p+q=8


2+6=8
3+5=8
4+4=8
5+3=8
6+2=8


P( p+q=8 ) = (5/36)


p+q=7


1+6=7
2+5=7
3+4=7
4+3=7
5+2=7
6+1=7


P( p+q=7 ) = (6/36) = (1/6)

limits de successions

lim ( oo+(-n) ) = 0
( oo+(-oo) ) = 0


lim ( (oo+p)+(-n) ) = p
( (oo+p)+(-oo) ) = p


lim ( (oo^{q}+p)+(-1)·n^{q} ) = p
( (oo^{q}+p)+(-1)·oo^{q}) ) = p


lim ( ( (oo+1)^{2}+(-1)(n^{2}+1) )/n ) = 2
( ( (oo+1)^{2}+(-1)(oo^{2}+1) )/oo ) = 2


lim ( ( (oo+1)^{2}+(-1)(n+(-1))^{2} )/n ) = 4
( ( (oo+1)^{2}+(-1)(oo+(-1))^{2} )/oo ) = 4


lim ( ( (oo+1)^{2}+(-1)(n+1) )/(n^{2}+n) ) = 1
( ( (oo+1)^{2}+(-1)(oo+1) )/(oo^{2}+oo) ) = 1


lim ( ( (oo+1)^{3}+(-1)(n^{3}+1) )/(n^{2}+n) ) = 3
( ( (oo+1)^{3}+(-1)(oo^{3}+1) )/(oo^{2}+oo) ) = 3


lim ( ( (oo+1)^{4}+(-1)(n^{2}+1)^{2} )/(n^{3}+n^{2}+n) ) = 4
( ( (oo+1)^{4}+(-1)(oo^{2}+1)^{2} )/(oo^{3}+oo^{2}+oo) ) = 4

progressions aritmética, geométrica y potencial

( a_{1}=b & a_{n+1} = a_{n}+b ) <==> a_{n+1}=(n+1)·b


( a_{1}=b & a_{n+1} = b·a_{n} ) <==> a_{n+1}=b^{(n+1)}


( a_{1}=b & a_{n+1} = (a_{n})^{m} ) <==> a_{n+1}=b^{m^{n+1}}

successions

si ( m >] 0 & a_{n}+m [< a_{n+1} ) ==> a_{n} és creishent
a_{n} [< a_{n}+m [< a_{n+1}


si ( m [< 0 & a_{n}+m >] a_{n+1} ) ==> a_{n} és decreishent
a_{n} >] a_{n}+m >] a_{n+1}

succesions

Si ( a_{1} >] 0 & a_{n+m} >] a_{n}+a_{m} ) ==> a_{n} és creishent
a_{n+1} >] a_{n}+a_{1} >] a_{n}


Si ( a_{1} [< 0 & a_{n+m} [< a_{n}+a_{m} ) ==> a_{n} és decreishent
a_{n+1} [< a_{n}+a_{1} [< a_{n}


Si a_{n+m} >] a_{n}+a_{m} ==> (a_{n}/n) >] a_{1}
a_{n} >] a_{1}+...(n)...+a_{1} = na_{1}


Si a_{n+m} [< a_{n}+a_{m} ==> (a_{n}/n) [< a_{1}
a_{n} [< a_{1}+...(n)...+a_{1} = na_{1}


Si ( n >] a_{n} & a_{n+m} >] a_{n}+a_{m} ) ==> 1 >] (a_{n}/n) >] a_{1}
n >] a_{n}


Si ( n [< a_{n} & a_{n+m} [< a_{n}+a_{m} ) ==> 1 [< (a_{n}/n) [< a_{1}
n [< a_{n}

circuits eléctrics amb bobines y condensadors II

( ∑ (1/L_{k}) )^{(-1)}·d_{tt}^{2}[q(t)] + ...
... ( ∑ a_{j} )·( ∑ (1/a_{k}) )·( ∑ a_{i} )·q(t) = V(t)


( ∑ L_{k} )·d_{tt}^{2}[q(t)] + ...
... ( ( ∑ (1/a_{j}) )·( ∑ a_{k} )·( ∑ (1/a_{i}) ) )^{(-1)}·q(t) = V(t)


( ∑ L_{j} )·( ∑ (1/L_{k}) )·( ∑ L_{i} )·d_{tt}^{2}[q(t)] + ...
... ( ∑ (1/a_{k}) )^{(-1)}·q(t) = V(t)


( ( ∑ (1/L_{j}) )·( ∑ L_{k} )·( ∑ (1/L_{i}) ) )^{(-1)}·d_{tt}^{2}[q(t)] + ...
... ( ∑ a_{k} )·q(t) = V(t)

circuits eléctrics amb bobines y condensadors

( L_{1}+...+L_{n} )·d_{tt}^{2}[q(t)] + ( a_{1}+...+a_{n} )·q(t) = V(t)


( (1/L_{1})+...+(1/L_{n}) )^{(-1)}·d_{tt}^{2}[q(t)] + ( (1/a_{1})+...+(1/a_{n}) )^{(-1)}·q(t) = V(t)


( ∑ L_{j} )·( ∑ (1/L_{k}) )·( ∑ L_{i} )·d_{tt}^{2}[q(t)] + ...
... ( ∑ a_{j} )·( ∑ (1/a_{k}) )·( ∑ a_{i} )·q(t) = V(t)


( ( ∑ (1/L_{j}) )·( ∑ L_{k} )·( ∑ (1/L_{i}) ) )^{(-1)}·d_{tt}^{2}[q(t)] + ...
... ( ( ∑ (1/a_{j}) )·( ∑ a_{k} )·( ∑ (1/a_{i}) ) )^{(-1)}·q(t) = V(t)

circuits eléctrics amb resistències y condensadors II

( ∑ (1/R_{k}) )^{(-1)}·d_{t}[q(t)] + ...
... ( ∑ a_{j} )·( ∑ (1/a_{k}) )·( ∑ a_{i} )·q(t) = V(t)


( ∑ R_{k} )·d_{t}[q(t)] + ...
... ( ( ∑ (1/a_{j}) )·( ∑ a_{k} )·( ∑ (1/a_{i}) ) )^{(-1)}·q(t) = V(t)


( ∑ R_{j} )·( ∑ (1/R_{k}) )·( ∑ R_{i} )·d_{t}[q(t)] + ...
... ( ∑ (1/a_{k}) )^{(-1)}·q(t) = V(t)


( ( ∑ (1/R_{j}) )·( ∑ R_{k} )·( ∑ (1/R_{i}) ) )^{(-1)}·d_{t}[q(t)] + ...
... ( ∑ a_{k} )·q(t) = V(t)

circuits electrics amb resistències y condensadors

( R_{1}+...+R_{n} )·d_{t}[q(t)] + ( a_{1}+...+a_{n} )·q(t) = V(t)


( (1/R_{1})+...+(1/R_{n}) )^{(-1)}·d_{t}[q(t)] + ( (1/a_{1})+...+(1/a_{n}) )^{(-1)}·q(t) = V(t)


( ∑ R_{j} )·( ∑ (1/R_{k}) )·( ∑ R_{i} )·d_{t}[q(t)] + ...
... ( ∑ a_{j} )·( ∑ (1/a_{k}) )·( ∑ a_{i} )·q(t) = V(t)


( ( ∑ (1/R_{j}) )·( ∑ R_{k} )·( ∑ (1/R_{i}) ) )^{(-1)}·d_{t}[q(t)] + ...
... ( ( ∑ (1/a_{j}) )·( ∑ a_{k} )·( ∑ (1/a_{i}) ) )^{(-1)}·q(t) = V(t)

vectors corrent-rotacionals de camps de variables separades

d_{t}[ F(x,y,z) ] = rot[ E(x,y,z) ]

F(x,y,z) = kq·(1/2)·( x_{k} )^{2} [o(t)o] ( x_{k} )^{[o(t)o](-1)} [o(t)o] ...
... ( ∫ [ f(y_{i}) ] d[t] + (-1)·∫ [ f(z_{j}) ] d[t] ).

d_{t}[ D(x,y,z) ] = rot[ B(x,y,z) ]

D(x,y,z) = (-1)·kq·(1/2)·( x_{k} )^{2} [o(t)o] ( x_{k} )^{[o(t)o](-1)} [o(t)o] ...
... ( ∫ [ f( d_{t}[y_{i}]·t ) ] d[t] + (-1)·∫ [ f( d_{t}[z_{j}]·t ) ] d[t] ).

div[ F(x,y,z) ] = kq·∑ ( x_{k}/d_{t}[x_{k}] )·( f(y_{i})+(-1)·f(z_{j}) ).

div[ D(x,y,z) ] = (-1)·kq·∑ ( x_{k}/d_{t}[x_{k}] )·( f( d_{t}[y_{i}]·t )+(-1)·f( d_{t}[z_{j}]·t ) ).

∯ [ F(x,y,z) ] d[(yz,zx,xy)] = ...

... kq·∑ ( x_{k} )^{2} [o(t)o] ( x_{k} )^{[o(x)o](-1)} [o(t)o] ... 

... ( ∫ [ ∫ [ f(y_{i}) ] d[t] ] d[y_{i}]·z_{j} + (-1)·∫ [ ∫ [ f(z_{j}) ] d[t] ] d[z_{j}]·y_{i} ).

∯ [ D(x,y,z) ] d[(yz,zx,xy)] = ...

... (-1)·kq·∑ ( x_{k} )^{2} [o(t)o] ( x_{k} )^{[o(x)o](-1)} [o(t)o] ... 

... ( ∫ [ ∫ [ f( d_{t}[y_{i}]·t ) ] d[t] ] d[y_{i}]·z_{j} + (-1)·∫ [ ∫ [ f( d_{t}[z_{j}]·t ) ] d[t] ] d[z_{j}]·y_{i} ).

m·d_{tt}^{2}[x_{k}] = p( F(x,y,z)+D(x,y,z) )

x_{k} = V_{k}·t

ones para-geométriques electro-magnétiques

m·d_{tt}^{2}[x] = kpq( (x^{m}+(-1))/(x^{n}+(-1))+(-1)·( ((d_{t}[x]·t)^{m}+(-1))/((d_{t}[x]·t)^{n}+(-1)) )


x(t) = vt


para-geométric-electro-magnetismo de dos medios,
olas en la intersección de los dos medios.

física: camp para-geométric-magnétic eléctric

B_{e}( d_{t}[x]·t , d_{t}[y]·t , d_{t}[z]·t ) = (-1)·kq·...
... < ((d_{t}[x]·t)^{m}+(-1))/((d_{t}[x]·t)^{n}+(-1)) , ...
... ((d_{t}[y]·t)^{m}+(-1))/((d_{t}[y]·t)^{n}+(-1)) , ...
... ((d_{t}[z]·t)^{m}+(-1))/((d_{t}[z]·t)^{n}+(-1)) >


flux[ B_{e}( d_{t}[x]·t , d_{t}[y]·t , d_{t}[z]·t ) ] = (-1)·kq·(...
... ((d_{t}[x]·t)^{m}+(-1))/(x((d_{t}[x]·t)^{n}+(-1))) + ...
... ((d_{t}[y]·t)^{m}+(-1))/(y((d_{t}[y]·t)^{n}+(-1))) + ...
... ((d_{t}[z]·t)^{m}+(-1))/(z((d_{t}[z]·t)^{n}+(-1)))
... )·xyz


div[ B_{e}( d_{t}[x]·t , d_{t}[y]·t , d_{t}[z]·t ) ] =  kq·( ...
... m(d_{t}[x]·t)^{m+(-1)}/((d_{t}[x]·t)^{n}+(-1)) + ...
... m(d_{t}[y]·t)^{m+(-1)}/((d_{t}[y]·t)^{n}+(-1)) + ...
... m(d_{t}[z]·t)^{m+(-1)}/((d_{t}[z]·t)^{n}+(-1)) ...
... )+...
... (-1)( ...
... n(d_{t}[x]·t)^{n+(-1)}((d_{t}[x]·t)^{m}+(-1))/((d_{t}[x]·t)^{n}+(-1))^{2} + ...
... n(d_{t}[y]·t)^{n+(-1)}((d_{t}[y]·t)^{m}+(-1))/((d_{t}[y]·t)^{n}+(-1))^{2} + ...
... n(d_{t}[z]·t)^{n+(-1)}((d_{t}[z]·t)^{m}+(-1))/((d_{t}[z]·t)^{n}+(-1))^{2}
... ) ...
... )

física: camps para-geométric-eléctric

E_{e}(x,y,z) = kq·< (x^{m}+(-1))/(x^{n}+(-1)) , (y^{m}+(-1))/(y^{n}+(-1)) , (z^{m}+(-1))/(z^{n}+(-1)) >
E_{e}(1,1,1) = kq·< (m/n) , (m/n) , (m/n) >


flux[ E_{e}(x,y,z) ] = ...
... kq·( (x^{m}+(-1))/(x^{n+1}+(-x)) + (y^{m}+(-1))/(y^{n+1}+(-y)) + (z^{m}+(-1))/(z^{n+1}+(-z))) )·xyz


flux[ E_{e}(1,1,1) ] = 3kq·(m/n)
flux[ E_{e}(0,1,1) ] = kq
flux[ E_{e}(0,0,1) ] = 0
flux[ E_{e}(0,0,0) ] = 0


div[ E_{e}(x,y,z) ] =  kq·( ...
... mx^{m+(-1)}/(x^{n}+(-1)) + ...
... my^{m+(-1)}/(y^{n}+(-1)) + ...
... mz^{m+(-1)}/(z^{n}+(-1)) ...
... )+...
... (-1)( ...
... nx^{n+(-1)}(x^{m}+(-1))/(x^{n}+(-1))^{2} + ...
... ny^{n+(-1)}(y^{m}+(-1))/(y^{n}+(-1))^{2} + ...
... nz^{n+(-1)}(z^{m}+(-1))/(z^{n}+(-1))^{2}
... ) ...
... )


div[ E_{e}(1,1,1) ] =  kq·( ...
... (m/n)(1/(x+(-1))) + ...
... (m/n)(1/(y+(-1))) + ...
... (m/n)(1/(z+(-1))) ...
... )+...
... (-1)( ...
... (nm/n^{2})(1/(x+(-1)) + ...
... (nm/n^{2})(1/(y+(-1)) + ...
... (nm/n^{2})(1/(z+(-1))
... ) ...
... ) = 3kq

vectors corrent de camps de variables separades

d_{t}[ E(x,y,z) ] + rot[ E(x,y,z) ] = J(x,y,z)
d_{t}[ B(x,y,z) ] + rot[ B(x,y,z) ] = H(x,y,z)

J(x,y,z) = ...

... kq·( d_{t}[f(x_{k})]+(x_{k})·( f(y_{i})+(-1)·f(z_{j}) ) ).

H(x,y,z) = ...

... (-1)·kq·( d_{t}[f( d_{t}[x_{k}]·t )]+(x_{k})·( f( d_{t}[y_{i}]·t )+(-1)·f( d_{t}[z_{j}]·t ) ) ).

∯ [ J(x,y,z) ] d[(yz,zx,xy)] = ...
... n·kq·∑ ( (1/(ct)^{n})·( x_{k} )^{(n+(-1))}·d_{t}[x_{k}] + ...
... (-1)·(c/(ct)^{(n+1)})·( x_{k} )^{n} )·(y_{i}z_{j})

∯ [ H(x,y,z) ] d[(yz,zx,xy)] = ...
... (-1)·n·kq·∑ ( (1/(ct)^{n})·( d_{t}[x_{k}]·t )^{(n+(-1))}·( d_{tt}^{2}[x_{k}]·t+d_{t}[x_{k}] ) +...
... (-1)·(c/(ct)^{(n+1)})·( d_{t}[x_{k}]·t )^{n} )·(y_{i}z_{j})

div[ J(x,y,z) ] = kq·∑ d_{tt}^{2}[ f(x_{k}) ]·( 1/d_{t}[x_{k}] )

div[ H(x,y,z) ] = (-1)·kq·∑ d_{tt}^{2}[ f( d_{t}[x_{k}]·t ) ]·( 1/d_{t}[x_{k}] )

m·d_{tt}^{2}[x_{k}] = p( J(x,y,z)+H(x,y,z) )

x_{k} = V_{k}·t

ecuacions de camps

∯ [ E(x,y,z) ] d[(yz,zx,xy)] = Q(x,y,z)
∯ [ B(x,y,z) ] d[(yz,zx,xy)] = A(x,y,z)

∯ [ E(x,y,z) ] d[(yz,zx,xy)] = ∭ [ div[ E(x,y,z) ] ] d[x]d[y]d[z]
∯ [ B(x,y,z) ] d[(yz,zx,xy)] = ∭ [ div[ B(x,y,z) ] ] d[x]d[y]d[z]

div[ E(x,y,z) ] = d_{xyz}^{3}[ Q(x,y,z) ]
div[ B(x,y,z) ] = d_{xyz}^{3}[ A(x,y,z) ]

d_{t}[ E(x,y,z) ] + rot[ E(x,y,z) ] = J(x,y,z)
d_{t}[ B(x,y,z) ] + rot[ B(x,y,z) ] = H(x,y,z)

∯ [ d_{t}[ E(x,y,z) ] ] d[(yz,zx,xy)] = ∯ [ J(x,y,z) ] d[(yz,zx,xy)]
∯ [ d_{t}[ B(x,y,z) ] ] d[(yz,zx,xy)] = ∯ [ H(x,y,z) ] d[(yz,zx,xy)]

div[ d_{t}[ E(x,y,z) ] ] = div[ J(x,y,z) ]
div[ d_{t}[ B(x,y,z) ] ] = div[ H(x,y,z) ]

rot[ E(x,y,z) ] = d_{t}[ F(x,y,z) ]
rot[ B(x,y,z) ] = d_{t}[ D(x,y,z) ]

∯ [ F(x,y,z) ] d[(yz,zx,xy)] = G(x,y,z)
∯ [ D(x,y,z) ] d[(yz,zx,xy)] = S(x,y,z)

∯ [ F(x,y,z) ] d[(yz,zx,xy)] = ∭ [ div[ F(x,y,z) ] ] d[x]d[y]d[z]
∯ [ D(x,y,z) ] d[(yz,zx,xy)] = ∭ [ div[ D(x,y,z) ] ] d[x]d[y]d[z]

div[ F(x,y,z) ] = d_{xyz}^{3}[ G(x,y,z) ]

div[ D(x,y,z) ] = d_{xyz}^{3}[ S(x,y,z) ]

Ecuacions d'ona electro-magnétiques y gravito-magnétiques:

d_{t}[ div[ E(x,y,z)+B(x,y,z) ] ] = Lap[ E(x,y,z)+B(x,y,z) ] [o] (d_{t}[x]+d_{t}[y]+d_{t}[z])

d_{t}[ div[ J(x,y,z)+H(x,y,z) ] ] = Lap[ J(x,y,z)+H(x,y,z) ] [o] (d_{t}[x]+d_{t}[y]+d_{t}[z])

d_{t}[ div[ F(x,y,z)+D(x,y,z) ] ] = Lap[ F(x,y,z)+D(x,y,z) ] [o] (d_{t}[x]+d_{t}[y]+d_{t}[z])

d_{tt}^{2}[ E(x,y,z)+B(x,y,z) ] = Lap[ E(x,y,z)+B(x,y,z) ] [o] (d_{t}[x]^{2}+d_{t}[y]^{2}+d_{t}[z]^{2})

d_{tt}^{2}[ J(x,y,z)+H(x,y,z) ] = Lap[ J(x,y,z)+H(x,y,z) ] [o] (d_{t}[x]^{2}+d_{t}[y]^{2}+d_{t}[z]^{2})

d_{tt}^{2}[ F(x,y,z)+D(x,y,z) ] = Lap[ F(x,y,z)+D(x,y,z) ] [o] (d_{t}[x]^{2}+d_{t}[y]^{2}+d_{t}[z]^{2})

lagranià magnétic eléctric

d_{tt}[r] =  ( d_{t}[r]^{n}/r^{n} )


r(t) = t^{(2+(-n))}


d_{tt}[r] = t^{(-n)}


d_{t}[r(t)]^{n} = t^{n(1+(-n))}


( r(t) )^{(-n)}= t^{(2(-n)+n^{2})}

lagranià eléctric

d_{tt}[r] =  ( 1/r^{n} )


r(t) =  t^{( 2/(1+n) )}


d_{tt}[r] = a^{(-2)n/(n+1)}·t^{( ((-2)n)/(1+n) )}

lagranià para-magnétic eléctric

d_{tt}^{2}[x]= ( d_{t}[x]^{n} )


d_{tt}[x(t)] = a^{(1/2)+(-1)(n/4)}·...
... ( a^{(-1)(1+(-n))/4)} )^{n/(1+(-n))}·a^{(-1)(1+(-n))/2)} = 1


( x(t) )^{n} = a^{(n/2)+(-1)(n^{2}/4)}·...
... ( a^{(-1)(1+(-n))(2+(-n))/4} )^{n/(1+(-n))} = 1


( x(t) ) = a^{(1/2)+(-1)(n/4)}·...
... ( a^{(-1)(1+(-n))/4} )^{(2+(-n))/(1+(-n))}·t^{( (2+(-n))/(1+(-n)) )}


d_{tt}[x(t)] = a^{(1/4)+(-1)(n/4)}·...
... ( a^{(-1)(1+(-n))/4) )^{n/(1+(-n))}·a^{(-1)(1+(-n))/2)} = a^{(-1)(1/4)}


( x(t) )^{n} = a^{(n/4)+(-1)(n^{2}/4)}·...
... ( a^{(-1)(1+(-n))(2+(-n))/4} )^{n/(1+(-n))} = a^{(-1)(n/4)}


( x(t) ) = a^{(-1)(1/4)+(-1)(n/4)}·...
... ( a^{(-1)(1+(-n))/4} )^{(2+(-n))/(1+(-n))}·t^{(2+(-n))/(1+(-n))}


d_{tt}^{2}[x]= ( a^{(n+(-1))(1/4))}·d_{t}[x]^{n}/c^{n} )
m·d_{tt}^{2}[x]= (k_{e}·pq)·d_{t}[x]^{n}/c^{n} )


a^{(n+(-1))(1/4))} = ( (k_{e}·pq)/(mc^{n}) )


a = ( (k_{e}·pq)/(mc^{n}) )^{( 1/(n+(-1))(1/4)) )}

lagranià para-eléctric

d_{tt}^{2}[x]= ( x^{n}/t^{n} )


d_{tt}[x(t)] = a^{(1/2)+(-1)(n/4)}·...
... ( a^{(-1)(1+(-n))/4)} )^{n/(1+(-n))}·a^{(-1)(1+(-n))/2)} = 1


( x(t) )^{n} = a^{(n/2)+(-1)(n^{2}/4)}·...
... ( a^{(-1)(1+(-n))(2+(-n))/4} )^{n/(1+(-n))} = 1


( x(t) ) = a^{(1/2)+(-1)(n/4)}·...
... ( a^{(-1)(1+(-n))/4} )^{(2+(-n))/(1+(-n))}·t^{(2+(-n))/(1+(-n))}


d_{tt}[x(t)] = a^{(1/4)+(-1)(n/4)}·...
... ( a^{(-1)(1+(-n))/4) )^{n/(1+(-n))}·a^{(-1)(1+(-n))/2)} = a^{(-1)(1/4)}


( x(t) )^{n} = a^{(n/4)+(-1)(n^{2}/4)}·...
... ( a^{(-1)(1+(-n))(2+(-n))/4} )^{n/(1+(-n))} = a^{(-1)(n/4)}


( x(t) ) = a^{(-1)(1/4)+(-1)(n/4)}·...
... ( a^{(-1)(1+(-n))/4} )^{(2+(-n))/(1+(-n))}·t^{(2+(-n))/(1+(-n))}


d_{tt}^{2}[x]= ( a^{(n+(-1))(1/4))}·x^{n}/t^{n} )
m·d_{tt}^{2}[x]= (k_{e}·pq)·x^{n}/(ct)^{n} )


a^{(n+(-1))(1/4))} = ( (k_{e}·pq)/(mc^{n}) )


a = ( (k_{e}·pq)/(mc^{n}) )^{( 1/(n+(-1))(1/4)) )}

ones para-electro-magnétiques y para-gravito-magnétiques

m·d_{tt}^{2}[x] = k·pq·( x^{n}/(ct)^{n} )+(-1)·kpq·( d_{t}[x]^{n}/c^{n} )
m·d_{tt}^{2}[x] = k·pq·( x^{n}/(ct)^{n} )+(-1)·kpq·( (d_{t}[x]^{n}·t^{n})/(ct)^{n} )


x(t) = vt


m·d_{tt}^{2}[x] = p·P[E]_{e}(x)+p·P[B]_{e}(x)


d_{tt}[ P[E]_{e}(x)+P[B]_{e}(x) ] = d_{xx}[ P[E]_{e}(x)+P[B]_{e}(x) ]·d_{t}[x]^{2}
d_{tt}[ P[E]_{e}(y)+P[B]_{e}(y) ] = d_{yy}[ P[E]_{e}(y)+P[B]_{e}(y) ]·d_{t}[y]^{2}
d_{tt}[ P[E]_{e}(z)+P[B]_{e}(z) ] = d_{zz}[ P[E]_{e}(z)+P[B]_{e}(z) ]·d_{t}[z]^{2}

para-magnetisme eléctric y para-magnetisme gravitatori

camp para-mangétic eléctric:
B_{e}(d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t) = ...
... (-1)·kq·< (d_{t}[x]·t)^{n}/(ct)^{n} , (d_{t}[y]·t)^{n}/(ct)^{n} , (d_{t}[z]·t)^{n}/(ct)^{n} >


flux[ B_{e}(d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t) ] = ...
... (-1)·kq·(1/(ct)^{n})·A[n]-[ (x_{i})^{(-1)} ](d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t)·xyz


div[ B_{e}(d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t) ] = ...
... (-1)·n·kq·( ...
... (1/(ct)^{n})·A[n+(-1)](d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t) + ...
... (-1)(c/(ct)^{n+1}A[n]-[ d_{t}[x_{i}]^{(-1)} ](d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t)) ...
... )


camp para-mangétic gravitatori:
B_{g}(d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t) = ...
... kq·< (d_{t}[x]·t)^{n}/(ct)^{n} , (d_{t}[y]·t)^{n}/(ct)^{n} , (d_{t}[z]·t)^{n}/(ct)^{n} >


flux[ B_{g}(d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t) ] = ...
... kq·(1/(ct)^{n})·A[n]-[ (x_{i})^{(-1)} ](d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t)·xyz


div[ B_{g}(d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t) ] = ...
... n·kq·( ...
... (1/(ct)^{n})·A[n+(-1)](d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t) + ...
... (-1)(c/(ct)^{n+1}A[n]-[ d_{t}[x_{i}]^{(-1)} ](d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t) ...
... )

potencial para-eléctric y para-gravitatori

potencial eléctric:
E_{e}(x,y,z) = kq·< x^{n}/(ct)^{n} , y^{n}/(ct)^{n} , z^{n}/(ct)^{n} >
V_{e}(x,y,z) = ( 1/(n+1) )·kq·(ct)·Q[n+1](x,y,z)
E_{e}(x,y,z) = grad[ V_{e}(x,y,z) ]


flux[ ∫ [ grad[ V_{e}(x,y,z) ] ]·< d[x],d[y],d[z]> ] = ( 1/(n+1) )·kq·Q[n](x,y,z)·xyz
div[ ∫ [ E_{e}(x,y,z) ]·< d[x],d[y],d[z]> ] = kq·Q[n](x,y,z)


potencial gravitatori:
E_{g}(x,y,z) = (-1)·kq·< x^{n}/(ct)^{n} , y^{n}/(ct)^{n} , z^{n}/(ct)^{n} >
V_{g}(x,y,z) = (-1)·( 1/(n+1) )·kq·(ct)·Q[n+1](x,y,z)
E_{g}(x,y,z) = grad[ V_{e}(x,y,z) ]


flux[ ∫ [ grad[ V_{g}(x,y,z) ] ]·< d[x],d[y],d[z]> ] = (-1)·( 1/(n+1) )·kq·Q[n](x,y,z)·xyz
div[ ∫ [ E_{g}(x,y,z) ]·< d[x],d[y],d[z]> ] = (-1)·kq·Q[n](x,y,z)


ecuacions de camp:
flux[ ∫ [ grad[ V_{e}(x,y,z) ] ]·< d[x],d[y],d[z]> ] = ∭ [ div[ ∫ [ E_{e}(x,y,z) ]·< d[x],d[y],d[z]> ] ] d[x]d[y]d[z]
flux[ ∫ [ grad[ V_{g}(x,y,z) ] ]·< d[x],d[y],d[z]> ] = ∭ [ div[ ∫ [ E_{g}(x,y,z) ]·< d[x],d[y],d[z]> ] ] d[x]d[y]d[z]

camps para-eléctrics y para-gravitatoris

camp eléctric:
E_{e}(x,y,z) = kq·< x^{n}/(ct)^{n} , y^{n}/(ct)^{n} , z^{n}/(ct)^{n} >


flux[ E_{e}(x,y,z) ] = kq·(1/(ct)^{n})·Q[n]-[ (x_{i})^{-1} ](x,y,z)·xyz
div[ E_{e}(x,y,z) ] = n·kq·( (1/(ct)^{n})·Q[n+(-1)](x,y,z) +...
... (-1)(c/(ct)^{n+1})·Q[n]-[ d_{t}[x_{i}]^{(-1)} ](x,y,z)) )


camp gravitatori:
E_{g}(x,y,z) = (-1)·kq·< x^{n}/(ct)^{n} , y^{n}/(ct)^{n} , z^{n}/(ct)^{n} >


flux[ E_{g}(x,y,z) ] = (-1)·kq·(1/(ct))·Q[n]-[ (x_{i})^{-1} ](x,y,z)·xyz
div[ E_{g}(x,y,z) ] = (-1)·n·kq·( (1/(ct)^{n})·Q[n+(-1)](x,y,z) + ...
... (-1)(c/(ct)^{n+1})·Q[n]-[ d_{t}[x_{i}]^{(-1)} ](x,y,z)) )


ecuacions de camp:
flux[ E_{e}(x,y,z) ] = ∭ [ div[ E_{e}(x,y,z) ] ] d[x]d[y]d[z]
flux[ E_{g}(x,y,z) ] = ∭ [ div[ E_{g}(x,y,z) ] ] d[x]d[y]d[z]


ecuacions de camp del temps:
d_{t}[ div[ E_{e}(x,y,z) ] ] = Lap[ E_{e}(x,y,z) ] [o] (d_{t}[x]+d_{t}[y]+d_{t}[z])
d_{t}[ div[ E_{g}(x,y,z) ] ] = Lap[ E_{g}(x,y,z) ] [o] (d_{t}[x]+d_{t}[y]+d_{t}[z])

ecuacions diferencials: binomi

d_{t}[x]^{n}+d_{t}[y]^{n} = (x+y)^{2}


x = ( ( 2^{(1/n)}·(n+(-2))/n )·t )^{( n/(n+(-2)) )}
y = ( ( 2^{(1/n)}·(n+(-2))/n )·t )^{( n/(n+(-2)) )}


d_{t}[x]^{n}+d_{t}[y]^{n} = (x+y)^{3}


x = ( ( 2^{(2/n)}·(n+(-3))/n )·t )^{( n/(n+(-3)) )}
y = ( ( 2^{(2/n)}·(n+(-3))/n )·t )^{( n/(n+(-3)) )}


d_{t}[x]^{n}+d_{t}[y]^{n} = (x+y)^{m}


x = ( ( 2^{((m+(-1))/n)}·(n+(-m))/n )·t )^{( n/(n+(-m)) )}
y = ( ( 2^{((m+(-1))/n)}·(n+(-m))/n )·t )^{( n/(n+(-m)) )}


d_{t}[x]^{n}+d_{t}[y]^{n} = (x+y)^{(1/m)}


x = ( ( 2^{((1/m)+(-1))}·(n+(-1)(1/m))/n )·t )^{( n/(n+(-1)(1/m)) )}
y = ( ( 2^{((1/m)+(-1))}·(n+(-1)(1/m))/n )·t )^{( n/(n+(-1)(1/m)) )}

pitagoras

1+1+1=3
3·(1+1+1)=9
9·(1+1+1)=27
27·(1+1+1)=81


3^{(n+(-1))}( x^{n}+y^{n}+z^{n} ) = R^{n}


2+2+2=6
3·(4+4+4)=36
9·(8+8+8)=216


(1/2)+(1/2)+(1/2)=(3/2)
3·((1/4)+(1/4)+(1/4))=(9/4)
9·((1/8)+(1/8)+(1/8))=(27/8)