Mecànica Cuàntica y Mecanisme de Gauge y Relativitat

Hamiltonià de Heisenberg-Srôdinguer:

Lley:

ihc·d_{r}[f(r)] = E(r)·P[ (-oo) [< r [< oo ]·f(r)

f(r) = e^{( 1/(ih) )·int[ (1/c)·E(r)·P[ (-oo) [< r [< oo ] ]d[r]}

Lagranià de Heisenberg-Srôdinguer:

Lley:

(-1)·( h^{2}/m )·d_{r}[f(r)]^{2} = E(r)·P[ (-oo) [< r [< oo ]·( f(r) )^{2}

f(r) = e^{( 1/(ih) )·int[ ( m·E(r)·P[ (-oo) [< r [< oo ] )^{(1/2)} ]d[r]}

Lleys de Heisenberg-Srôdinguer-Newton:

Lley:

ihc·d_{r}[f(r)] = pqg·r·(1/pi)·( 1/(1+(ar)^{2}) )·f(r)

f(r) = e^{( 1/(ih) )·(1/c)·(1/a)·(1/pi)·( ...

... ( ( pq·(g/a)·(1/2)·(ar)^{2} ) [o(ar)o] arc-tan(ar) ) ...

... )

Lley:

(-1)·( h^{2}/m )·d_{r}[f(r)]^{2} = pqg·r·(1/pi)·( 1/(1+(ar)^{2}) )·( f(x) )^{2}

f(r) = e^{( 1/(ih) )·(1/a)·(1/pi)·( ...

... ( m·( pq·(g/a)·(1/2)·(ar)^{2} ) [o(ar)o] arc-tan(ar) )^{[o(ar)o](1/2)}} ...

... )

Lleys de Heisenberg-Srôdinguer-Newton-Parabólic:

Lley:

ihc·d_{r}[f(r)] = ( (-1)·pqg·r+E )·(1/pi)·( 1/(1+(ar)^{2}) )·f(x)

f(x,y) = e^{( 1/(ih) )·(1/c)·(1/a)·(1/pi)·( ...

... ( (-1)·pq·(g/a)·(1/2)·(ar)^{2}+E·ar ) [o(ar)o] arc-tan(ar) ...

... )

Lley:

(-1)·( h^{2}/m )·d_{r}[f(r)]^{2} = ( (-1)·pqg·r+E )·(1/pi)·( 1/(1+(ar)^{2}) )·( f(x) )^{2}

f(r) = e^{( 1/(ih) )·(1/a)·(1/pi)·( ...

... ( m·( (-1)·pq·(g/a)·(1/2)·(ar)^{2}+E·ar ) [o(ar)o] arc-tan(ar) )^{[o(ar)o](1/2)}} ...

... )

Mecanisme de Heisenberg-Higgs:

Lley:

(ih)^{n}·d_{x}[f_{1}(x)]·...(n)...·d_{x}[f_{n}(x)] = ...

... ( p(1) )·...(n)...( p(n) )·f_{1}(x)·...(n)...f_{n}(x)

f_{k}(x) = e^{( 1/(ih) )·int[ p(k) ]d[x]}

Mecanisme de Srôdinguer-Higgs:

Lley:

( (ih)/c )^{n}·d_{t}[f_{1}(t)]·...(n)...·d_{t}[f_{n}(t)] = ...

... ( p(1) )·...(n)...( p(n) )·f_{1}(t)·...(n)...f_{n}(t)

f_{k}(t) = e^{( 1/(ih) )·int[ c·p(k) ]d[t]}

Invariant Gauge de Heisenberg-Higgs:

Lley:

Si ( A_{1}(x) = f_{1}(x)·B_{1}(x) & ...(n)... & A_{n}(x) = f_{n}(x)·B_{n}(x) ) ==> ...

... A_{1}(x)·...(n)...·A_{n}(x) = B_{1}(x)·...(n)...·B_{n}(x)

Invariant Gauge de Srôdinguer-Higgs:

Lley:

Si ( A_{1}(t) = f_{1}(t)·B_{1}(t) & ...(n)... & A_{n}(t) = f_{n}(t)·B_{n}(t) ) ==> ...

... A_{1}(t)·...(n)...·A_{n}(t) = B_{1}(t)·...(n)...·B_{n}(t)

Teoría electro-débil:

SU(2):

3 Neutrins:

Lley:

(-1)·h^{2}·d_{x}[f(x)]·d_{x}[g(x)] = p(f)·p(g)·f(x)·g(x)

p(f)·p(g) = (a+(-b))·(b+(-a))·( mc·x )^{2}

f(x) = e^{( 1/(ih) )·mc·(1/2)·x^{2}·(a+(-b))}

g(x) = e^{( 1/(ih) )·mc·(1/2)·x^{2}·(b+(-a))}

Bosó nuclear de Wienenberg:

Lley:

(-1)·(h/c)^{2}·d_{t}[f(t)]·d_{t}[g(t)] = p(f)·p(g)·f(t)·g(t)

p(f)·p(g) = (u+(-v))·(v+(-u))·( mc·t )^{2}

f(t) = e^{( 1/(ih) )·mc^{2}·(1/2)·t^{2}·(u+(-v))}

g(t) = e^{( 1/(ih) )·mc^{2}·(1/2)·t^{2}·(v+(-u))}

U(2,1):

3 Leptons:

Lley:

(-1)·h^{2}·d_{x}[f(x)]·d_{x}[g(x)] = p(f)·p(g)·f(x)·g(x)

p(f)·p(g) = (-1)·( mc·ax )^{2}

f(x) = e^{( 1/(ih) )·mc·(1/2)·x^{2}·a}

g(x) = e^{(-1)·( 1/(ih) )·mc·(1/2)·x^{2}·a}

Bosó eléctric de Glashow:

Lley:

(-1)·(h/c)^{2}·d_{t}[f(t)]·d_{t}[g(t)] = p(f)·p(g)·f(t)·g(t)

p(f)·p(g) = (-1)·( mc·ut )^{2}

f(t) = e^{( 1/(ih) )·mc^{2}·(1/2)·t^{2}·u}

g(t) = e^{(-1)·( 1/(ih) )·mc^{2}·(1/2)·t^{2}·u}

Teoría gravito-electro-forta:

SU(3):

9 Quarks:

Lley:

(-i)·h^{3}·d_{x}[f(x)]·d_{x}[g(x)]·d_{x}[h(x)] = p(f)·p(g)·p(h)·f(x)·g(x)·h(x)

p(f)·p(g)·p(h) = (a+(-b))·(b+(-d))·(d+(-a))·( mc·x )^{3}

f(x) = e^{( 1/(ih) )·mc·(1/2)·x^{2}·(a+(-b))}

g(x) = e^{( 1/(ih) )·mc·(1/2)·x^{2}·(b+(-d))}

h(x) = e^{( 1/(ih) )·mc·(1/2)·x^{2}·(d+(-a))}

Lley:

(-i)·(h/c)^{3}·d_{t}[f(t)]·d_{t}[g(t)]·d_{t}[h(t)] = p(f)·p(g)·p(h)·f(t)·g(t)·h(t)

p(f)·p(g)·p(h) = (u+(-v))·(v+(-w))·(w+(-u))·( mc·t )^{3}

f(t) = e^{( 1/(ih) )·mc^{2}·(1/2)·t^{2}·(u+(-v))}

g(t) = e^{( 1/(ih) )·mc^{2}·(1/2)·t^{2}·(v+(-w))}

h(t) = e^{( 1/(ih) )·mc^{2}·(1/2)·t^{2}·(w+(-u))}

Lley:

(-i)·h^{3}·d_{x}[f(x)]·d_{x}[g(x)]·d_{x}[h(x)] = p(f)·p(g)·p(h)·f(x)·g(x)·h(x)

p(f)·p(g)·p(h) = (-1)·(a+(-b))·(b+(-d))·(d+(-a))·( mc·x )^{3}

f(x) = e^{(-1)·( 1/(ih) )·mc·(1/2)·x^{2}·(a+(-b))}

g(x) = e^{(-1)·( 1/(ih) )·mc·(1/2)·x^{2}·(b+(-d))}

h(x) = e^{(-1)·( 1/(ih) )·mc·(1/2)·x^{2}·(d+(-a))}

Lley:

(-i)·(h/c)^{3}·d_{t}[f(t)]·d_{t}[g(t)]·d_{t}[h(t)] = p(f)·p(g)·p(h)·f(t)·g(t)·h(t)

p(f)·p(g)·p(h) = (-1)·(u+(-v))·(v+(-w))·(w+(-u))·( mc·t )^{3}

f(t) = e^{(-1)·( 1/(ih) )·mc^{2}·(1/2)·t^{2}·(u+(-v))}

g(t) = e^{(-1)·( 1/(ih) )·mc^{2}·(1/2)·t^{2}·(v+(-w))}

h(t) = e^{(-1)·( 1/(ih) )·mc^{2}·(1/2)·t^{2}·(w+(-u))}

U(3,2):

9 Hexatrons:

Lley:

(-i)·h^{3}·d_{x}[f(x)]·d_{x}[g(x)]·d_{x}[h(x)] = p(f)·p(g)·p(h)·f(x)·g(x)·h(x)

p(f)·p(g)·p(h) = (a+b)·ab·( mc·x )^{3}

f(x) = e^{(-1)·( 1/(ih) )·mc·(1/2)·x^{2}·a}

g(x) = e^{( 1/(ih) )·mc·(1/2)·x^{2}·(a+b)}

h(x) = e^{(-1)·( 1/(ih) )·mc·(1/2)·x^{2}·b}

Lley:

(-i)·(h/c)^{3}·d_{t}[f(t)]·d_{t}[g(t)]·d_{t}[h(t)] = p(f)·p(g)·p(h)·f(t)·g(t)·h(t)

p(f)·p(g)·p(h) = (u+v)·uv·( mc·t )^{3}

f(t) = e^{(-1)·( 1/(ih) )·mc^{2}·(1/2)·t^{2}·u}

g(t) = e^{( 1/(ih) )·mc^{2}·(1/2)·t^{2}·(u+v)}

h(t) = e^{(-1)·( 1/(ih) )·mc^{2}·(1/2)·t^{2}·v}

Lley:

(-i)·h^{3}·d_{x}[f(x)]·d_{x}[g(x)]·d_{x}[h(x)] = p(f)·p(g)·p(h)·f(x)·g(x)·h(x)

p(f)·p(g)·p(h) = (-1)·(a+b)·ab·( mc·x )^{3}

f(x) = e^{( 1/(ih) )·mc·(1/2)·x^{2}·a}

g(x) = e^{(-1)·( 1/(ih) )·mc·(1/2)·x^{2}·(a+b)}

h(x) = e^{( 1/(ih) )·mc·(1/2)·x^{2}·b}

Lley:

(-i)·(h/c)^{3}·d_{t}[f(t)]·d_{t}[g(t)]·d_{t}[h(t)] = p(f)·p(g)·p(h)·f(t)·g(t)·h(t)

p(f)·p(g)·p(h) = (-1)·(u+v)·uv·( mc·t )^{3}

f(t) = e^{( 1/(ih) )·mc^{2}·(1/2)·t^{2}·u}

g(t) = e^{(-1)·( 1/(ih) )·mc^{2}·(1/2)·t^{2}·(u+v)}

h(t) = e^{( 1/(ih) )·mc^{2}·(1/2)·t^{2}·v}

Estructura de l'àtom:

( x_{k} = x || x_{k} = y || x_{k} = z )

Nucli de l'àtom:

Lley:

Protó x_{k}:

1 hexatró eléctric x_{k} = 1

3 quarks x_{k}

Proto-Neutró x_{k}

1 hexatró gravito-eléctric x_{k} = (-2)

3 quarks x_{k}

Neutró x_{k}:

1 hexatró gravitori x_{k} = 1

3 quarks x_{k}

Órbita de l'àtom:

Lley:

1 leptó x_{k} = (-1)

1 anti-leptó x_{k} = 1

Radio-activitat de l'àtom:

1 neutrí x_{k}

1 anti-neutrí x_{k}

Álgebra lineal y Geometría diferencial de Quaternions:

Teorema:

det( < 1,j >,< i,k > ) = 0

det( < 1,i >,< j,k > ) = 0

Teorema:

det( < 1,x^{n} >,< y^{n},z^{n} > ) = 0 <==> ...

... ( x = i^{(1/n)} & y = j^{(1/n)} & z = k^{(1/n)} )

det( < 1,y^{n} >,< x^{n},z^{n} > ) = 0 <==> ...

... ( x = i^{(1/n)} & y = j^{(1/n)} & z = k^{(1/n)} )

Teorema:

det( < (1+i),(k+(-j)) >,< (k+(-j)),(1+i) > ) = det( < (j+k),(1+(-i)) >,< (1+(-i)),(j+k) > )

Forma Fonamental escalar:

Definició:

< a,b > [o] < x,y > ) = ax+by

Teorema:

< 1,i > [o] < 1,i > = 0

Teorema:

< d_{u}[uv],d_{v}[i·uv] > [o] < d_{u}[uv],d_{v}[i·uv] > = 0

Teorema:

< k,j > [o] < k,j > = 0

Teorema:

< d_{u}[k·uv],d_{v}[j·uv] > [o] < d_{u}[k·uv],d_{v}[j·uv] > = 0

Definició:

< a,b,c,d > [o] < x,y,z,ct > ) = ax+by+cz+dct

Teorema:

< 1,i,k,j > [o] < 1,i,k,j > = 0

Teorema:

< 1,x^{n},y^{n},z^{n} > [o] < 1,x^{n},y^{n},z^{n} > = 0 <==> ...

... ( x = i^{(1/n)} & y = j^{(1/n)} & z = k^{(1/n)} )

Teorema:

< d_{x}[xyz·ct],d_{y}[i·xyz·ct],d_{z}[k·xyz·ct],d_{ct}[j·xyz·ct] > [o] ...

... d_{x}[xyz·ct],d_{y}[i·xyz·ct],d_{z}[k·xyz·ct],d_{ct}[j·xyz·ct] = 0

Forma Bilineal Quaterniónica de dimesió 3:

Definició:

< x,y,z > [-|o|-] < u,w,v > = (yv+(-1)·zw)+(xv+zu)+(xw+(-1)·yu) 

Teorema:

< a+x,b+y,c+z > [-|o|-] < u,w,v > = ...

... ((b+y)·v+(-1)·(c+z)·w)+((a+x)·v+(c+z)·u)+((a+x)·w+(-1)·(b+y)·u) = ...

... ( < a,b,c > [-|o|-] < u,w,v > )+( < x,y,z > [-|o|-] < u,w,v > )

Teorema:

< ax,ay,az > [-|o|-] < u,w,v > = ...

... (ayv+(-1)·azw)+(axv+azu)+(axw+(-1)·ayu) = a·( < x,y,z > [-|o|-] < u,w,v > )

Teorema:

< 1,i,1 > [-|o|-] < j,w,k > = 0

< (-1),i,(-1) > [-|o|-] < k,w,j > = 0

Teorema:

< x,y,z > [-|o|-] < u,w,v > = (yv+(-1)·zw)+(xv+zu)+(xw+(-1)·yu) = ...

... < x,y,z > o ( < 0,1,1 >,< (-1),0,1 >,< 1,(-1),0 > ) o < u,w,v >

Geometría Diferencial de Quaternions:

Forma Fonamental Quaterniónica de dimensió 3:

Teorema:

... < d_{x}[f(x,y,z)],d_{y}[g(x,y,z)],d_{z}[h(x,y,z)] > ...

... [-|o|-] ...

... < d_{x}[F(x,y,z)],d_{y}[G(x,y,z)],d_{z}[H(x,y,z)] > ...

... = ...

... < d_{x}[f(x,y,z)],d_{y}[g(x,y,z)],d_{z}[h(x,y,z)] > ...

... o ...

... ( < 0,d[y]d[x],d[z]d[x] >,< (-1)·d[x]d[y],0,d[z]d[y] >,< d[x]d[z],(-1)·d[y]d[z],0 > ) ...

... o ...

... < d_{x}[F(x,y,z)],d_{y}[G(x,y,z)],d_{z}[H(x,y,z)] >

Teorema:

< d_{x}[yxz],d_{y}[i·yzx],d_{z}[zyx] > [-|o|-] ...

... < d_{x}[j·xyz],d_{y}[w·xyz],d_{z}[k·zyx] > = 0

Teorema:

< d_{x}[(-1)·yxz],d_{y}[i·yzx],d_{z}[(-1)·zyx] > [-|o|-] ...

... < d_{x}[k·xyz],d_{y}[w·xyz],d_{z}[j·zyx] > = 0

Demostració:

int-int[ d_{y}[xyz]·d_{z}[xyz] ]d[y]d[z] = (1/4)·(xyz)^{2}

Forma Fonamental Binómica:

Definició:

< a,b > [-(2)-] < x,y > ) = (a+b)·(x+y)

Teorema:

< 1,(-1) > [-(2)-] < 1,(-1) > ) = 0

Teorema:

< d_{u}[uv],d_{v}[(-1)·uv] > [-(2)-] < d_{u}[uv],d_{v}[(-1)·uv] > = 0

Métrica de Minkowski invariant Lorentz-Newton-LaGrange:

Sistema de coordenades A:

Lley:

( < x,y,z > [o] < x,y,z > ) = (ct)^{2} <==> ...

... ( x = ct·cos(u)·cos(v) & y = ct·sin(u)·cos(v) & z = ct·sin(v) )

v = velocitat del sistema de coordenades.

Lley:

( 1/(1+(-1)·(v/c)^{2}) )·( < x,y,z,i·vt > [o] < x,y,z,i·vt > ) = ...

... ( 1/(1+(-1)·(v/c)^{2}) )·( x^{2}+y^{2}+z^{2}+(-1)·(vt)^{2} )

Sistema de coordenades B:

Lley:

( 1/(1+(-1)·(v/c)^{2}) )·( < x,y,z,i·vt > [o] < x,y,z,i·vt > )  = (ct)^{2} <==> ...

... ( x = ct·cos(u)·cos(v) & y = ct·sin(u)·cos(v) & z = ct·sin(v) )

Métrica de Minkowski invariant Lorentz-Hamilton:

Sistema de coordenades A:

Lley:

( < x,y,z > [o] < ct,ct,ct > ) = (ct)^{2} <==> ...

... ( x = ct·( cos(u) )^{2}·( cos(v) )^{2} & ...

... y = ct·( sin(u) )^{2}·( cos(v) )^{2} & z = ct·( sin(v) )^{2} )

v = velocitat del sistema de coordenades.

Lley:

( 1/(1+(-1)·(v/c)) )·( < x,y,z,i·vt > [o] < ct,ct,ct,i·ct > ) = ...

... ( 1/(1+(-1)·(v/c)) )·( x·ct+y·ct+z·ct+(-1)·(vt)·(ct) )

Sistema de coordenades B:

Lley:

( 1/(1+(-1)·(v/c)) )·( < x,y,z,i·vt > [o] < ct,ct,ct,i·ct > )  = (ct)^{2} <==> ...

... ( x = ct·( cos(u) )^{2}·( cos(v) )^{2} & ...

... y = ct·( sin(u) )^{2}·( cos(v) )^{2} & z = ct·( sin(v) )^{2} )

Métrica de Klein-Gordon invariant Lorentz-Newton-LaGrange:

Lley:

( d[x]d[x]+d[y]d[y]+d[z]d[z] ) = d[ct]d[ct] <==> ...

... ( x = ct·cos(u)·cos(v) & y = ct·sin(u)·cos(v) & z = ct·sin(v) )

Lley:

( 1/(1+(-1)·(v/c)^{2}) )·( d[x]d[x]+d[y]d[y]+d[z]d[z]+(-1)·d[vt]d[vt] ) = d[ct]d[ct] <==> ...

... ( x = ct·cos(u)·cos(v) & y = ct·sin(u)·cos(v) & z = ct·sin(v) )

Métrica de Dirac invariant Lorentz-Hamilton:

Lley:

( d[x]d[ct]+d[y]d[ct]+d[z]d[ct] ) = d[ct]d[ct] <==> ...

... ( x = ct·( cos(u) )^{2}·( cos(v) )^{2} & ...

... y = ct·( sin(u) )^{2}·( cos(v) )^{2} & z = ct·( sin(v) )^{2} )

Lley:

( 1/(1+(-1)·(v/c)) )·( d[x]d[ct]+d[y]d[ct]+d[z]d[ct]+(-1)·d[vt]d[ct] ) = d[ct]d[ct] <==> ...

... ( x = ct·( cos(u) )^{2}·( cos(v) )^{2} & ...

... y = ct·( sin(u) )^{2}·( cos(v) )^{2} & z = ct·( sin(v) )^{2} )

Ecuacions de Klein-Gordon invariants Lorentz:

Lley:

( int-int[ P[ 0 [< t [< oo ]·f(t) ]d[x]d[x]+...

... int-int[ P[ 0 [< t [< oo ]·f(t) ]d[y]d[y]+int-int[ P[ 0 [< t [< oo ]·f(t) ]d[z]d[z] ) = ...

... i·(h/m)·int[f(t)]d[t]

P[ 0 [< t [< oo ]·f(t)·( d_{t}[x]^{2}+d_{t}[y]^{2}+d_{t}[z]^{2} ) =  i·(h/m)·d_{t}[f(t)]

( x = ct·cos(u)·cos(v) & y = ct·sin(u)·cos(v) & z = ct·sin(v) )

f(t) = e^{(1/(ih))·( int[ P[ 0 [< t [< oo ] ]d[t] [o(t)o] mc^{2}·t )}

Lley:

( 1/(1+(-1)·(v/c)^{2}) )·( ...

... int-int[ P[ 0 [< t [< oo ]·f(t) ]d[x]d[x]+...

... int-int[ P[ 0 [< t [< oo ]·f(t) ]d[y]d[y]+int-int[ P[ 0 [< t [< oo ]·f(t) ]d[z]d[z]+...

... (-1)·int-int[ P[ 0 [< t [< oo ]·f(t) ]d[vt]d[vt] ) = i·(h/m)·int[f(t)]d[t]

( 1/(1+(-1)·(v/c)^{2}) )·( ...

... P[ 0 [< t [< oo ]·f(t)·( d_{t}[x]^{2}+d_{t}[y]^{2}+d_{t}[z]^{2}+(-1)·d_{t}[vt]^{2} ) = ...

... i·(h/m)·d_{t}[f(t)]

( x = ct·cos(u)·cos(v) & y = ct·sin(u)·cos(v) & z = ct·sin(v) )

f(t) = e^{(1/(ih))·( int[ P[ 0 [< t [< oo ] ]d[t] [o(t)o] mc^{2}·t )}

Ecuació de Klein-Gordon-Newton

Lley:

(2/pi)·( 1/(1+(a·r(t))^{2}) )·f(t)·( d_{t}[x]^{2}+d_{t}[y]^{2}+d_{t}[z]^{2} ) = ...

... i·(h/m)·d_{t}[f(t)]

... ( ...

... x = (-1)·pq·(g/m)·(1/2)·t^{2}·cos(u)·cos(v) & ...

... y = (-1)·pq·(g/m)·(1/2)·t^{2}·sin(u)·cos(v) & ...

... z = (-1)·pq·(g/m)·(1/2)·t^{2}·sin(v) ...

... )

f(t) = e^{(1/(ih))·m·(-1)·(1/a)^{2}·(2/pi)·( ...

... ( a·(-1)·pq·(g/m)·(1/2)·t^{2} ) [o(t)o] arc-tan(a·(-1)·pq·(g/m)·(1/2)·t^{2}) ...

...)}

Lley:

( 1/(1+(-1)·(v/c)^{2}) )·( ...

(2/pi)·( 1/(1+(a·r(t))^{2}) )·f(t)·( ...

... d_{t}[x]^{2}+d_{t}[y]^{2}+d_{t}[z]^{2}+(-1)·d_{2}[vt]^{2} ) = ...

... i·(h/m)·d_{t}[f(t)]

... ( ...

... x = (-1)·pq·(g/m)·(1/2)·t^{2}·cos(u)·cos(v) & ...

... y = (-1)·pq·(g/m)·(1/2)·t^{2}·sin(u)·cos(v) & ...

... z = (-1)·pq·(g/m)·(1/2)·t^{2}·sin(v) ...

... )

f(t) = e^{(1/(ih))·m·( 1/(1+(-1)·(v/c)^{2}) )·(2/pi)·( ...

... (-1)·(1/a)^{2}·( ...

... ( a·(-1)·pq·(g/m)·(1/2)·t^{2} ) [o(t)o] arc-tan(a·(-1)·pq·(g/m)·(1/2)·t^{2}) ...

... )+...

... (-1)·v^{2}·t [o(t)o] arc-tan(a·(-1)·pq·(g/m)·(1/2)·t^{2}) [o(t)o] ...

... ln(t) [o(t)o] (1/a)·(-1)·(1/(pq))·(m/g)·t

... )}

Ecuacions de Dirac invariants Lorentz:

Lley:

( int-int[ P[ 0 [< t [< oo ]·f(t) ]d[x]d[ct]+...

... int-int[ P[ 0 [< t [< oo ]·f(t) ]d[y]d[ct]+int-int[ P[ 0 [< t [< oo ]·f(t) ]d[z]d[ct] ) = ...

... i·(h/m)·int[f(t)]d[t]

P[ 0 [< t [< oo ]·f(t)·( d_{t}[x]·c+d_{t}[y]·c+d_{t}[z]·c ) =  i·(h/m)·d_{t}[f(t)]

( x = ct·( cos(u) )^{2}·( cos(v) )^{2} & y = ct·( sin(u) )^{2}·( cos(v) )^{2} & ...

... z = ct·( sin(v) )^{2} )

r(t) = x+y+z

f(t) = e^{(1/(ih))·( int[ P[ 0 [< t [< oo ] ]d[t] [o(t)o] mc^{2}·t )}

Lley:

( 1/(1+(-1)·(v/c)) )·( ...

... int-int[ P[ 0 [< t [< oo ]·f(t) ]d[x]d[ct]+...

... int-int[ P[ 0 [< t [< oo ]·f(t) ]d[y]d[ct]+int-int[ P[ 0 [< t [< oo ]·f(t) ]d[z]d[ct]+...

... (-1)·int-int[ P[ 0 [< t [< oo ]·f(t) ]d[vt]d[ct] ) = i·(h/m)·int[f(t)]d[t]

( 1/(1+(-1)·(v/c)) )·( ...

... P[ 0 [< t [< oo ]·f(t)·( d_{t}[x]·c+d_{t}[y]·c+d_{t}[z]·c+(-1)·vc ) = ...

... i·(h/m)·d_{t}[f(t)]

( x = ct·( cos(u) )^{2}·( cos(v) )^{2} & y = ct·( sin(u) )^{2}·( cos(v) )^{2} & ...

... z = ct·( sin(v) )^{2} )

r(t) = x+y+z

f(t) = e^{(1/(ih))·( int[ P[ 0 [< t [< oo ] ]d[t] [o(t)o] mc^{2}·t )}

Ecuació de Dirac-Hamilton

Lley:

( ((-d)/pi)·( 1/(1+(a·r(t))^{2}) )·f(t)·( d_{t}[x]·c+d_{t}[y]·c+d_{t}[z]·c ) = ...

... i·(h/m)·d_{t}[f(t)]

... ( ...

... x = x_{0}·e^{(-2)·( (pq·g)/(mc) )·t}·( cos(u) )^{2}·( cos(v) )^{2} & ...

... y = x_{0}·e^{(-2)·( (pq·g)/(mc) )·t}·( sin(u) )^{2}·( cos(v) )^{2} & ...

... z = x_{0}·e^{(-2)·( (pq·g)/(mc) )·t}·( sin(v) )^{2} ...

... )

r(t) = x+y+z

f(t) = e^{(1/(ih))·mc·(1/a)·((-d)/pi)·( ...

... arc-tan(a·x_{0}·e^{(-2)·( (pq·g)/(mc) )·t}) ...

... )}

Lley:

( 1/(1+(-1)·(v/c)) )·( ...

((-d)/pi)·( 1/(1+(a·r(t))^{2}) )·f(t)·( ...

... d_{t}[x]·c+d_{t}[y]·c+d_{t}[z]·c+(-1)·d_{t}[vt]c ) = ...

... i·(h/m)·d_{t}[f(t)]

... ( ...

... x = x_{0}·e^{(-2)·( (pq·g)/(mc) )·t}·( cos(u) )^{2}·( cos(v) )^{2} & ...

... y = x_{0}·e^{(-2)·( (pq·g)/(mc) )·t}·( sin(u) )^{2}·( cos(v) )^{2} & ...

... z = x_{0}·e^{(-2)·( (pq·g)/(mc) )·t}·( sin(v) )^{2} ...

... )

r(t) = x+y+z

f(t) = e^{(1/(ih))·m·( 1/(1+(-1)·(v/c)^{2}) )·((-d)/pi)·( ...

... (1/a)·c·( ...

... arc-tan(a·x_{0}·e^{(-2)·( (pq·g)/(mc) )·t}) ...

... )+...

... (-1)·v^{2}·t [o(t)o] arc-tan(a·x_{0}·e^{(-2)·( (pq·g)/(mc) )·t}) [o(t)o] ...

... (-1)·(1/4)·( 1/(a·x_{0}) )·e^{2·( (pq·g)/(mc) )·t} [o(t)o] ( (mc)/(pq·g) )^{2}·t

... )}

Mecàniques:

Newton-LaGrange:

Lley:

(m/2)·( d_{t}[x]^{2}+d_{t}[y]^{2}+d_{t}[z]^{2} ) = (1/2)·mc^{2}

x(t) = ct·cos(u)·cos(v) & y(t) = ct·sin(u)·cos(v) & z(t) = ct·sin(v)

Métrica de Newton-LaGrange bilineal: 

d[r(t)]d[r(t)] = ...

... < d[x],d[y],d[z] > o ( < 1,0,0 >,< 0,1,0 >,< 0,0,1 > ) o < d[x],d[y],d[z] >

Hamilton:

Lley:

(m/2)·c·( d_{t}[x]+d_{t}[y]+d_{t}[z] ) = (1/2)·mc^{2}

x(t) = ct·( cos(u) )^{2}·( cos(v) )^{2} & ...

... y(t) = ct·( sin(u) )^{2}·( cos(v) )^{2} & z(t) = ct·( sin(v) )^{2}

Métrica de Hamilton lineal:

d[r(t)] = < 1,1,1 > o < d[x],d[y],d[z] >

No ser malvado o seguir al Diablo,

no es de deficiente mental,

porque toda-alguna gente no es y hay condenación según dice el Diablo.

Ser malvado y no seguir al Diablo,

es de deficiente mental,

aunque quizás toda-alguna gente no es y hay condenación según dice el Diablo.

Francisco chupa un Jalisco es ley del mundo,

porque es un chocho y se comete adulterio.

Francisca chupa una Jalisca es ley del mundo,

porque es una polla y se comete adulterio.

Matrius de Dirac:

s_{0} = ( < (-1),0,0,0 >,< 0,(-1),0,0 >,< 0,0,(-1),0 >,< 0,0,0,(-1) > )

s_{x} = ( < 0,0,0,1 >,< 0,0,1,0 >,< 0,1,0,0 >,< 1,0,0,0 > )

s_{y} = ( < 0,0,1,0 >,< 0,0,0,1 >,< 1,0,0,0 >,< 0,1,0,0 > )

s_{z} = ( < 0,1,0,0 >,< 1,0,0,0 >,< 0,0,0,1 >,< 0,0,1,0 > )

Ecuació de Klein-Gordon invariant Lorentz-Newton-LaGrange de funció d'ona 4 dimesional:

Lley:

sum[k = 1]-[3][ s_{k}·d_{t}[x_{k}]^{2} ] o ...

... ( < 1,1,1,1> (1/E)·E(x)·f(x),< 1,1,1,1 >·(1/E)·E(y)·f(y),< 1,1,1,1 >·(1/E)·E(z)·f(z),...

... < 1,1,1,1 >·f(t) > = ...

... i·(h/m)·...

... ( < 1,1,1,1> d_{t}[f(x)],< 1,1,1,1 >·d_{t}[f(y)],< 1,1,1,1 >·d_{t}[f(z)],...

... < 1,1,1,1 >·d_{t}[f(t)] >

Lley:

( 1/(1+(-1)·(v/c)^{2}) )·( s_{0}·v^{2}+sum[k = 1]-[3][ s_{k}·d_{t}[x_{k}]^{2} ] ) o ...

... ( < 1,1,1,1> (1/E)·E(x)·f(x),< 1,1,1,1 >·(1/E)·E(y)·f(y),< 1,1,1,1 >·(1/E)·E(z)·f(z),...

... < 1,1,1,1 >·f(t) > = ...

... i·(h/m)·...

... ( < 1,1,1,1> d_{t}[f(x)],< 1,1,1,1 >·d_{t}[f(y)],< 1,1,1,1 >·d_{t}[f(z)],...

... < 1,1,1,1 >·d_{t}[f(t)] >

Ecuació de Dirac invariant Lorentz-Hamilton de funció d'ona 4 dimesional:

Lley:

sum[k = 1]-[3][ s_{k}·d_{t}[x_{k}]·c ] o ...

... ( < 1,1,1,1> (1/E)·E(x)·f(x),< 1,1,1,1 >·(1/E)·E(y)·f(y),< 1,1,1,1 >·(1/E)·E(z)·f(z),...

... < 1,1,1,1 >·f(t) > = ...

... i·(h/m)·...

... ( < 1,1,1,1> d_{t}[f(x)],< 1,1,1,1 >·d_{t}[f(y)],< 1,1,1,1 >·d_{t}[f(z)],...

... < 1,1,1,1 >·d_{t}[f(t)] >

Lley:

( 1/(1+(-1)·(v/c)) )·( s_{0}·cv+sum[k = 1]-[3][ s_{k}·d_{t}[x_{k}]·c ] ) o ...

... ( < 1,1,1,1> (1/E)·E(x)·f(x),< 1,1,1,1 >·(1/E)·E(y)·f(y),< 1,1,1,1 >·(1/E)·E(z)·f(z),...

... < 1,1,1,1 >·f(t) > = ...

... i·(h/m)·...

... ( < 1,1,1,1> d_{t}[f(x)],< 1,1,1,1 >·d_{t}[f(y)],< 1,1,1,1 >·d_{t}[f(z)],...

... < 1,1,1,1 >·d_{t}[f(t)] >

Lley de Einstein-Newton-LaGrange:

Lley:

(m/2)·d_{t}[r]^{2} = mc^{2}·( 1/( 1+(-1)·(d_{t}[r]/c)^{2} )^{(1/2)} )

r(t) = (2/i)^{( 1/(2+(1/2)·]2[) )}·ct

Deducció:

(d_{t}[r]/c)^{2} = (2/i)·(d_{t}[r]/c)^{(-1)·(1/2)·]2[}

(d_{t}[r]/c)^{2+(1/2)·]2[} = (2/i)

Lley:

(2/i)^{( 1/(2+(1/2)·]2[) )} = a <==> (-4) = a^{6}+(-1)·a^{4}

a^{2} = b <==> (-4) = b^{3}+(-1)·b^{2}

b = y+(1/3)

y^{3}+y^{2}+(1/3)·y+(1/27)+(-1)·y^{2}+(-1)·(2/3)·y+(-1)·(1/9)+4 = 0

y^{3}+(-1)·(1/3)·y+(-1)·(2/27)+4 = 0

Lley de Einstein-Hamilton:

Lley:

(m/2)·c·d_{t}[r] = mc^{2}·( 1/( 1+(-1)·(d_{t}[r]/c) )^{(1/2)} )

r(t) = (2/i)^{( 1/(1+(1/2)·]1[) )}·ct

Deducció:

(d_{t}[r]/c) = (2/i)·(d_{t}[r]/c)^{(-1)·(1/2)·]1[}

(d_{t}[r]/c)^{1+(1/2)·]1[} = (2/i)

Lley:

(2/i)^{( 1/(1+(1/2)·]1[) )} = a <==> (-4) = a^{3}+(-1)·a^{2}

a = y+(1/3)

y^{3}+y^{2}+(1/3)·y+(1/27)+(-1)·y^{2}+(-1)·(2/3)·y+(-1)·(1/9)+4 = 0

y^{3}+(-1)·(1/3)·y+(-1)·(2/27)+4 = 0

Energía cinética y energía en repós de Einstein:

Lley:

E(t)+(-1)·mc^{2} = mc^{2}·( 1+(1/2)·(d_{t}[r]/c)^{2}+(-1) )

Lley:

E(t)+(-1)·mc^{2} = mc^{2}·( 1+(1/2)·(d_{t}[r]/c)+(-1) )

Força de Einstein:

Lley:

m·d_{tt}^{2}[r] = m·d_{tt}^{2}[r]·( 1/( 1+(-1)·(d_{t}[r]/c)^{2} )^{(3/2)} ) = 0

Lley:

(m/2)·c·( d_{tt}^{2}[r]/d_{t}[r] ) = ...

... (m/2)·c·( d_{tt}^{2}[r]/d_{t}[r] )·( 1/( 1+(-1)·(d_{t}[r]/c) )^{(3/2)} ) = 0

Moment de Einstein:

Lley:

m·d_{t}[r] = mc^{2}·( ( t/o(t)o/x ) [o(t)o] ( 1/( 1+(-1)·(d_{t}[r]/c)^{2} )^{(1/2)} ) )

Lley:

(m/2)·c·ln(d_{t}[r]) = mc^{2}·( ( t/o(t)o/x ) [o(t)o] ( 1/( 1+(-1)·(d_{t}[r]/c) )^{(1/2)} ) )

Caminad con la luz,

mientras tengáis luz,

para que vos sorprendan las tinieblas,

porque el que camina con la luz sin saber a donde vatchnar,

no le sorprenden las tinieblas.

Caminad con el sonido,

mientras tengáis sonido,

para que vos sorprenda el silencio,

porque el que camina con el sonido sin saber a donde vatchnar,

no le sorprende el silencio.

Caminad con la luz,

mientras tengáis luz,

para que vos sorprendan las tinieblas,

porque el que camina por las tinieblas,

no ve a donde va.

Caminad con el sonido,

mientras tengáis sonido,

para que vos sorprenda el silencio,

porque el que camina por el silencio,

no oye a donde va.

Relativitat ampliada:

Lley:

(m/2)·d_{t}[r]^{2} = i·mc^{2}·( 1+(-i)·(d_{t}[r]/c)^{2} )^{(1/2)}

r(t) = (2k)^{( 1/(2+(-1)·(1/2)[...(i)...[2]...(i)...]) )}·ct

Lley:

(m/2)·c·d_{t}[r] = i·mc^{2}·( 1+(-i)·(d_{t}[r]/c) )^{(1/2)}

r(t) = (2k)^{( 1/(1+(-1)·(1/2)[...(i)...[1]...(i)...]) )}·ct

Invariants Lorentz inversos:

Lley:

( 1+(-i)·(v/c)^{2} )·( 1/( < x,y,z,j·vt >[o]< x,y,z,j·vt > ) ) = (1/ct)^{2}

Lley:

( 1+(-i)·(v/c) )·( 1/( < x,y,z,j·vt >[o]< ct,ct,ct,j·ct > ) ) = (1/ct)^{2}

Los hombres con la polla grande,

si son hombres fieles,

se van a extinguir y no ver nunca más,

y tienen que andar con la luz sin saber a donde vatchnar,

porque Dios les está concediendo.

Los hombres fieles con la polla grande,

tienen el otro vector del par de vectores de centros del alma de hombre no renovado,

y tienen la polla como un hombre infiel,

en tener solo un vector de centros como un hombre infiel.

Las mujeres con el chocho grande,

si son mujeres fieles,

se van a extinguir y no oír nunca más,

y tienen que andar con el sonido sin saber a donde vatchnar,

porque Diosa les está concediendo.

Las mujeres fieles con el chocho grande,

tienen el otro vector del par de vectores de centros del alma de mujer no renovado,

y tienen el chocho como una mujer infiel,

en tener solo un vector de centros como una mujer infiel.

Teorema: [ de Poisson ]

int[x = (-oo)]-[oo][ ( 1/(2x) ) ]d[x] = (1/6)·pi^{2}

Demostració:

int[ ( 1/(2x) ) ]d[x]+(-1)·int[ ( 1/(2·(-x)) ) ]d[(-x)] = C

lim[x = oo][ x [o(x)o] ( ( x^{2} )^{[o(x)o](-1)}+...+( (x+(-k))^{2} )^{[o(x)o](-1)} )+...

(-x) [o(x)o] ( ( (-x)^{2} )^{[o(x)o](-1)}+...+( ((-x)+k)^{2} )^{[o(x)o](-1)} ) ] = ...

... (1/6)·pi^{2}·oo+(1/6)·pi^{2}·(-oo) = (1/6)·pi^{2}

Teorema: [ de Euler-Mascheroni ]

ln(oo)+(-1)·ln(oo) = ln(-1)+(1/3)·pi^{2}

Demostració:

int[x = (-oo)]-[oo][ ( 1/(2x) ) ]d[x] = (1/2)·( ln(oo)+(-1)·ln(-oo) ) = (1/6)·pi^{2}

Teorema:

ln(oo^{p})+(-1)·ln(oo^{p}) = p·( ln(-1)+(1/3)·pi^{2} )

Teorema:

ln(n) no és convergent

Demostració:

|ln(oo)+(-1)·ln(oo)| = |ln(-1)+(1/3)·pi^{2}| = |pi|·|i+(1/3)·pi| >] s

|a+bi| = |a|+(-1)·|b|

Teorema:

(ln(n)/n) és convergent

Demostració:

|(ln(oo)/oo)+(-1)·(ln(oo)/oo)| = (1/oo)·|ln(oo)+(-1)·ln(oo)| = (1/oo)·|ln(-1)+(1/3)·pi^{2}| = ...

... (1/oo)·|pi|·|i+(1/3)·pi| < s

Teoría de viatges en el temps:

Te dos dimensions el temps ( it & t ) = ( p(-t) || 1 )

(-t) [< 0 [< t

Lley:

¬p(t) = p(-t) <==> p(t) = ¬p(-t)

Lley:

( ( p(-t) || 1 ) & p(t) ) <==> p(t)

Lley:

( ( p(t) & ( q(-t) || 1 ) ) & ( ( p(t) & ( q(-t) || 1 ) ) ==> w(t) )

( q(-t) != ¬p(t) || q(-t) != p(-t) )

( No pot ser matar || No pot ser viure )

Lley:

( ( ( p(t) || p(t) està en el Pare ) & ( q(-t) || 1 ) ) & ...

... ( ( ( p(t) || p(t) està en el Pare ) & ( q(-t) || 1 ) ) ==> w(t) )

Lley:

a [< t [< b <==> (-b) [< (-t) [< (-a)

p(a) = p(-b) & p(-a) = p(b)

A star-trek 4 la tripulació del enteprise està en el Pare en el segle XX.

Les ulleres del capità Kirk es destrueishen cuant es fabriquen les ulleres del present.

L'arma del Pavel Checkov no funciona perque no pot matar.

L'arma del capità Kirk funciona només perque no mata.

La científica se'n pot vaitxnar al present,

perque el temps està bifurcat en it.

La bifurcació acaba cuant tornen al present.

Lley:

L(x,u,v,t) = ...

... pqk·(1/r)^{2}·r(u,v,t)+...

... (-h)·( (c/s)^{(1/3)}·( (3/2^{(1/2)})·V·t )^{(2/3)} )·( e^{iut}+e^{ivt} )

m·(c/s)^{(2/3)}·V^{2}·( (3/2^{(1/2)})·V·t )^{(-1)·(2/3)} = ...

... pqk·( ( pq·(k/m) )^{(1/2)}·t )^{(-1)·(2/3)}

Lley:

L(x,u,v,t) = ...

... (-1)·pqk·(1/r)^{2}·r(u,v,t)+...

... (-h)·( (c/s)^{(1/3)}·( (3/2^{(1/2)})·V·it )^{(2/3)} )·( e^{iut}+e^{ivt} )

m·(c/s)^{(2/3)}·V^{2}·( (3/2^{(1/2)})·V·it )^{(-1)·(2/3)} = ...

... pqk·( (3/2^{(1/2)})·( pq·(k/m) )^{(1/2)}·it )^{(-1)·(2/3)}

Viatge al passat:

e^{(1/3)·pi·i} || (-1) || e^{(-1)·(1/3)·pi·i}

Viatge al present:

e^{(2/3)·pi·i} || 1 || e^{(-1)·(2/3)·pi·i}

Matrius de Pauli:

s_{0} = ( < (-1),0 >,< 0,(-1)> )

s_{x} = ( < 0,1 >,< 1,0 > )

Matrius de Dirac-Pauli:

s_{0} = ( < (-1),0,0 >,< 0,(-1),0 >,< 0,0,(-1) > )

s_{x} = ( < 0,1,0 >,< 0,0,1 >,< 1,0,0 > )

s_{y} = ( < 0,0,1 >,< 1,0,0 >,< 0,1,0 > )

I havere-kate-maruto menjjet-yuto-yamed mutchet-muto,

and I not querere-kate-maruto smash-muto.

I havere-kate-maruto menjjet-yuto-yamed pocket-muto,

and I not querere-kate-maruto smensh-muto.

Kino-yute I vare-kate-maruto drinket-yuto-yam mutchet-muto.

Kino-yute I vare-kate-maruto drinket-yuto-yam pocket-muto.

Asa-yute I wil-kate-maruto drinket-yuto-yam mutchet-muto.

Asa-yute I wil-kate-maruto drinket-yuto-yam pocket-muto.

I querere-kate-maruto a cotet-yuto-yamed wizh miruku.

I querere-kate-maruto a cotet-yuto-yamed wizhwat miruku.

ingenieria técnica en llum y so y calor

1r curs de electrónica analógica

i^{2} = (-1)

q(t) = pe^{at} || q(t) = pe^{(-1)·at}

q(t) = pe^{i·at} || q(t) = pe^{(-i)·at}

2n curs de electrónica virtual

k^{2} = i

j^{2} = (-i)

q(t) = pe^{k·at} || q(t) = pe^{(-k)·at}

q(t) = pe^{j·at} || q(t) = pe^{(-j)·at}

Teoría matemàtica:

Teorema:

d_{t}[ f(t)+g(t) ] = d_{t}[f(t)]+d_{t}[g(t)]

d_{t}[ s·f(t) ] = s·d_{t}[f(t)]

Teorema:

d_{t}[g( f(t) )] = d_{f(t)}[g( f(t) )]·d_{t}[f(t)]

Teorema:

d_{t}[t] = 1

d_{t}[e^{t}] = e^{t}

Teorema:

d_{t}[e^{at}] = ae^{at}

d_{tt}^{2}[e^{at}] = a^{2}·e^{at}

Teorema:

d_{tt}^{2}[ f(t)+g(t) ] = d_{tt}^{2}[f(t)]+d_{tt}^{2}[g(t)]

Teorema:

int[s]d[t] = st

Teorema:

d_{t}[ ln(f(t)) ] = ( d_{t}[f(t)]/f(t) )

Teorema:

d_{t}[ f(t) [o(t)o] g(t) ] = d_{t}[f(t)]·d_{t}[g(t)]

Teoría física:

1r curs de electrónica analógica:

Osciladors:

Lley:

R·d_{t}[q(t)]+(-C)·q(t) = 0

q(t) = pe^{(C/R)·t}

R·d_{t}[q(t)]+C·q(t) = 0

q(t) = pe^{(-1)·(C/R)·t}

Lley:

R·d_{t}[q(t)]+i·(-C)·q(t) = 0

q(t) = pe^{i·(C/R)·t}

R·d_{t}[q(t)]+i·C·q(t) = 0

q(t) = pe^{(-i)·(C/R)·t}

Bifurcadors:

Lley:

L·d_{tt}^{2}[q(t)]+(-C)·q(t) = 0

q(t) = pe^{(C/L)^{(1/2)}·t} || q(t) = pe^{(-1)·(C/L)^{(1/2)}·t}

L·d_{tt}^{2}[q(t)]+C·q(t) = 0

q(t) = pe^{i·(C/L)^{(1/2)}·t} || q(t) = pe^{(-i)·(C/L)^{(1/2)}·t}

Lley:

L·d_{tt}^{2}[f(t)+g(t)]+(-C)·(f(t)+g(t)) = 0

f(t) = pe^{(C/L)^{(1/2)}·t} & g(t) = pe^{(-1)·(C/L)^{(1/2)}·t}

L·d_{tt}^{2}[f(t)+g(t)]+C·(f(t)+g(t)) = 0

f(t) = pe^{i·(C/L)^{(1/2)}·t} & g(t) = pe^{(-i)·(C/L)^{(1/2)}·t}

Distorsoniados:

Lley:

R·d_{t}[q(t)]+(-C)·q(t) = Ae^{st}

q(t) = A·( 1/(s·R+(-C)) )·e^{st}

R·d_{t}[q(t)]+C·q(t) = Ae^{(-1)·st}

q(t) = A·( 1/((-s)·R+C) )·e^{(-1)·st}

Lley:

R·d_{t}[q(t)]+i·(-C)·q(t) = Ae^{i·st}

q(t) = A·(1/i)·( 1/(s·R+(-C)) )·e^{i·st}

R·d_{t}[q(t)]+i·C·q(t) = Ae^{(-i)·st}

q(t) = A·(1/i)·( 1/((-s)·R+C) )·e^{(-i)·st}

Amplificadors:

Lley:

L·d_{tt}^{2}[q(t)]+(-C)·q(t) = Ae^{st}

q(t) = A·( 1/(s^{2}·L+(-C)) )·e^{st}

L·d_{tt}^{2}[q(t)]+(-C)·q(t) = Ae^{(-1)·st}

q(t) = A·( 1/(s^{2}·L+(-C)) )·e^{(-1)·st}

Lley:

L·d_{tt}^{2}[q(t)]+C·q(t) = Ae^{i·st}

q(t) = A·( 1/((-1)·s^{2}·L+C) )·e^{i·st}

L·d_{tt}^{2}[q(t)]+C·q(t) = Ae^{(-i)·st}

q(t) = A·( 1/((-1)·s^{2}·L+C) )·e^{(-i)·st}

Lley:

n resistencies en serie <==> R = ( R_{1}+...+R_{n} )

n resistencies en paralel <==> R = ( (1/R_{1})+...+(1/R_{n}) )^{(-1)}

Lley:

R = ( R_{1}+...+R_{n} )·( (1/R_{1})+...+(1/R_{m}) )·( R_{1}+...+R_{n} )

R = ( ( (1/R_{1})+...+(1/R_{n}) )·( R_{1}+...+R_{m} )·( (1/R_{1})+...+(1/R_{n}) ) )^{(-1)}

Principi:

E(x) = qk·(1/r^{2})·(x/r)

B(x) = (-1)·qk·(1/r^{2})·(d_{t}[x]/r)

Lley:

m·d_{tt}^{2}[h(t)] = p·( E( h(t) )+int[ B( h(t) ) ]d[t] ) = 0

h(t) = ct

(-1)·h(t) = (-c)·t

Microfons y Altavoxums mono:

Lley:

int[ c·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{(C/R)·t}

int[ (-c)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{(C/R)·t}

Lley:

int[ (-c)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{(-1)·(C/R)·t}

int[ c·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{(-1)·(C/R)·t}

Cámares y Pantalles mono:

Lley:

int[ c·(1/i)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{i·(C/R)·t}

int[ (-c)·(1/i)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{i·(C/R)·t}

Lley:

int[ (-c)·(1/i)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{(-i)·(C/R)·t}

int[ c·(1/i)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{(-i)·(C/R)·t}

Microfons y Altavoxums stereos:

Lley:

int[ c·(L/C)·( d_{tt}^{2}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{(C/L)^{(1/2)}·t} || q(t) = pe^{(-1)·(C/L)^{(1/2)}·t}

int[ (-c)·(L/C)·( d_{tt}^{2}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{(C/L)^{(1/2)}·t} || q(t) = pe^{(-1)·(C/L)^{(1/2)}·t}

Lley:

int[ c·(L/C)·( d_{tt}^{2}[f(t)+g(t)]/(f(t)+g(t)) ) ]d[t] = h(t)

f(t) = pe^{(C/L)^{(1/2)}·t} & g(t) = pe^{(-1)·(C/L)^{(1/2)}·t}

int[ (-c)·(L/C)·( d_{tt}^{2}[f(t)+g(t)]/(f(t)+g(t)) ) ]d[t] = (-1)·h(t)

f(t) = pe^{(C/L)^{(1/2)}·t} & g(t) = pe^{(-1)·(C/L)^{(1/2)}·t}

Cámares y Pantalles stereos:

Lley:

int[ (-c)·(L/C)·( d_{tt}^{2}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{i·(C/L)^{(1/2)}·t} || q(t) = pe^{(-i)·(C/L)^{(1/2)}·t}

int[ c·(L/C)·( d_{tt}^{2}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{i·(C/L)^{(1/2)}·t} || q(t) = pe^{(-i)·(C/L)^{(1/2)}·t}

Lley:

int[ (-c)·(L/C)·( d_{tt}^{2}[f(t)+g(t)]/(f(t)+g(t)) ) ]d[t] = h(t)

f(t) = pe^{i·(C/L)^{(1/2)}·t} & g(t) = pe^{(-i)·(C/L)^{(1/2)}·t}

int[ c·(L/C)·( d_{tt}^{2}[f(t)+g(t)]/(f(t)+g(t)) ) ]d[t] = (-1)·h(t)

f(t) = pe^{i·(C/L)^{(1/2)}·t} & g(t) = pe^{(-i)·(C/L)^{(1/2)}·t}

Reproductors de disc:

Lley:

h(t) = ( ln(1/p)+ln(q(t)) ) [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (C/R)·r(t)

q(t) = pe^{(C/R)·t}

h(t) = ( ln(1/p)+ln(q(t)) ) [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (-1)·(C/R)·r(t)

q(t) = pe^{(-1)·(C/R)·t}

Lley:

h(t) = ( ln(1/p)+ln(q(t)) ) [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = i·(C/R)·r(t)

q(t) = pe^{i·(C/R)·t}

h(t) = ( ln(1/p)+ln(q(t)) ) [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (-i)·(C/R)·r(t)

q(t) = pe^{(-i)·(C/R)·t}

Reproductor de disc amb pitch:

Lley:

h(t) = int[ (C/L)^{(1/2)}·( ln(1/p)+ln(q(t)) ) ]d[t] [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (C/L)·t·r(t)

d_{tt}^{2}[h(t)] = (C/L)·r(t)+(C/L)·t·d_{t}[r(t)]

q(t) = pe^{(C/L)^{(1/2)}·t}

h(t) = int[ (C/L)^{(1/2)}·( ln(1/p)+ln(q(t)) ) ]d[t] [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (-1)·(C/L)·t·r(t)

d_{tt}^{2}[h(t)] = (-1)·(C/L)·r(t)+(-1)·(C/L)·t·d_{t}[r(t)]

q(t) = pe^{(-1)·(C/L)^{(1/2)}·t}

Lley:

h(t) = int[ (C/L)^{(1/2)}·( ln(1/p)+ln(q(t)) ) ]d[t] [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = i·(C/L)·t·r(t)

d_{tt}^{2}[h(t)] = i·(C/L)·r(t)+i·(C/L)·t·d_{t}[r(t)]

q(t) = pe^{i·(C/L)^{(1/2)}·t}

h(t) = int[ (C/L)^{(1/2)}·( ln(1/p)+ln(q(t)) ) ]d[t] [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (-i)·(C/L)·t·r(t)

d_{tt}^{2}[h(t)] = (-i)·(C/L)·r(t)+(-i)·(C/L)·t·d_{t}[r(t)]

q(t) = pe^{(-i)·(C/L)^{(1/2)}·t}

Pitch:

Lley:

P = < R_{1},L_{1},...,R_{n},L_{n}>

Q = < L_{1},R_{1},...,L_{n},R_{n}>

2n curs de electrónica virtual:

Osciladors virtuals:

Lley:

R·d_{t}[q(t)]+k·(-C)·q(t) = 0

q(t) = pe^{k·(C/R)·t}

R·d_{t}[q(t)]+k·C·q(t) = 0

q(t) = pe^{(-k)·(C/R)·t}

Lley:

R·d_{t}[q(t)]+j·(-C)·q(t) = 0

q(t) = pe^{j·(C/R)·t}

R·d_{t}[q(t)]+j·C·q(t) = 0

q(t) = pe^{(-j)·(C/R)·t}

Bifurcadors virtuals:

Lley:

L·d_{tt}^{2}[q(t)]+i·(-C)·q(t) = 0

q(t) = pe^{k·(C/L)^{(1/2)}·t} || q(t) = pe^{(-k)·(C/L)^{(1/2)}·t}

L·d_{tt}^{2}[q(t)]+i·C·q(t) = 0

q(t) = pe^{j·(C/L)^{(1/2)}·t} || q(t) = pe^{(-j)·(C/L)^{(1/2)}·t}

Lley:

L·d_{tt}^{2}[f(t)+g(t)]+i·(-C)·(f(t)+g(t)) = 0

f(t) = pe^{k·(C/L)^{(1/2)}·t} & g(t) = pe^{(-k)·(C/L)^{(1/2)}·t}

L·d_{tt}^{2}[f(t)+g(t)]+i·C·(f(t)+g(t)) = 0

f(t) = pe^{j·(C/L)^{(1/2)}·t} & g(t) = pe^{(-j)·(C/L)^{(1/2)}·t}

Distorsoniados virtuals:

Lley:

R·d_{t}[q(t)]+k·(-C)·q(t) = Ae^{k·st}

q(t) = A·(1/k)·( 1/(s·R+(-C)) )·e^{k·st}

R·d_{t}[q(t)]+k·C·q(t) = Ae^{(-k)·st}

q(t) = A·(1/k)·( 1/((-s)·R+C) )·e^{(-k)·st}

Lley:

R·d_{t}[q(t)]+j·(-C)·q(t) = Ae^{j·st}

q(t) = A·(1/j)·( 1/(s·R+(-C)) )·e^{j·st}

R·d_{t}[q(t)]+j·C·q(t) = Ae^{(-j)·st}

q(t) = A·(1/j)·( 1/((-s)·R+C) )·e^{(-j)·st}

Amplificadors virtuals:

Lley:

L·d_{tt}^{2}[q(t)]+i·(-C)·q(t) = Ae^{k·st}

q(t) = A·(1/i)·( 1/(s^{2}·L+(-C)) )·e^{k·st}

L·d_{tt}^{2}[q(t)]+i·(-C)·q(t) = Ae^{(-k)·st}

q(t) = A·(1/i)·( 1/(s^{2}·L+(-C)) )·e^{(-k)·st}

Lley:

L·d_{tt}^{2}[q(t)]+i·C·q(t) = Ae^{j·st}

q(t) = A·(1/i)·( 1/((-1)·s^{2}·L+C) )·e^{j·st}

L·d_{tt}^{2}[q(t)]+i·C·q(t) = Ae^{(-j)·st}

q(t) = A·(1/i)·( 1/((-1)·s^{2}·L+C) )·e^{(-j)·st}

Microfons y Altavoxums virtuals mono:

Lley:

int[ c·(1/k)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{k·(C/R)·t}

int[ (-c)·(1/k)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{k·(C/R)·t}

Lley:

int[ (-c)·(1/k)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{(-k)·(C/R)·t}

int[ c·(1/k)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{(-k)·(C/R)·t}

Cámares y Pantalles virtuals mono:

Lley:

int[ c·(1/j)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{j·(C/R)·t}

int[ (-c)·(1/j)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{j·(C/R)·t}

Lley:

int[ (-c)·(1/j)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{(-j)·(C/R)·t}

int[ c·(1/j)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{(-j)·(C/R)·t}

Microfons y Altavoxums virtuals stereos:

Lley:

int[ c·(1/i)·(L/C)·( d_{tt}^{2}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{k·(C/L)^{(1/2)}·t} || q(t) = pe^{(-k)·(C/L)^{(1/2)}·t}

int[ (-c)·(1/i)·(L/C)·( d_{tt}^{2}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{k·(C/L)^{(1/2)}·t} || q(t) = pe^{(-k)·(C/L)^{(1/2)}·t}

Lley:

int[ c·(1/i)·(L/C)·( d_{tt}^{2}[f(t)+g(t)]/(f(t)+g(t)) ) ]d[t] = h(t)

f(t) = pe^{k·(C/L)^{(1/2)}·t} & g(t) = pe^{(-k)·(C/L)^{(1/2)}·t}

int[ (-c)·(1/i)·(L/C)·( d_{tt}^{2}[f(t)+g(t)]/(f(t)+g(t)) ) ]d[t] = (-1)·h(t)

f(t) = pe^{k·(C/L)^{(1/2)}·t} & g(t) = pe^{(-k)·(C/L)^{(1/2)}·t}

Cámares y Pantalles virtuals stereos:

Lley:

int[ (-c)·(1/i)·(L/C)·( d_{tt}^{2}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{j·(C/L)^{(1/2)}·t} || q(t) = pe^{(-j)·(C/L)^{(1/2)}·t}

int[ c·(1/i)·(L/C)·( d_{tt}^{2}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{j·(C/L)^{(1/2)}·t} || q(t) = pe^{(-j)·(C/L)^{(1/2)}·t}

Lley:

int[ (-c)·(1/i)·(L/C)·( d_{tt}^{2}[f(t)+g(t)]/(f(t)+g(t)) ) ]d[t] = h(t)

f(t) = pe^{j·(C/L)^{(1/2)}·t} & g(t) = pe^{(-j)·(C/L)^{(1/2)}·t}

int[ c·(1/i)·(L/C)·( d_{tt}^{2}[f(t)+g(t)]/(f(t)+g(t)) ) ]d[t] = (-1)·h(t)

f(t) = pe^{j·(C/L)^{(1/2)}·t} & g(t) = pe^{(-j)·(C/L)^{(1/2)}·t}

Reproductors de disc virtuals:

Lley:

h(t) = ( ln(1/p)+ln(q(t)) ) [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = k·(C/R)·r(t)

q(t) = pe^{k·(C/R)·t}

h(t) = ( ln(1/p)+ln(q(t)) ) [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (-k)·(C/R)·r(t)

q(t) = pe^{(-k)·(C/R)·t}

Lley:

h(t) = ( ln(1/p)+ln(q(t)) ) [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = j·(C/R)·r(t)

q(t) = pe^{j·(C/R)·t}

h(t) = ( ln(1/p)+ln(q(t)) ) [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (-j)·(C/R)·r(t)

q(t) = pe^{(-j)·(C/R)·t}

Reproductor de disc virtuals amb pitch:

Lley:

h(t) = int[ (C/L)^{(1/2)}·( ln(1/p)+ln(q(t)) ) ]d[t] [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = k·(C/L)·t·r(t)

d_{tt}^{2}[h(t)] = k·(C/L)·r(t)+k·(C/L)·t·d_{t}[r(t)]

q(t) = pe^{k·(C/L)^{(1/2)}·t}

h(t) = int[ (C/L)^{(1/2)}·( ln(1/p)+ln(q(t)) ) ]d[t] [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (-k)·(C/L)·t·r(t)

d_{tt}^{2}[h(t)] = (-k)·(C/L)·r(t)+(-k)·(C/L)·t·d_{t}[r(t)]

q(t) = pe^{(-k)·(C/L)^{(1/2)}·t}

Lley:

h(t) = int[ (C/L)^{(1/2)}·( ln(1/p)+ln(q(t)) ) ]d[t] [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = j·(C/L)·t·r(t)

d_{tt}^{2}[h(t)] = j·(C/L)·r(t)+j·(C/L)·t·d_{t}[r(t)]

q(t) = pe^{j·(C/L)^{(1/2)}·t}

h(t) = int[ (C/L)^{(1/2)}·( ln(1/p)+ln(q(t)) ) ]d[t] [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (-j)·(C/L)·t·r(t)

d_{tt}^{2}[h(t)] = (-j)·(C/L)·r(t)+(-j)·(C/L)·t·d_{t}[r(t)]

q(t) = pe^{(-j)·(C/L)^{(1/2)}·t}

3r curs de tecnología del calor:

T(x) = (R·q)(x)

T(x) = (P+(-Q))(x)

Increment de temperatura <==> T(x) >] 0

Decrement de temperatura <==> T(x) [< 0

Lley:

v·d_{x}[T(x)]+(-u)·T(x) = 0

T(x) = we^{(u/v)·x}

v·d_{x}[T(x)]+u·T(x) = 0

T(x) = we^{(-1)·(u/v)·x}

Lley:

v·d_{x}[T(x)]+i·(-u)·T(x) = 0

T(x) = we^{i·(u/v)·x}

v·d_{x}[T(x)]+i·u·T(x) = 0

T(x) = we^{(-i)·(u/v)·x}

Lley:

(h/m)·d_{xx}^{2}[T(x)]+(-u)·T(x) = 0

T(x) = we^{( u·(m/h) )^{(1/2)}·x} || T(x) = we^{(-1)·( u·(m/h) )^{(1/2)}·x}

(h/m)·d_{xx}^{2}[T(x)]+u·T(x) = 0

T(x) = we^{i·( u·(m/h) )^{(1/2)}·x} || T(x) = we^{(-i)·( u·(m/h) )^{(1/2)}·x}

Lley:

(h/m)·d_{xx}^{2}[f(x)+g(x)]+(-u)·(f(x)+g(x)) = 0

f(x) = we^{( u·(m/h) )^{(1/2)}·x} & g(x) = we^{(-1)·( u·(m/h) )^{(1/2)}·x}

(h/m)·d_{xx}^{2}[f(x)+g(x)]+u·(f(x)+g(x)) = 0

f(x) = we^{i·( u·(m/h) )^{(1/2)}·x} & g(x) = we^{(-i)·( u·(m/h) )^{(1/2)}·x}

Lley:

v·d_{x}[T(x)]+(-u)·T(x) = Ae^{ax}

T(x) = A·( 1/(av+(-u)) )·e^{ax}

v·d_{x}[T(x)]+u·T(x) = Ae^{(-1)·ax}

T(x) = A·( 1/((-a)·v+u) )·e^{(-1)·ax}

Lley:

v·d_{x}[T(x)]+i·(-u)·T(x) = Ae^{i·ax}

T(x) = A·(1/i)·( 1/(av+(-u)) )·e^{i·ax}

v·d_{x}[T(x)]+i·u·T(x) = Ae^{(-i)·ax}

T(x) = A·(1/i)·( 1/((-a)·v+u) )·e^{(-i)·ax}

Lley:

(h/m)·d_{xx}^{2}[T(x)]+(-u)·T(x) = Ae^{ax}

T(x) = A·( 1/(a^{2}·(h/m)+(-u)) )·e^{ax}

(h/m)·d_{xx}^{2}[T(x)]+(-u)·T(x) = Ae^{(-1)·ax}

T(x) = A·( 1/(a^{2}·(h/m)+(-u)) )·e^{(-1)·ax}

Lley:

(h/m)·d_{xx}^{2}[T(x)]+u·T(x) = Ae^{i·ax}

T(x) = A·( 1/((-1)·a^{2}·(h/m)+u) )·e^{i·ax}

(h/m)·d_{xx}^{2}[T(x)]+u·T(x) = Ae^{(-i)·ax}

T(x) = A·( 1/((-1)·a^{2}·(h/m)+u) )·e^{(-i)·ax}

Lley:

int[ c·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = h(t)

T(x) = we^{(u/v)·x}

int[ (-c)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = (-1)·h(t)

T(x) = we^{(u/v)·x}

Lley:

int[ (-c)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = h(t)

T(x) = we^{(-1)·(u/v)·x}

int[ c·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = (-1)·h(t)

T(x) = we^{(-1)·(u/v)·x}

Lley:

int[ c·(1/i)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = h(t)

T(x) = we^{i·(u/v)·x}

int[ (-c)·(1/i)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = (-1)·h(t)

T(x) = we^{i·(u/v)·x}

Lley:

int[ (-c)·(1/i)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = h(t)

T(x) = we^{(-i)·(u/v)·x}

int[ c·(1/i)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = (-1)·h(t)

T(x) = we^{(-i)·(u/v)·x}

4r curs de tecnología virtual del calor:

Lley:

v·d_{x}[T(x)]+k·(-u)·T(x) = 0

T(x) = we^{k·(u/v)·x}

v·d_{x}[T(x)]+k·u·T(x) = 0

T(x) = we^{(-k)·(u/v)·x}

Lley:

v·d_{x}[T(x)]+j·(-u)·T(x) = 0

T(x) = we^{j·(u/v)·x}

v·d_{x}[T(x)]+j·u·T(x) = 0

T(x) = we^{(-j)·(u/v)·x}

Lley:

(h/m)·d_{xx}^{2}[T(x)]+i·(-u)·T(x) = 0

T(x) = we^{k·( u·(m/h) )^{(1/2)}·x} || T(x) = we^{(-k)·( u·(m/h) )^{(1/2)}·x}

(h/m)·d_{xx}^{2}[T(x)]+i·u·T(x) = 0

T(x) = we^{j·( u·(m/h) )^{(1/2)}·x} || T(x) = we^{(-j)·( u·(m/h) )^{(1/2)}·x}

Lley:

(h/m)·d_{xx}^{2}[f(x)+g(x)]+i·(-u)·(f(x)+g(x)) = 0

f(x) = we^{k·( u·(m/h) )^{(1/2)}·x} & g(x) = we^{(-k)·( u·(m/h) )^{(1/2)}·x}

(h/m)·d_{xx}^{2}[f(x)+g(x)]+i·u·(f(x)+g(x)) = 0

f(x) = we^{j·( u·(m/h) )^{(1/2)}·x} & g(x) = we^{(-j)·( u·(m/h) )^{(1/2)}·x}

Lley:

v·d_{x}[T(x)]+k·(-u)·T(x) = Ae^{k·ax}

T(x) = A·(1/k)·( 1/(av+(-u)) )·e^{k·ax}

v·d_{x}[T(x)]+k·u·T(x) = Ae^{(-k)·ax}

T(x) = A·(1/k)·( 1/((-a)·v+u) )·e^{(-k)·ax}

Lley:

v·d_{x}[T(x)]+j·(-u)·T(x) = Ae^{j·ax}

T(x) = A·(1/j)·( 1/(av+(-u)) )·e^{j·ax}

v·d_{x}[T(x)]+j·u·T(x) = Ae^{(-j)·ax}

T(x) = A·(1/j)·( 1/((-a)·v+u) )·e^{(-j)·ax}

Lley:

(h/m)·d_{xx}^{2}[T(x)]+i·(-u)·T(x) = Ae^{k·ax}

T(x) = A·(1/i)·( 1/(a^{2}·(h/m)+(-u)) )·e^{k·ax}

(h/m)·d_{xx}^{2}[T(x)]+i·(-u)·T(x) = Ae^{(-k)·ax}

T(x) = A·(1/i)·( 1/(a^{2}·(h/m)+(-u)) )·e^{(-k)·ax}

Lley:

(h/m)·d_{xx}^{2}[T(x)]+i·u·T(x) = Ae^{j·ax}

T(x) = A·(1/i)·( 1/((-1)·a^{2}·(h/m)+u) )·e^{j·ax}

(h/m)·d_{xx}^{2}[T(x)]+i·u·T(x) = Ae^{(-j)·ax}

T(x) = A·(1/i)·( 1/((-1)·a^{2}·(h/m)+u) )·e^{(-j)·ax}

Lley:

int[ c·(1/k)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = h(t)

T(x) = we^{k·(u/v)·x}

int[ (-c)·(1/k)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = (-1)·h(t)

T(x) = we^{k·(u/v)·x}

Lley:

int[ (-c)·(1/k)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = h(t)

T(x) = we^{(-k)·(u/v)·x}

int[ c·(1/k)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = (-1)·h(t)

T(x) = we^{(-k)·(u/v)·x}

Lley:

int[ c·(1/j)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = h(t)

T(x) = we^{j·(u/v)·x}

int[ (-c)·(1/j)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = (-1)·h(t)

T(x) = we^{j·(u/v)·x}

Lley:

int[ (-c)·(1/j)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = h(t)

T(x) = we^{(-j)·(u/v)·x}

int[ c·(1/j)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = (-1)·h(t)

T(x) = we^{(-j)·(u/v)·x}

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