Teoría matemática preliminar:
int[ f(x)·d_{u}[x] ] d[u]·int[ g(x)·d_{v}[x] ] d[v] = ...
... int[f(x)] d[x]·int[g(x)] d[x]
d_{u}[ int[f(x)] d[x]·int[g(x)] d[x] ] = ...
... d_{u}[ int[ f(x)·d_{u}[x] ] d[u]·int[g(x)] d[x] ] = ...
... d_{u}[ int[ f(x)·d_{u}[x] ] d[u] ]·int[g(x)] d[x] = ...
... f(x)·d_{u}[x]·int[g(x)] d[x]
d_{v}[ f(x)·d_{u}[x]·int[g(x)] d[x] ] = ...
... d_{v}[ f(x)·d_{u}[x]·int[g(x)·d_{v}[x] ] d[v] ] = ...
... f(x)·d_{u}[x]·d_{v}[ int[g(x)·d_{v}[x] ] d[v] ] = ...
... f(x)·d_{u}[x]·g(x)·d_{v}[x]
d_{uu}^{2}[ ...
... int-int[ f(x) ] d[x]d[u] [o( ux )o] int-int[ g(x) ] d[x]d[u] ...
... ] = ...
d_{uu}^{2}[ ...
... int-int[ f(x)·d_{u}[x] ] d[u]d[u] [o( (1/2)·u^{2} )o] int-int[ g(x)·d_{u}[x] ] d[u]d[u] ...
... ] = ...
... f(x)·d_{u}[x]·g(x)·d_{u}[x]
( f(u) )^{[o(x)o]n} = ...
... int[ ( f(u) )^{n} ] d[x] = int[ ( f(u) )^{n} ] d_{u}[x] d[u] = ...
... ( int[f(u)] d[u] )^{[o(u)o]n} [o(u)o] x
( f(u) )^{[o(x)o]n} [o(x)o] int[ g(x) ] d[x] = ...
... int[ ( f(u) )^{n} ] d[x] [o(x)o] int[ g(x) ] d[x] = int[ ( f(u) )^{n}·g(x) ] d[x] = ...
... int[ ( f(u) )^{n}·g(x) ] d_{u}[x] d[u] = ...
... ( int[f(u)] d[u] )^{[o(u)o]n} [o(u)o] int[g(x)] d[x]
( d_{x}[ ( f(u) )^{[o(x)o]n} ] )^{(1/n)} = f(x)
Métrica de cuerda:
M_{uv} = int-int[ R_{111}^{2} ] d_{u}[x]d_{u}[x] d[u]d[v]
M_{vu} = int-int[ R_{222}^{1} ] d_{v}[x]d_{v}[x] d[v]d[u]
Métrica de cuerda diagonal:
M_{uu} = int-int[ R_{111}^{2} ] d_{v}[x]d_{u}[x] d[u]d[u]
M_{vv} = int-int[ R_{222}^{1} ] d_{v}[u]d_{v}[x] d[v]d[v]
Acción de cuerda:
S_{uv} = int-int[ E(u,u)·F(u,v) ] d[u]d[v]
S_{vu} = int-int[ G(v,v)·F(v,u) ] d[v]d[u]
Acción de cuerda diagonal:
S_{uu} = int-int[ E(u,u) ] d[u]d[u]
S_{vv} = int-int[ G(v,v) ] d[v]d[v]
E(u,u)·F(u,v) = ( d_{u}[f(u,v)]·d_{u}[f(u,v)] )·( d_{u}[f(u,v)]·d_{v}[f(u,v)] )
G(v,v)·F(v,u) = ( d_{v}[f(u,v)]·d_{v}[f(u,v)] )·( d_{v}[f(u,v)]·d_{u}[f(u,v)] )
S_{uu} = d_{x}[ E_{u}( d_{u}[x] ) ]
S_{vv} = d_{x}[ E_{v}( d_{v}[x] ) ]
S_{uu} = d_{x}[ ( H_{u}( d_{u}[x] ) ) ]^{(1/2)}
S_{vv} = d_{x}[ ( H_{v}( d_{v}[x] ) ) ]^{(1/2)}
L(x,u,v) = E(x)+h·f(u,v) = 0
R_{111}^{2} = ( E(u,u)·F(u,v) )^{(-1)}
R_{222}^{1} = ( G(v,v)·F(v,u) )^{(-1)}
cuerda exponencial-compleja:
h = (-1)·qg
L(x,u,v) = qg·x(u,v)+h·( e^{iu}+e^{(-i)v} ) = 0
x(u,v)= e^{iu}+e^{(-i)v}
S_{uu} = int-int[ (-1)·e^{i·2u} ] d[u]d[u] = (1/4)·e^{i·2u} = ...
... (-1)·(1/4)·d_{u}[x(u,v)]^{2}
S_{vv} = int-int[ (-1)·e^{(-i)·2v} ] d[v]d[v] = (1/4)·e^{(-i)·2v} = ...
... (-1)·(1/4)·d_{v}[x(u,v)]^{2}
d_{x}[ E_{u}( d_{u}[x] ) ] = (-1)·(1/4)·d_{u}[x]^{2}
E_{u}( d_{u}[x] ) = (-1)·(1/4)·int[ d_{u}[x]^{2} ] d[x] = ...
... (-1)·(1/4)·int[ d_{u}[x]^{3} ] d[u] = (-1)·(1/4)·int[ (-i)·e^{i·3u} ] d[u] = ...
... (1/12)·e^{i·3u} = ...
E_{u}( d_{u}[x] ) = i·(1/12)·d_{u}[x]^{3}
d_{x}[ i·(1/12)·d_{u}[x]^{3} ] = ...
... d_{d_{u}[x]}[ i·(1/12)·d_{u}[x]^{3} ]·d_{x}[ d_{u}[x] ] = ...
... d_{d_{v}[x]}[ i·(1/12)·d_{v}[x]^{3} ]·d_{uu}^{2}[x]·( 1/(i·e^{iv}) ) = ...
... (-1)·(1/4)·d_{u}[x]^{2}
d_{x}[ E_{v}( d_{v}[x] ) ] = (-1)·(1/4)·d_{v}[x]^{2}
E_{v}( d_{v}[x] ) = (-1)·(1/4)·int[ d_{v}[x]^{2} ] d[x] = ...
... (-1)·(1/4)·int[ d_{v}[x]^{3} ] d[v] = (-1)·(1/4)·int[ i·e^{(-i)·3v} ] d[v] = ...
... (1/12)·e^{(-i)·3v}
E_{v}( d_{v}[x] ) = (-i)·(1/12)·d_{v}[x]^{3}
d_{x}[ (-i)·(1/12)·d_{v}[x]^{3} ] = ...
... d_{d_{v}[x]}[ (-i)·(1/12)·d_{v}[x]^{3} ]·d_{x}[ d_{v}[x] ] = ...
... d_{d_{v}[x]}[ (-i)·(1/12)·d_{v}[x]^{3} ]·d_{vv}^{2}[x]·( 1/((-i)·e^{(-i)v}) ) = ...
... (-1)·(1/4)·d_{v}[x]^{2}
d_{x}[ ( H_{u}( d_{u}[x] ) ) ]^{(1/2)} = (-1)·(1/4)·d_{u}[x]^{2}
H_{u}( d_{u}[x] ) = ( (-1)·(1/4)·d_{u}[x]^{2} )^{[o(x)o]2} = ...
... ( (1/4)·e^{i·2u} )^{[o(x)o]2}
H_{u}( d_{u}[x] ) = ( (1/4)·( 1/(2i) )·e^{i·2u} )^{[o(u)o]2} [o(u)o] x = ...
... ( (1/16)·(1/(4i))·e^{i·4u} ) [o(u)o] x
... ( (1/16)·(1/(4i))·d_{u}[x]^{4} ) [o(u)o] int[ d_{u}[x] ] d[u]
d_{x}[ ( (1/16)·(1/(4i))·d_{u}[x]^{4} ) [o(u)o] int[ d_{u}[x] ] d[u] ] = ...
... d_{u}[ ( (1/16)·(1/(4i))·d_{u}[x]^{4} ) [o(u)o] int[ d_{u}[x] ] d[u] ]·(1/d_{u}[x]) = ...
... (1/16)·(1/i)·d_{u}[x]^{3}d_{uu}^{2}[x] = (1/16)·e^{i·3u}·e^{iu} = (1/16)·e^{i·4u} = ...
... (1/16)·d_{u}[x]^{4} = ( (-1)·(1/4)·d_{u}[x]^{2} )^{2}
d_{x}[ ( H_{v}( d_{v}[x] ) ) ]^{(1/2)} = (-1)·(1/4)·d_{v}[x]^{2}
H_{v}( d_{v}[x] ) = ( (-1)·(1/4)·d_{v}[x]^{2} )^{[o(x)o]2} = ...
... ( (1/4)·e^{(-i)·2v} )^{[o(x)o]2}
H_{v}( d_{v}[x] ) = ( (1/4)·( 1/(2(-i)) )·e^{(-i)·2v} )^{[o(v)o]2} [o(v)o] x = ...
... ( (1/16)·( 1/(4(-i)) )·e^{(-i)·4v} ) [o(v)o] x
... ( (1/16)·( 1/(4(-i)) )·d_{v}[x]^{4} ) [o(v)o] int[ d_{v}[x] ] d[v]
d_{x}[ ( (1/16)·(1/(4i))·d_{v}[x]^{4} ) [o(v)o] int[ d_{v}[x] ] d[v] ] = ...
... d_{v}[ ( (1/16)·(1/(4i))·d_{v}[x]^{4} ) [o(v)o] int[ d_{v}[x] ] d[v] ]·(1/d_{v}[x]) = ...
... (1/16)·(1/i)·d_{v}[x]^{3}d_{vv}^{2}[x] = (1/16)·e^{i·3v}·e^{iv} = (1/16)·e^{i·4v} = ...
... (1/16)·d_{v}[x]^{4} = ( (-1)·(1/4)·d_{v}[x]^{2} )^{2}
S_{uv} = int-int[ (-1)·e^{i·3u}·e^{(-i)v} ] d[u]d[v] = (-1)·(1/3)·e^{i·3u}·e^{(-i)v}
... (1/3)·d_{u}[x(u,v)]^{3}·d_{v}[x(u,v)]
S_{vu} = int-int[ (-1)·e^{(-i)·3v}·e^{iu} ] d[v]d[u] = (-1)·(1/3)·e^{(-i)·3v}·e^{iu} = ...
... (1/3)·d_{v}[x(u,v)]^{3}·d_{u}[x(u,v)]
R_{111}^{2} = ...
... (-1)·d_{u}[x(u,v)]^{(-3)}·d_{v}[x(u,v)]^{(-1)} = ...
... (-1)·e^{i·( (-3)u+v )}
R_{222}^{1} = ...
... (-1)·d_{v}[x(u,v)]^{(-3)}·d_{u}[x(u,v)]^{(-1)} = ...
... (-1)·e^{i·( 3v+(-u) )}
M_{uv} = ...
... int-int[ (-1)·d_{u}[x]^{(-3)}·d_{v}[x]^{(-1)} ] d_{u}[x]d_{u}[x] d[u]d[v] = ...
... (-1)·( x )^{[o(u)o](-1)}·( x )^{[o(v)o](-1)}
M_{vu} = int-int[ (-1)·d_{v}[x]^{(-3)}·d_{u}[x]^{(-1)} ] d_{v}[x]d_{v}[x] d[v]d[u] = ...
... (-1)·( x )^{[o(v)o](-1)}·( x )^{[o(u)o](-1)}
M_{uu} = ...
... int-int[ (-1)·d_{u}[x]^{(-3)}·d_{v}[x]^{(-1)} ] d_{v}[x]d_{u}[x] d[u]d[u] = ...
... (-1)·...
... ( int[ x ] d[u] )^{[o( (1/2)·u^{2} )o](-1)} ...
... [o( (1/2)·u^{2} )o] ...
... ( int[ x ] d[u] )^{[o( (1/2)·u^{2} )o](-1)}
M_{vv} = ...
... int-int[ (-1)·d_{v}[x]^{(-3)}·d_{u}[x]^{(-1)} ] d_{u}[x]d_{v}[x] d[v]d[v] = ...
... (-1)·...
... ( int[ x ] d[v] )^{[o( (1/2)·v^{2} )o](-1)} ...
... [o( (1/2)·v^{2} )o] ...
... ( int[ x ] d[v] )^{[o( (1/2)·v^{2} )o](-1)}
cuerda exponencial-compleja de oscilador harmónico:
h = (-1)·(1/2)·a^{2}·b
L(x,u,v) = (1/2)·a^{2}·( x(u,v) )^{2}+h·( e^{iu}+e^{(-i)v} ) = 0
x(u,v)= ( b·( e^{iu}+e^{(-i)v}) )^{(1/2)}
S_{uu} = int-int[ (-1)·e^{i·2u} ] d[u]d[u] = (1/4)·e^{i·2u} = ...
... (-1)·(1/4)·( 1/b^{2} )·d_{u}[( x(u,v) )^{2}]^{2}
... (-1)·( 1/b^{2} )·( x(u,v)·d_{u}[x(u,v)] )^{2}
S_{vv} = int-int[ (-1)·e^{(-i)·2v} ] d[v]d[v] = (1/4)·e^{(-i)·2v} = ...
... (-1)·(1/4)·( 1/b^{2} )·d_{v}[( x(u,v) )^{2}]^{2}
... (-1)·( 1/b^{2} )·( x(u,v)·d_{v}[x(u,v)] )^{2}
H_{u}( d_{u}[x] ) = ( (1/4)·( 1/(2i) )·e^{i·2u} )^{[o(u)o]2} [o(u)o] x = ...
... ( (1/16)·(1/(4i))·e^{i·4u} ) [o(u)o] x
( (1/(4i))·(1/b^{4})·( int[ d_{u}[x] ] d[u]·d_{u}[x] )^{4} ) [o(u)o] int[ d_{u}[x] ] d[u]
d_{u}[ ( (1/(4i))·(1/b^{4})·( int[ d_{u}[x] ] d[u]·d_{u}[x] )^{4} ) ] = ...
... (1/i)·(1/b^{4})·( xd_{x}[x] )^{3}( d_{u}[x]^{2}+xd_{uu}^{2}[x] ) =
... (1/i)·(1/b^{4})·(1/8)·( b^{3}e^{i·3u} )( ...
... (1/4)·( be^{i·2u}/(e^{iu}+e^{(-i)v}) )+...
... (1/2)·bie^{iu}+(-1)·(1/4)·( be^{i·2u}/(e^{iu}+e^{(-i)v}) ) = ...
... ( (1/16)·e^{i·4u} ) = ( (1/4)·e^{i·2u} )^{2} = ...
... ( (-1)·( 1/b^{2} )·( x(u,v)·d_{v}[x(u,v)] )^{2} )^{2}
H_{v}( d_{v}[x] ) = ( (1/4)·( 1/(2i) )·e^{i·2v} )^{[o(v)o]2} [o(v)o] x = ...
... ( (1/16)·(1/(4i))·e^{i·4v} ) [o(v)o] x
( (1/(4i))·(1/b^{4})·( int[ d_{v}[x] ] d[v] )^{4}·d_{v}[x]^{4} ) [o(v)o] int[ d_{v}[x] ] d[v]
R_{111}^{2} = ...
... (-1)·b^{4}...
... d_{u}[( x(u,v) )^{2}]^{(-3)}·...
... d_{v}[( x(u,v) )^{2}]^{(-1)} = ...
... (-1)·e^{i( (-3)u+v )}
R_{222}^{1} = ...
... (-1)·b^{4}...
... d_{v}[( x(u,v) )^{2}]^{(-3)}·...
... d_{u}[( x(u,v) )^{2}]^{(-1)} = ...
... (-1)·e^{i( 3v+(-u) )}
M_{uv} = ...
... int-int[ (-1)·b^{4}·d_{u}[x^{2}]^{(-3)}·d_{v}[x^{2}]^{(-1)} ] ...
... d_{u}[x]d_{u}[x] d[u]d[v] = ...
... (-1)·(1/16)·b^{4}·( (1/3)·x^{3} )^{[o(u)o](-1)}·( (1/3)·x^{3}) )^{[o(v)o](-1)}
M_{vu} = ...
... int-int[ (-1)·b^{4}·d_{v}[x^{2}]^{(-3)}·d_{u}[x^{2}]^{(-1)} ] ...
... d_{v}[x]d_{v}[x] d[v]d[u] = ...
... (-1)·(1/16)·b^{4}·( (1/3)·x^{3} )^{[o(v)o](-1)}·( (1/3)·x^{3} )^{[o(u)o](-1)}
M_{uu} = ...
... int-int[ (-1)·b^{4}·d_{u}[x^{2}]^{(-3)}·d_{v}[x^{2}]^{(-1)} ] ...
... d_{v}[x]d_{u}[x] d[u]d[u] = ...
... (-1)·(1/16)·b^{4}·...
... ( int[ (1/3)·x^{3} ] d[u] )^{[o( (1/2)·u^{2} )o](-1)} ...
... [o( (1/2)·u^{2} )o] ...
... ( int[ (1/3)·x^{3} ] d[u] )^{[o( (1/2)·u^{2} )o](-1)}
M_{uu} = ...
... int-int[ (-1)·b^{4}·d_{v}[x^{2}]^{(-3)}·d_{u}[x^{2}]^{(-1)} ] ...
... d_{u}[x]d_{v}[x] d[v]d[v] = ...
... (-1)·(1/16)·b^{4}·...
... ( int[ (1/3)·x^{3} ] d[v] )^{[o( (1/2)·v^{2} )o](-1)} ...
... [o( (1/2)·v^{2} )o] ...
... ( int[ (1/3)·x^{3} ] d[v] )^{[o( (1/2)·v^{2} )o](-1)}
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