teoría de cordes

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)}

geometría diferencial general

H_{jk}·f(x^{i}) = int-int-int[ R_{ijk}^{i} ] d[x^{i}]d[x^{j}]d[x^{k}]

M_{jj} = int-int[ R_{jjj}^{j} ] d[x^{j}]d[x^{j}]

M_{kk} = int-int[ R_{kkk}^{k} ] d[x^{k}]d[x^{k}]

H_{jk}·f(x^{i}) = ...

... int-int-int[ R_{ijk}^{i} = ( (x^{i}+1)/x^{i} )·( x^{j} )^{0}·( x^{k} )^{0} ] ...

... d[x^{i}]d[x^{j}]d[x^{k}]

H_{23}·f(x) = ( x+ln(x) )·yz

H_{31}·f(y) = ( y+ln(y) )·zx

H_{12}·f(z) = ( z+ln(z) )·xy

M_{11} = (1/2)·x^{2}+( ln(x) )^{2}·[er-h]_{k!:2}( ln(x) )

M_{22} = (1/2)·y^{2}+( ln(y) )^{2}·[er-h]_{k!:2}( ln(y) )

M_{33} = (1/2)·z^{2}+( ln(z) )^{2}·[er-h]_{k!:2}( ln(z) )

H_{jk}·f(x^{i}) = ...

... int-int-int[ R_{ijk}^{i} = ( (( x^{i} )^{n}+1)/x^{i} )·( x^{j} )^{0}·( x^{k} )^{0} ] ...

... d[x^{i}]d[x^{j}]d[x^{k}]

H_{23}·f(x) = ( (1/n)·x^{n}+ln(x) )·yz

H_{31}·f(y) = ( (1/n)·y^{n}+ln(y) )·zx

H_{12}·f(z) = ( (1/n)·z^{n}+ln(z) )·xy

M_{11} = (1/n)·( 1/(n+1) )·x^{n+1}+( ln(x) )^{2}·[er-h]_{k!:2}( ln(x) )

M_{22} = (1/n)·( 1/(n+1) )·y^{n+1}+( ln(y) )^{2}·[er-h]_{k!:2}( ln(y) )

M_{33} = (1/n)·( 1/(n+1) )·z^{n+1}+( ln(z) )^{2}·[er-h]_{k!:2}( ln(z) )

Tensor de curvatura y relatividad general

Principio:

Ecuación fundamental de la relatividad general: 

H_{jk}+(1/2)·( M_{jj}+M_{kk} ) = k^{2}·( T_{jk}+(1/2)·( T_{jj}+T_{kk} ) )

Contracción tensorial del tensor de curvatura:

H_{jk}·x^{i} = int-int-int[ R_{ijk}^{i} ] d[x^{i}]d[x^{j}]d[x^{k}]

H_{jk} = ( 1/x^{i} )·int-int-int[ R_{ijk}^{i} ] d[x^{i}]d[x^{j}]d[x^{k}]

Tensor métrico:

M_{jj} = int-int[ R_{jjj}^{j} ] d[x^{j}]d[x^{j}]

M_{kk} = int-int[ R_{kkk}^{k} ] d[x^{k}]d[x^{k}]

Tensor de impulsión-energía:

T_{jk} = d_{jk}^{2}[ E(x^{j},x^{k}) ]

Tensor de impulsión-energía Laplaciano:

T_{jj} = d_{jj}^{2}[ E(x^{j}) ]

T_{kk} = d_{kk}^{2}[ E(x^{k}) ]

Exemple:

Curvatura eléctrica:

H_{jk}·x^{i} = ...

... int-int-int[ R_{ijk}^{i} = (x^{i}/x^{i})·( ( x^{j} )^{0}·( x^{k} )^{0} ) ] ...

... d[x^{i}]d[x^{j}]d[x^{k}]

Curvatura gravitatoria:

H_{jk}·x^{i} = ...

... int-int-int[ R_{ijk}^{i} = (x^{i}/x^{i})·( (-1)·( x^{j} )^{0}·( x^{k} )^{0} ) ] ...

... d[x^{i}]d[x^{j}]d[x^{k}]

Contracción tensorial de curvatura:

H_{23}·x = ...

... int-int-int[ R_{123}^{1} = (x/x)·( y^{0}·z^{0} ) ] d[x]d[y]d[z] = x·( yz )

H_{31}·y = ...

... int-int-int[ R_{231}^{2} = (y/y)·( z^{0}·x^{0} ) ] d[x]d[y]d[z] = y·( zx )

H_{12}·z = ...

... int-int-int[ R_{312}^{3} = (z/z)·( x^{0}·y^{0} ) ] d[x]d[y]d[z] = z·( xy )

Campo eléctrico:

E(y,z) = (1/k)·(1/4)·( yz )^{2}

E(z,x) = (1/k)·(1/4)·( zx )^{2}

E(x,y) = (1/k)·(1/4)·( xy )^{2}

Campo gravitatorio:

E(y,z) = (-1)·(1/k)·(1/4)·( yz )^{2}

E(z,x) = (-1)·(1/k)·(1/4)·( zx )^{2}

E(x,y) = (-1)·(1/k)·(1/4)·( xy )^{2}

Métrica eléctrica:

M_{jj} = ...

... int-int-[ R_{jjj}^{j} = (x^{j}/x^{j})·( ( x^{j} )^{0}·( x^{j} )^{0} ) ] d[x^{j}]d[x^{j}]

Métrica gravitatoria:

M_{jj} = ...

... int-int-[ R_{jjj}^{j} = (x^{j}/x^{j})·( (-1)·( x^{j} )^{0}·( x^{j} )^{0} ) ] d[x^{j}]d[x^{j}]

Métrica:

M_{11} = ...

... int-int[ R_{111}^{1} = (x/x)·( x^{0}·x^{0} ) ] d[x]d[x] = (1/2)·x^{2}

M_{22} = ...

... int-int[ R_{222}^{2} = (y/y)·( y^{0}·y^{0} ) ] d[y]d[y] = (1/2)·y^{2}

M_{33} = ...

... int-int[ R_{333}^{3} = (z/z)·( z^{0}·z^{0} ) ] d[z]d[z] = (1/2)·z^{2}

Campo eléctrico:

E(x) = (1/k)·(1/24)·x^{4}

E(y) = (1/k)·(1/24)·y^{4}

E(z) = (1/k)·(1/24)·z^{4}

Campo gravitatorio:

E(x) = (-1)·(1/k)·(1/24)·x^{4}

E(y) = (-1)·(1/k)·(1/24)·y^{4}

E(z) = (-1)·(1/k)·(1/24)·z^{4}

Problema-Ley:

Contracción tensorial de curvatura:

H_{jk}·x^{i} = ...

... int-int-int[ R_{ijk}^{i} = (x^{i}/x^{i})·( x^{j}·x^{k} ) ] d[x^{i}]d[x^{j}]d[x^{k}]

H_{23}·x = ...

... int-int-int[ R_{123}^{1} = (x/x)·( y·z ) ] d[x]d[y]d[z] = x·(1/4)·( yz )^{2}

H_{31}·y = ...

... int-int-int[ R_{231}^{2} = (y/y)·( z·x ) ] d[x]d[y]d[z] = y·(1/4)·( zx )^{2}

H_{12}·z = ...

... int-int-int[ R_{312}^{3} = (z/z)·( x·y ) ] d[x]d[y]d[z] = z·(1/4)·( xy )^{2}

Campo eléctrico:

E(y,z) = (1/k)·(1/36)·( yz )^{3}

E(z,x) = (1/k)·(1/36)·( zx )^{3}

E(x,y) = (1/k)·(1/36)·( xy )^{3}

Métrica:

M_{jj} = ...

... int-int-[ R_{jjj}^{j} = (x^{j}/x^{j})·( x^{j}·x^{j} ) ] d[x^{j}]d[x^{j}]

M_{11} = ...

... int-int[ R_{111}^{1} = (x/x)·( x^{2} ) ] d[x]d[x] = (1/12)·x^{4}

M_{22} = ...

... int-int[ R_{222}^{2} = (y/y)·( y^{2} ) ] d[y]d[y] = (1/12)·y^{4}

M_{33} = ...

... int-int[ R_{333}^{3} = (z/z)·( z^{2} ) ] d[z]d[z] = (1/12)·z^{4}

Campo eléctrico:

E(x) = (1/k)·(1/360)·x^{6}

E(y) = (1/k)·(1/360)·y^{6}

E(z) = (1/k)·(1/360)·z^{6}

Problema-Ley:

Contracción tensorial de curvatura:

H_{jk}·x^{i} = ...

... int-int-int[ R_{ijk}^{i} = (x^{i}/x^{i})·( x^{j}+x^{k} ) ] d[x^{i}]d[x^{j}]d[x^{k}]

H_{23}·x = ...

... int-int-int[ R_{123}^{1} = (x/x)·( y+z ) ] d[x]d[y]d[z] = x·(1/2)·( y^{2}z+yz^{2} )

H_{31}·y = ...

... int-int-int[ R_{231}^{2} = (y/y)·( z+x ) ] d[x]d[y]d[z] = y·(1/2)·( z^{2}x+zx^{2} )

H_{12}·z = ...

... int-int-int[ R_{312}^{3} = (z/z)·( x+y ) ] d[x]d[y]d[z] = z·(1/2)·( x^{2}y+xy^{2} )

Campo eléctrico:

E(y,z) = (1/k)·(1/12)·( y^{3}z^{2}+y^{2}z^{3} )

E(z,x) = (1/k)·(1/12)·( z^{3}x^{2}+z^{2}x^{3} )

E(x,y) = (1/k)·(1/12)·( x^{3}y^{2}+x^{2}y^{3} )

Métrica:

M_{jj} = ...

... int-int-[ R_{jjj}^{j} = (x^{j}/x^{j})·( x^{j}+x^{j} ) ] d[x^{j}]d[x^{j}]

M_{11} = ...

... int-int[ R_{111}^{1} = (x/x)·( 2x ) ] d[x]d[x] = (1/3)·x^{3}

M_{22} = ...

... int-int[ R_{222}^{2} = (y/y)·( 2y ) ] d[y]d[y] = (1/3)·y^{3}

M_{33} = ...

... int-int[ R_{333}^{3} = (z/z)·( 2z ) ] d[z]d[z] = (1/3)·z^{3}

Campo eléctrico:

E(x) = (1/k)·(1/60)·x^{5}

E(y) = (1/k)·(1/60)·y^{5}

E(z) = (1/k)·(1/60)·z^{5}

electricitat y economia

A(t) = 220 Volts

mv·d_{t}[x(t)] = 220·q

B(t) = pt+(-n)·( 220·( q/(mv) )·t ) [o(t)o] (1/k)·( 1/3600 )·(1/2)·t^{2}

t = horas

n = (precio/kilo-metro) [ precio de la red ]

k = tipo de tarifa según hora

k = 1 hora punta

k = 2 hora valle

6 electrodomésticos:

220V·6C = 1320 J = 1.320 KW·s

d_{t}[B(t)] = p+(-1)·220·(6/11)·(1/3600)·t = p+(-1)·(1/30)·t

d_{t}[B(t)] = p+(-1)·220·(6/22)·(1/3600)·t = p+(-1)·(1/60)·t

social-democracia:

hora punta:

24p = (1/30)·24h = 0.80€ al día

hora valle:

24p = (1/60)·24h = 0.40€ al día

q = 6 electrodomésticos

impuestos:

(mv/n)·(1/100) = 0.11€ al día

socialismo:

hora punta:

24p = (1/60)·24h = 0.40€ al día

hora valle:

24p = (1/120)·24h = 0.20€ al día

q = 6 electrodomésticos

impuestos:

(mv/n)·(1/100) = 0.22€ al día

12 electrodomésticos:

220V·12C = 2640 J = 2.640 KW·s

d_{t}[B(t)] = p+(-1)·220·(12/11)·(1/3600)·t = p+(-1)·(1/15)·t

d_{t}[B(t)] = p+(-1)·220·(12/22)·(1/3600)·t = p+(-1)·(1/30)·t

social-democracia:

hora punta:

24p = (1/15)·24h = 1.60€ al día

hora valle:

24p = (1/30)·24h = 0.80€ al día

q = 12 electrodomésticos

impuestos:

(mv/n)·(1/100) = 0.11€ al día

socialismo:

hora punta:

24p = (1/30)·24h = 0.80€ al día

hora valle:

24p = (1/60)·24h = 0.40€ al día

q = 12 electrodomésticos

impuestos:

(mv/n)·(1/100) = 0.22€ al día

q(t) = at

I = d_{t}[q(t)] = a

R = (110/a)

220 = RI+110 & (-110)+(-R)I = (-220)

110 = (-R)I+220 & (-220)+RI = (-110)

330 = 2RI+110 & (-110)+2(-R)I = (-330)

110 = 2(-R)I+330 & (-330)+2RI = (-110)

Gas:

T(t) = 110 Celsius

mv·d_{t}[x(t)] = 110·k

B(t) = pt+(-n)·( 110·( k/(mv) )·t ) [o(t)o] (1/k)·( 1/3600 )·(1/2)·t^{2}

6 gas-domésticos:

d_{t}[B(t)] = p+(-1)·110·(6/11)·(1/3600)·t = p+(-1)·(1/60)·t

d_{t}[B(t)] = p+(-1)·110·(6/22)·(1/3600)·t = p+(-1)·(1/120)·t

social-democracia:

hora punta:

24p = (1/60)·24h = 0.40€ al día

k = 6

hora valle:

24p = (1/120)·24h = 0.20€ al día

k = 6

impuestos:

( (mv)/n )·(1/100) = 0.11€

socialismo:

hora punta:

24p = (1/120)·24h = 0.20€ al día

k = 6

hora valle

24p = (1/240)·24h = 0.10€ al día

k = 6

impuestos:

( (mv)/n )·(1/100) = 0.22€

música

[00+01][00+01][00+02][00+02] = 06k = 1·2·3·k

[00+04][00+04][00+05][00+05] = 18k = 3·2·3·k

[00+07][00+07][00+08][00+08] = 30k = 5·2·3·k

[00+10][00+10][00+11][00+11] = 42k = 7·2·3·k

42 = 30+12 = (18+12)+12 = ( (6+12)+12 )+12

límits no acotats y límits acotats

Definició:

[As][ s > 0 ==> [En_{0}][ n_{0}€N & [An][ n > n_{0} ==> a_{n} > s ] ] ]

[As][ s < 0 ==> [En_{0}][ n_{0}€N & [An][ n > n_{0} ==> a_{n} < s ] ] ]

Exemple:

[Ak][ k€N ==> n+k no és acotada-límit superiorment ].

Sigui s > 0 ==>

Es defineish n_{0} > s ==>

Sigui n > n_{0} ==>

n > s

Sigui k€N ==>

n+k >] n > s

Exemple:

[Ak][ k€N ==> (-n)+(-k) no és acotada-límit inferiorment ].

Sigui s < 0 ==>

Es defineish n_{0} > (-s) ==>

Sigui n > n_{0} ==>

n > (-s) 

(-n) < s

Sigui k€N ==>

(-n)+(-k) [< (-n) < s

Teorema:

Si [En_{0}][ n_{0}€N & [An][ n > n_{0} ==> a_{n} > n ] ] ==> ...

... a_{n} no és acotada-límit superiorment.

Sigui s > 0 ==>

Es defineish n_{0} = max{n_{1},n_{2}} & ...

... n_{1} > s & [An][ n > n_{2} ==> a_{n} > n ]

Sigui n > n_{0} ==>

a_{n} > n > s

Teorema:

Si [En_{0}][ n_{0}€N & [An][ n > n_{0} ==> a_{n} < (-n) ] ]  ==> ...

... a_{n} no és acotada-límit inferiorment.

Sigui s < 0 ==>

Es defineish n_{0} = max{n_{1},n_{2}} & ...

... n_{1} > (-s) & [An][ n > n_{2} ==> a_{n} < (-n) ]

Sigui n > n_{0} ==>

a_{n} < (-n) < s

Teorema:

Siguin a_{1},...,a_{n} > 0 ==> a_{1}+...+a_{n} no és acotada-límit superiorment.

Sigui s > 0 ==>

Es defineish M = min{a_{1},...,a_{n}} ==>

Es defienish n_{0} > (s/M) ==>

Sigui n > n_{0} ==>

a_{1}+...+a_{n} >] Mn > Mn_{0} > s

Teorema:

Siguin a_{1},...,a_{n} < 0 ==> a_{1}+...+a_{n} no és acotada-límit inferiorment.

Sigui s < 0 ==>

Es defineish M = max{a_{1},...,a_{n}} ==>

Es defienish n_{0} > (s/M) ==>

Sigui n > n_{0} ==>

a_{1}+...+a_{n} [< Mn < Mn_{0} < s

Teorema:

Si a_{n} & b_{n} no son acotades-límit superiorment ==> ...

... a_{n}+b_{n} no és acotada-límit superiorment.

Sigui s > 0 ==>

Sigui s_{1} = (s/2) & s_{2} = (s/2) ==>

Es defienish n_{0} = max{n_{1},n_{2}} & ...

... [An][ n > n_{1} ==> a_{n} > s_{1} ] & [An][ n > n_{2} ==> b_{n} > s_{2} ] ==>

Sigui n > n_{0} ==>

a_{n}+b_{n} > s_{1}+s_{2} = s

Teorema:

Si a_{n} & b_{n} no son acotades-límit inferiorment ==> ...

... a_{n}+b_{n} no és acotada-límit inferiorment.

Sigui s < 0 ==>

Sigui s_{1} = (s/2) & s_{2} = (s/2) ==>

Es defienish n_{0} = max{n_{1},n_{2}} & ...

... [An][ n > n_{1} ==> a_{n} < s_{1} ] & [An][ n > n_{2} ==> b_{n} < s_{2} ] ==>

Sigui n > n_{0} ==>

a_{n}+b_{n} < s_{1}+s_{2} = s

Problema-lema:

[Ak][ k€N ==> ( (n+k)^{2}/n ) no és acotada-límit superiorment ].

Sigui s > 0 ==>

Es defineish n_{0} > s ==>

Sigui n > n_{0} ==>

n > s

Sigui k€N ==>

( (n+k)^{2}/n ) = ( (n^{2}+2nk+k^{2})/n ) >] (n^{2}/n) > n > s

Problema-lema:

[Ak][ k€N ==> (-1)·( (n+k)^{2}/n ) no és acotada-límit inferiorment ].

Sigui s < 0 ==>

Es defineish n_{0} > (-s) ==>

Sigui n > n_{0} ==>

n > (-s)

(-n) < s

Sigui k€N ==>

(-1)·( (n+k)^{2}/n ) = (-1)·( (n^{2}+2nk+k^{2})/n ) [< (-1)·(n^{2}/n) < (-n) < s

Problema-lema:

[Ak][ k€N ==> ( n^{2}/(n+(-k)) ) no és acotada-límit superiorment ].

Sigui s > 0 ==>

Sigui k€N ==>

Es defineish n_{0} = max{n_{1},k} & n_{1} > s ==>

Sigui n > n_{0} ==>

n > s

( n^{2}/(n+(-k)) ) = ( ((n+(-k))^{2}+2(n+(-k))k+k^{2})/(n+(-k)) ) >] ...

... (n+(-k))+2k+( k^{2}/(n+(-k)) ) > n+k > n > s

Problema-lema:

[Ak][ k€N ==> (-1)·( n^{2}/(n+(-k)) ) no és acotada-límit inferiorment ].

Sigui s < 0 ==>

Sigui k€N ==>

Es defineish n_{0} = max{n_{1},k} & n_{1} > (-s) ==>

Sigui n > n_{0} ==>

n > (-s)

(-n) < s

(-1)·( n^{2}/(n+(-k)) ) = (-1)·( ((n+(-k))^{2}+2(n+(-k))k+k^{2})/(n+(-k)) ) = ...

... (-1)·( (n+(-k))+2k+( k^{2}/(n+(-k)) ) ) < (-n)+(-k) < (-n) < s

Definició:

[Es][ s > 0 ==> [An_{0}][ n_{0}€N ==> [En][ n > n_{0} & a_{n} [< s ] ] ]

[Es][ s < 0 ==> [An_{0}][ n_{0}€N ==> [En][ n > n_{0} & a_{n} >] s ] ] ]

Exemple:

[Ak][ k€N ==> ( 1/(n+k) ) està acotada superiorment ]

Es defienish s >] 1 ==>

Sigui n_{0}€N ==>

Es defineixh n = n_{0}+1 ==>

n > n_{0}

1 [< n_{0}+1 = n

( 1/(n+k) ) [< (1/n) [< 1 [< s

Exemple:

[Ak][ k€N ==> (-1)·( 1/(n+k) ) està acotada inferiorment ]

Es defienish s [< (-1) ==>

Sigui n_{0}€N ==>

Es defineixh n = n_{0}+1 ==>

n > n_{0}

1 [< n_{0}+1 = n

(-n) [< (-1)

(-1)·( 1/(n+k) ) >] (-1)·(1/n) >] (-1) >] s

Problema-lema:

[Ak][ k€N ==> ( 1/(n+(-k)) ) està acotada superiorment ]

Es defienish s >] 1 ==>

Sigui n_{0}€N ==>

Es defineixh n = (n_{0}+1)+k ==>

n > n_{0}

1 [< n_{0}+1

( 1/(n+(-k)) ) = ( 1/(n_{0}+1) ) [< 1 [< s

Problema-lema:

[Ak][ k€N ==> ( 1/(n+(-k)) ) està acotada superiorment ]

Es defienish s [< (-1) ==>

Sigui n_{0}€N ==>

Es defineixh n = (n_{0}+1)+k ==>

n > n_{0}

1 [< n_{0}+1

(-1)( n_{0}+1 ) [< (-1)

(-1)·( 1/(n+(-k)) ) = (-1)·( 1/(n_{0}+1) ) >] (-1) >] s

Teorema:

Si a_{n} & b_{n} son acotades-límit superiorment ==> ...

... a_{n}+b_{n} és acotada-límit superiorment.

Es defineish s > 0 & s_{1} = (s/2) & s_{2} = (s/2) ==>

Sigui n_{0}€N ==>

Siguin n_{1},n_{2}€N ==> 

Es defineish n = max{n_{0}+1,n_{1}+1,n_{2}+1} & ...

... [En][ n > n_{1} ==> a_{n} [< s_{1} ] & [En][ n > n_{2} ==> b_{n} [< s_{2} ] ==>

a_{n}+b_{n} [< s_{1}+s_{2} = s

Teorema:

Si a_{n} & b_{n} son acotades-límit inferiorment ==> ...

... a_{n}+b_{n} és acotada-límit inferiorment.

Es defineish s < 0 & s_{1} = (s/2) & s_{2} = (s/2) ==>

Sigui n_{0}€N ==>

Siguin n_{1},n_{2}€N ==> 

Es defineish n = max{n_{0}+1,n_{1}+1,n_{2}+1} & ...

... [En][ n > n_{1} ==> a_{n} >] s_{1} ] & [En][ n > n_{2} ==> b_{n} >] s_{2} ] ==>

a_{n}+b_{n} >] s_{1}+s_{2} = s

álgebra lineal: sub-espais y combinacions lineals

Axioma:

[As][ s€K ==> s·0 = 0 ]

<0,x>+<0,y> = <2·0,x+y> = <0,x+y>

s·<0,x> = <s·0,s·x> = <0,s·x>

i·<0,x>+j·<0,x> = <0,0> ==> ( i = i & j = (-i) ) || ( i = (-j) & j = j )

i·<0,x>+j·<0,x> = <0,x> ==> ( i = a & j = 1+(-a) ) || ( i = 1+(-a) & j = a )

i·<0,x>+j·<0,y> = <0,0> ==> ( i = (y/x) & j = (-1) ) || ( i = (-1) & j = (x/y) )

i·<0,x>+j·<0,y> = <0,k> ==> ...

... ( i = (1+(-a))·(k/x) & j = a·(k/y) ) || ( i = a·(k/x) & j = (1+(-a))·(k/y) )

Axioma:

[As][ s€K ==> 1^{s} = 1 ]

<1,x>·<1,y> = <1^{2},x·y> = <1,x·y>

<1,x>^{s} = <1^{s},x^{s}> = <1,x^{s}>

<1,x>^{i}·<1,x>^{j} = <1,1> ==> ( i = i & j = (-i) ) || ( i = (-j) & j = j )

<1,x>^{i}·<1,x>^{j} = <1,x> ==> ( i = a & j = 1+(-a) ) || ( i = 1+(-a) & j = a )

<1,x>^{i}·<1,y>^{j} = <1,1> ==> ( i = log_{x}(y) & j = (-1) ) || ( i = (-1) & j = log_{y}(x) )

<1,x>^{i}·<1,y>^{j} = <1,k> ==> ...

... ( i = (1+(-a))·log_{x}(k) & j = a·log_{y}(k) ) || ( i = a·log_{x}(k) & j = (1+(-a))·log_{y}(k) )

integrals

int[ ln(x) ] d[x] = int[ y·e^{y} ] d[y] = y^{2}·[er-h]_{k!:2}(y) = ...

... ( ln(x) )^{2}·[er-h]_{k!:2}( ln(x) )

d_{x}[ ( ln(x) )^{2}·[er-h]_{k!:2}( ln(x) ) ] = ...

... ln(x)·(e^{ln(x)}/x) = ln(x)

int[ ln(1/x) ] d[x] = int[ (-1)·y·e^{(-y)} ] d[y] = (-1)·y^{2}·[er]_{k!:2}(y) = ...

... (-1)·( ln(1/x) )^{2}·[er]_{k!:2}( ln(1/x) )

d_{x}[ (-1)·( ln(1/x) )^{2}·[er]_{k!:2}( ln(1/x) ) ] = ...

... (-1)·ln(1/x)·( (-1)/x )·e^{(-1)·ln(1/x)} = ln(1/x)

int[ (1/ln(x)) ] d[x] = int[ (e^{y}/y) ] d[y] = [er-h]_{k!:0}(y) = ...

... [er-h]_{k!:0}( ln(x) )

d_{x}[ [er-h]_{k!:0}( ln(x) ) ] = ...

... (1/ln(x))·(e^{ln(x)}/x) = (1/ln(x))

int[ (1/ln(1/x)) ] d[x] = int[ (-1)·(e^{(-y)}/y) ] d[y] = (-1)·[er]_{k!:0}(y) = ...

... (-1)·[er]_{k!:0}( ln(1/x) )

d_{x}[ (-1)·[er]_{k!:0}( ln(1/x) ) ] = ...

... (-1)·(1/ln(1/x))·( (-1)/x )·e^{(-1)·ln(1/x)} = (1/ln(1/x))

int[ ln(ax+b) ] d[x] = ...

... ( ln(ax+b) )^{2}·[er-h]_{k!:2}( ln(ax+b) ) [o(x)o] (1/a)·x

int[ ln(1/(ax+b)) ] d[x] = ...

... (-1)·( ln(1/(ax+b)) )^{2}·[er]_{k!:2}( ln(1/(ax+b)) ) [o(x)o] (1/a)·x

int[ ln(ax^{2}+bx+c) ] d[x] = ...

... ( ln(ax^{2}+bx+c) )^{2}·[er-h]_{k!:2}( ln(ax^{2}+bx+c) ) [o(x)o] ...

... ln(2ax+b) [o(x)o] (1/(2a))·x

int[ ln(1/(ax^{2}+bx+c)) ] d[x] = ...

... (-1)·( ln(1/(ax^{2}+bx+c)) )^{2}·[er]_{k!:2}( ln(1/(ax^{2}+bx+c)) ) [o(x)o] ...

... ln(2ax+b) [o(x)o] (1/(2a))·x

int[ arc-sin(x) ] d[x] = int[ y·cos(y) ] d[y] = ...

... y^{2}·[er-cos]_{(2k)!:2}(y) = ...

... ( arc-sin(x) )^{2}·[er-cos]_{(2k)!:2}( arc-sin(x) )

d_{x}[ cos(arc-sin(x)) ] = ( (-x)/(1+(-1)·x^{2})^{(1/2)} ) = d_{x}[ (1+(-1)·x^{2})^{(1/2)} ]

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

d_{x}[ ( arc-sin(x) )^{2}·[er-cos]_{(2k)!:2}( arc-sin(x) ) ] = ...

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

int[ arc-cos(x) ] d[x] = int[ (-1)·y·sin(y) ] d[y] = ...

... (-1)·y^{2}·[er-sin]_{(2k+1)!:2}(y) = ...

... (-1)·( arc-cos(x) )^{2}·[er-sin]_{(2k+1)!:2}( arc-cos(x) )

d_{x}[ sin(arc-cos(x)) ] = ( (-x)/(1+(-1)·x^{2})^{(1/2)} ) = d_{x}[ (1+(-1)·x^{2})^{(1/2)} ]

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

d_{x}[ ( arc-cos(x) )^{2}·[er-sin]_{(2k+1)!:2}( arc-cos(x) ) ] = ...

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

int[ x^{p}·arc-sin(x) ] d[x] = ...

... (1/(p+1))·x^{p+1} [o(x)o] ( arc-sin(x) )^{2}·[er-cos]_{(2k)!:2}( arc-sin(x) )

int[ x^{p}·arc-cos(x) ] d[x] = ...

... (1/(p+1))·x^{p+1} [o(x)o] (-1)·( arc-cos(x) )^{2}·[er-sin]_{(2k+1)!:2}( arc-cos(x) )

int[ arc-sinh(x) ] d[x] = int[ y·cosh(y) ] d[y] = ...

... y^{2}·[er-cosh]_{(2k)!:2}(y) = ...

... ( arc-sinh(x) )^{2}·[er-cosh]_{(2k)!:2}( arc-sinh(x) )

d_{x}[ cosh(arc-sinh(x)) ] = ( x/(1+x^{2})^{(1/2)} ) = d_{x}[ (1+x^{2})^{(1/2)} ]

cosh(arc-sinh(x)) = (1+x^{2})^{(1/2)}

d_{x}[ ( arc-sinh(x) )^{2}·[er-cosh]_{(2k)!:2}( arc-sinh(x) ) ] = ...

... arc-sinh(x)·cosh(arc-sinh(x))·( 1/(1+x^{2})^{(1/2)} ) = arc-sinh(x)

int[ arc-cosh(x) ] d[x] = int[ y·sinh(y) ] d[y] = ...

... y^{2}·[er-sinh]_{(2k+1)!:2}(y) = ...

... ( arc-cosh(x) )^{2}·[er-sinh]_{(2k+1)!:2}( arc-cosh(x) )

d_{x}[ sinh(arc-cosh(x)) ] = ( x/((-1)+x^{2})^{(1/2)} ) = d_{x}[ ((-1)+x^{2})^{(1/2)} ]

sinh(arc-cosh(x)) = ((-1)+x^{2})^{(1/2)}

d_{x}[ ( arc-cosh(x) )^{2}·[er-sinh]_{(2k+1)!:2}( arc-cosh(x) ) ] = ...

... arc-cosh(x)·sinh(arc-cosh(x))·( 1/((-1)+x^{2})^{(1/2)} ) = arc-cosh(x)

int[ tan(x) ] d[x] = int[ y/(1+(-1)·y^{2}) ]d[y] = (-1)·ln( (1+(-1)·y^{2})^{(1/2)} ) = ...

... (-1)·ln(cos(x))

int[ cot(x) ] d[x] = int[ (-y)/(1+(-1)·y^{2}) ]d[y] = ln( (1+(-1)·y^{2})^{(1/2)} ) = ...

... ln(sin(x))

integrals: sinus y cosinus

int[ ( sin(x) )^{n} ] d[x] = ...

... (1/(n+1))·( sin(x) )^{n+1} [o(x)o] ( sin(x)+( x [o(x)o] ln( cos(x) ) [o(x)o] cos(x) ) )

int[ ( cos(x) )^{n} ] d[x] = ...

... (1/(n+1))·( cos(x) )^{n+1} [o(x)o] ( cos(x)+( (-x) [o(x)o] ln( sin(x) ) [o(x)o] sin(x) ) )

int[ ( sin(ax+b) )^{n} ] d[x] = ...

... (1/(n+1))·( sin(ax+b) )^{n+1} [o(x)o] ...

... ( sin(ax+b)+( (1/a)·x [o(x)o] ln( cos(ax+b) ) [o(x)o] cos(ax+b) ) ) [o(x)o] (1/a^{2})·x

int[ ( cos(ax+b) )^{n} ] d[x] = ...

... (1/(n+1))·( cos(ax+b) )^{n+1} [o(x)o] ...

... ( cos(ax+b)+( (1/a)·(-x) [o(x)o] ln( sin(ax+b) ) [o(x)o] sin(ax+b) ) ) [o(x)o] (1/a^{2})·x

int[ ( sin(ax^{2}+bx+c) )^{n} ] d[x] = ...

... (1/(n+1))·( sin(ax^{2}+bx+c) )^{n+1} [o(x)o] ...

... ( ...

... sin(ax^{2}+bx+c)+...

... ( ... 

... (1/(2a))·x [o(x)o] ln(2ax+b) [o(x)o] ...

... ln( cos(ax^{2}+bx+c) ) [o(x)o] cos(ax^{2}+bx+c) ...

... ) ...

... ) ...

... [o(x)o] ...

... (2ax+b)^{(-1)} [o(x)o] (1/(2a))·(-x)

int[ ( cos(ax^{2}+bx+c) )^{n} ] d[x] = ...

... (1/(n+1))·( cos(ax^{2}+bx+c) )^{n+1} [o(x)o] ...

... ( ...

... cos(ax^{2}+bx+c)+...

... ( ... 

... (1/(2a))·(-x) [o(x)o] ln(2ax+b) [o(x)o] ...

... ln( sin(ax^{2}+bx+c) ) [o(x)o] sin(ax^{2}+bx+c) ...

... ) ...

... ) ...

... [o(x)o] ...

... (2ax+b)^{(-1)} [o(x)o] (1/(2a))·(-x)

int[ sin(x)·e^{x} ] d[x] = (-x) [o(x)o] cos(x) [o(x)o] e^{x} = ...

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

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

int[ cos(x)·e^{x} ] d[x] = x [o(x)o] sin(x) [o(x)o] e^{x} = ...

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

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

int[ x^{p}·sin(x) ] d[x] = (-x) [o(x)o] (1/(p+1))·x^{p+1} [o(x)o] cos(x) = ...

... x^{p+1}·[sn-er]_{(2k+1)!:p+1}(x)

[sn-er]_{(2k+1)!:q}(x) = sum[ (-1)^{k}·( 1/(2k+1)! )·( 1/((2k+1)+q) )·x^{2k+1} ]

int[ x^{p}·cos(x) ] d[x] = x [o(x)o] (1/(p+1))·x^{p+1} [o(x)o] sin(x) = ...

... x^{p+1}·[sn-er]_{(2k)!:p+1}(x)

[sn-er]_{(2k)!:q}(x) = sum[ (-1)^{k}·( 1/(2k)! )·( 1/((2k)+q) )·x^{2k} ]

ciclons

ciclons:

f(x,y) = P+( x+(-1)·c_{x} )^{2n+2}+( y+(-1)·c_{y} )^{2n+2}

fronts:

g(x,y) = ...

... (P/2)+( x+(-1)·c_{x} )·sin[2n+1]( x+(-1)·c_{x} )+...

... (P/2)+( y+(-1)·c_{y} )·cos[2n+1]( y+(-1)·c_{y} )

g(x,y) = ...

... (P/2)+( x+(-1)·c_{x} )·cos[2n+1]( x+(-1)·c_{x} )+...

... (P/2)+( y+(-1)·c_{y} )·sin[2n+1]( y+(-1)·c_{y} )