fisica: electro-magnetisme y gravito-magnetisme

H-E_{e}(r) = k_{e,h}q_{e}·( r^{n}/d_{t}[r]^{n} )
H-E_{g}(r) = (-1)·k_{g,h}q_{g}·( r^{n}/d_{t}[r]^{n} )


H-B_{e}(r) = k_{e,h,m}q_{e}·( d_{tt}^{2}[r]^{n}/d_{t}[r]^{n} )
H-B_{g}(r) = (-1)·k_{g,h,m}q_{g}·( d_{tt}^{2}[r]^{n}/d_{t}[r]^{n} )


ones de so sense massa: m=0
camp emisor:
H-E_{e}(r) + H-B_{e}(r) = 0 <==> ...
... r(t)  = ( sinh( e^{(1/n)·(pi·i)}(kq/kq)t )+ cosh( e^{(1/n)·(pi·i)}(kq/kq)t ) )
camp receptor:
H-E_{e}(r) + H-B_{g}(r) = 0 <==> ...
... r(t)  = ( sinh( e^{(1/n)·(2pi·i)}(kq/kq)t )+ cosh( e^{(1/n)·(2pi·i)}(kq/kq)t ) )
camp receptor:
H-E_{g}(r) + H-B_{e}(r) = 0 <==> ...
... r(t)  = ( sinh( e^{(1/n)·(2pi·i)}(kq/kq)t )+ cosh( e^{(1/n)·(2pi·i)}(kq/kq)t ) )
camp emisor:
H-E_{g}(r) + H-B_{g}(r) = 0 <==> ...
... r(t)  = ( sinh( e^{(1/n)·(pi·i)}(kq/kq)t )+ cosh( e^{(1/n)·(pi·i)}(kq/kq)t ) )

algebra: suma per diferencia

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


( 1/(x^{2}+a^{2}) ) = (1/2ai)·( 1/(x+(-a)i) )+(-1)(1/2ai)·( 1/(x+ai) )


(x^{2}+(-1)·a^{2}) = (x+a)(x+(-a))
(x^{2}+a^{2}) = (x+ai)(x+(-a)i)


( c/(x^{2}+(-1)·a^{2}) ) = (c/2a)·( 1/(x+(-a)) )+(-1)(c/2a)·( 1/(x+a) )


( c/(x^{2}+a^{2}) ) = (c/2ai)·( 1/(x+(-a)i) )+(-1)(c/2ai)·( 1/(x+ai) )


( (cx)/(x^{2}+(-1)·a^{2}) ) = (c/2)·( 1/(x+(-a)) )+(c/2)·( 1/(x+a) )


( (cx)/(x^{2}+a^{2}) ) = (c/2)·( 1/(x+(-a)i) )+(c/2)·( 1/(x+ai) )

limits de funcions II

lim [ x --> a ]-[ ( (x^{2}+(-1)·a^{2})/(x+(-a)) ) ] = 2a
lim [ x --> (-a) ]-[ ( (x^{2}+(-1)·a^{2})/(x+a) ) ] = (-2)a


lim [ x --> ai ]-[ ( (x^{2}+a^{2})/(x+(-a)i) ) ] = 2ai
lim [ x --> (-a)i ]-[ ( (x^{2}+a^{2})/(x+ai) ) ] = (-2)ai


lim [ x --> 1 ]-[ ( ( a_{1}x+...(n)...+a_{n}x^{n}+(-1)·∑ ( a_{k} ) )/(x+(-1)) ) ] = ∑ ( a_{k}·k )
lim [ x --> 1 ]-[ ( ( a_{1}x+...(n)...+a_{n}x^{(1/n)}+(-1)·∑ ( a_{k} ) )/(x+(-1)) ) ] = ∑ ( a_{k}·(1/k) )

limits de funcions

lim [ x --> 1 ]-[ ( (x^{n}+(-1))/(x+(-1)) ) ] = n
lim [ x --> 1 ]-[ ( (x^{(1/n)}+(-1))/(x+(-1)) ) ] = (1/n)


lim [ x --> 1 ]-[ ( (x^{n}+(-1))/(x^{m}+(-1)) ) ] = (n/m)
lim [ x --> 1 ]-[ ( (x^{(1/n)}+(-1))/(x^{(1/m)}+(-1)) ) ] = (m/n)


lim [ x --> 1 ]-[ ( ( (x+(-1))·...(n)...·(x^{n}+(-1)) )/(x+(-1))^{n} ) ] = n!
lim [ x --> 1 ]-[ ( ( (x+(-1))·...(n)...·(x^{(1/n)}+(-1)) )/(x+(-1))^{n} ) ] = (1/n!)


lim [ x --> 1 ]-[ ( ( x+...(n)...+x^{n}+(-n) )/(x+(-1)) ) ] = ∑ k = (1/2)·n(n+1)
lim [ x --> 1 ]-[ ( ( x+...(n)...+x^{(1/n)}+(-n) )/(x+(-1)) ) ] = ∑ (1/k)


lim [ x --> 0 ]-[ ( ( 1^{x}+...(n)...+n^{x}+(-n) )/x ) ] = ln(n!)
lim [ x --> 0 ]-[ ( ( 1^{x}+...(n)...+(1/n)^{x}+(-n) )/x ) ] = (-1)·ln(n!)


indicacions:
(x^{n}+(-1)) = (x+(-1))·(1+...(n)...+x^{(n+(-1))})
(x+(-1)) = (x^{(1/n)}+(-1))·(1+...(n)...+x^{( (n+(-1))/n )})
a^{x} = e^{ln(a)·x}

series númeriques

teorema:
Si ( a_{n} >] 0 & ∑ ( n·a_{n} ) < oo ) ==> ∑ ( a_{n} ) < oo


demostració:
∑ ( a_{n} ) [< ∑ ( n·a_{n} ) < oo


teorema:
Si ( a_{n} >] 0 & ∑ ( a_{n} ) < oo ) ==> ∑ ( (1/n)·a_{n} ) < oo


demostració:
∑ ( (1/n)·a_{n} ) [< ∑ ( a_{n} ) < oo

series numériques

teorema:
Si 0 < a_{n} < 1 ==> ∑ ( a_{n} ) < oo


demostració:
∑ ( a_{n} ) < 1+...(oo)...+1 = oo


teorema:
Si ( m€N & 0 < a_{n} < m ) ==> (1/m)·∑ ( a_{n} ) < oo


demostració:
∑ ( a_{n} ) < m+...(oo)...+m = m·oo

series numériques

teorema:
Si ( a_{n} >] 0 &  m€N & ∑ ( m·a_{n} ) < oo ) ==> ∑ ( a_{n} ) < oo


demostració:
m·∑ ( a_{n} ) = ∑ ( m·a_{n} ) < oo


∑ ( a_{n} ) < (oo/m) < oo


teorema:
Si ( a_{n} >] 0 & ∑ ( oo·a_{n} ) < oo ) ==> ∑ ( a_{n} ) < 1


demostració:
oo·∑ ( a_{n} ) = ∑ ( oo·a_{n} ) < oo


∑ ( a_{n} ) < (oo/oo) = 1

series númeriques mitjes

teorema:
Si ( a_{n} >] 0 & b_{n} >] 0 &  ∑ ( a_{n} ) < oo & ∑ ( b_{n} ) < oo ) ==> ...
... ∑ ( (a_{n}+b_{n})/2 ) < oo


demostració:
∑ ( a_{n}+b_{n} ) = ∑ ( a_{n} ) + ∑ ( b_{n} )  < 2·oo


teorema:
Si ( a_{n} >] 0 & b_{n} >] 0 & ∑ ( a_{n} ) < oo & ∑ ( b_{n} ) < oo ) ==> ...
... ∑ ( (a_{n}·b_{n})^{(1/2)} ) < oo


demostració:
∑ ( (a_{n}·b_{n})^{(1/2)} ) [< ∑ ( (a_{n}+b_{n})/2 ) < oo

álgebra: ecuacions polinomiques II

ax^{p}+ax^{q} = ax^{[..(1)..[ p+(-q) ]..(1)..]+q}


ax^{p}+ax^{q} = c


x = (c/a)^{( 1/( [..(1)..[ p+(-q) ]..(1)..]+q ) )}


ax^{p}+bx^{q} = ax^{[..(b/a)..[ p+(-q) ]..(b/a)..]+q}


ax^{p}+bx^{q} = c


x = (c/a)^{( 1/( [..(b/a)..[ p+(-q) ]..(b/a)..]+q ) )}


x^{p}+mx = c


x = c^{( 1/( [..(m)..[ p+(-1) ]..(m)..]+1 ) )}


x = c^{( 1/( log[x](x^{(p+(-1))}+m)+1 ) )}


x^{p}+mx = (1+(-m))^{(1/(p+(-1)))} <==> ...
... log[x](x^{(p+(-1))}+m) = 0 <==> x = (1+(-m))^{(1/(p+(-1)))}


x^{p}+mx = 2m^{(p/(p+(-1)))} <==> ...
... log[x](x^{(p+(-1))}+m) = log[x](2m) <==> x = m^{(1/(p+(-1)))}


k·x^{p}+(mk)·x^{q} = c


x = (c/k)^{( 1/( [..(m)..[ p+(-q) ]..(m)..]+q ) )}


k·x^{p}+(mk)·x^{q} = k·(1+(-m))^{(q/(p+(-q)))} <==> ...
... log[x](x^{(p+(-q))}+m) = 0 <==> x = (1+(-m))^{(1/(p+(-q)))}


k·x^{p}+(mk)·x^{q} = (2k)·m^{(p/(p+(-q)))} <==> ...
... log[x](x^{(p+(-q))}+m) = log[x](2m) <==> x = m^{(1/(p+(-q)))}


x^{4}+4x = c


x = c^{( 1/( [..(4)..[3]..(4)..]+1 ) )}


x = c^{( 1/( log[x](x^{3}+4)+1 ) )}


x^{4}+4x = (-3)^{(1/3)} <==> log[x](x^{3}+4) = 0 <==> x = (-3)^{(1/3)}


x^{4}+4x = 2·4^{(4/3)} <==> log[x](x^{3}+4) = log[x](8) <==> x = 4^{(1/3)}


x^{4}+x = 0 <==> x = e^{(1/3)·(pi·i)}


4x^{8}+12x = c


x = (c/4)^{( 1/( [..(3)..[7]..(3)..]+1 ) )}


x = (c/4)^{( 1/( log[x](x^{7}+3)+1 ) )}


4x^{8}+12x = 4·(-2)^{(1/7)} <==> log[x](x^{7}+3) = 0 <==> x = (-2)^{(1/7)}


4x^{8}+12x = 8·3^{(8/7)} <==> log[x](x^{7}+3) = log[x](6) <==> x = 3^{(1/7)}


x^{8}+x = 0 <==> x = e^{(1/7)·(pi·i)}


8x^{8}+32x^{2} = c


x = (c/8)^{( 1/( [..(4)..[6]..(4)..]+2 ) )}


x = (c/8)^{( 1/( log[x](x^{6}+4)+2 ) )}


8x^{8}+32x^{2} = 8·(-3)^{(1/3)} <==> log[x](x^{6}+4) = 0 <==> x = (-3)^{(1/6)}


8x^{8}+32x^{2} = 16·4^{(4/3)} <==> log[x](x^{6}+4) = log[x](8) <==> x = 4^{(1/6)}


x^{8}+x^{2} = 0 <==> x = e^{(1/6)·(pi·i)}

álgebra: ecuacions polinomiques

x^{p}+x^{q} = 0 <==> x = e^{(1/(p+(-q)))·(pi·i)}


x^{[..(1)..[n]..(1)..]} = x^{n}+1


x^{p}+x^{q} = x^{[..(1)..[p+(-q)]..(1)..]+q}


x^{p}+x^{q} = c


x = c^{( 1/( [..(1)..[p+(-q)]..(1)..]+q ) )}


c^{( p/( [..(1)..[p+(-q)]..(1)..]+q ) )}+c^{( q/( [..(1)..[p+(-q)]..(1)..]+q ) )} = ...
c^{( q/( [..(1)..[p+(-q)]..(1)..]+q ) )}( c^{( (p+(-q))/( [..(1)..[p+(-q)]..(1)..]+q ) )}+1 ) = ...
c^{( q/( [..(1)..[p+(-q)]..(1)..]+q ) )}·c^{( [..(1)..[p+(-q)]..(1)..]/( [..(1)..[p+(-q)]..(1)..]+q ) )} = ...
c^{( ( [..(1)..[p+(-q)]..(1)..]+q )/( [..(1)..[p+(-q)]..(1)..]+q ) )} = c


x^{[..(1)..[0]..(1)..]} = 2
x^{[..(1)..[0]..(1)..]} = x^{0}+1 = 2


x^{[..(m)..[0]..(m)..]} = x^{0}+m = m+1

logaritme suma

ln(a+b) = ln( e^{ln(a)}+e^{ln(b)} ) = [ln(a)+(-1)·ln(b)]+ln(b)


ln(a+a) = [ln(a)+(-1)·ln(a)]+ln(a)
ln(a+a) = ln(2)+ln(a)


ln(a+b+c) = [ ln(a)+(-1)( [ln(b)+(-1)·ln(c)]+ln(c) )]+[ln(b)+(-1)·ln(c)]+ln(c)


ln(a+a+a) = [ ln(a)+(-1)( [ln(a)+(-1)·ln(a)]+ln(a) )]+[ln(a)+(-1)·ln(a)]+ln(a)
ln(a+a+a) = ln(3/2)+ln(2)+ln(a)


ln( x^{p}+y^{q} ) = z  <==> ...
... z = [ ln(p!)+e[( 1/p! )]+(-1)( ln(q!)+e[( 1/q! )] ) ]+( ln(q!)+e[( 1/q! )] ) ...
... x = e[( 1/p! )] ...
... y = e[( 1/q! )]


ln( x^{p}+y^{q} ) = z^{n}  <==> ...
... z = ( [ ln(p!)+e[( 1/p! )]+(-1)( ln(q!)+e[( 1/q! )] ) ]+( ln(q!)+e[( 1/q! )] ) )^{(1/n)} ...
... x = e[( 1/p! )] ...
... y = e[( 1/q! )]


ln( ax^{p}+by^{q} ) = cz^{n}  <==> ...
... ax^{p}+by^{q} = e^{cz^{n}}  <==> ...
... z = ( (1/c)·( [ ln(a·p!)+e[( 1/p! )]+(-1)( ln(b·q!)+e[( 1/q! )] ) ]+( ln(b·q!)+e[( 1/q! )] ) ) )^{(1/n)} ...
... x = e[( 1/p! )] ...
... y = e[( 1/q! )]


ln( ax^{p}+by^{q} ) = ln(c)+z^{n}  <==> ...
... ax^{p}+by^{q} = c·e^{z^{n}}  <==> ...
... z = ( (-1)·ln(c)+[ ln(a·p!)+e[( 1/p! )]+(-1)( ln(b·q!)+e[( 1/q! )] ) ]+( ln(b·q!)+e[( 1/q! )] ) )^{(1/n)} ...
... x = e[( 1/p! )] ...
... y = e[( 1/q! )]


ln( ax^{p}+by^{q} ) = ln(s)+cz^{n}  <==> ...
... ax^{p}+by^{q} = s·e^{cz^{n}}  <==> ...
... z = ( (1/c)·( (-1)·ln(s)+[ ln(a·p!)+e[( 1/p! )]+(-1)( ln(b·q!)+e[( 1/q! )] ) ]+( ln(b·q!)+e[( 1/q! )] ) ) )^{(1/n)}
... x = e[( 1/p! )] ...
... y = e[( 1/q! )]

ecuacions para-exponencial

e^{[..(m)..[x]..(m)..]} = (e^{x}+m)


[..(m)..[0]..(m)..] = ln(m+1)


e^{[..(m+1)..[0]..(m+1)..]} = e^{[..(m)..[0]..(m)..]}+1 = (m+1)+1 = m+2


e^{[..(m)..[x]..(m)..]} = (n+m) <==> x = ln(n)


e^{[..(m)..[x]..(m)..]} = n <==> x = ln(n+(-m))


e^{[..(m)..[z]..(m)..]} = (1/n!)·x^{n} <==> ...
... z = ln( e^{e[( 1/n! )]}+(-m) ) ...
... x = e[( 1/n! )]


e^{[..(m)..[z]..(m)..]} = x^{n} <==> ...
... z = ln( e^{ln(n!)+e[( 1/n! )]}+(-m) ) ...
... x = e[( 1/n! )]


e^{[..(m)..[z]..(m)..]} = (1/p!)·x^{p}+(1/q!)·y^{q} <==> ...
... z = ln( e^{[ e[( 1/p! )]+(-1)·e[( 1/q! )] ]+e[( 1/q! )]}+(-m) ) ...
... x = e[( 1/p! )] ...
... y = e[( 1/q! )]


e^{[..(m)..[z]..(m)..]} = x^{p}+y^{q} <==> ...
... z = ln( e^{[ ln(p!)+e[( 1/p! )]+(-1)·( ln(q!)+e[( 1/q! )] ) ]+ln(q!)+e[( 1/q! )]}+(-m) ) ...
... x = e[( 1/p! )] ...
... y = e[( 1/q! )]

ecuació exponencial III

e^{x}+e^{y}+e^{z}+e^{t} = e^{[x+(-1)( [y+(-1)( [z+(-t)]+t ) ]+[z+(-t)]+t )]+[y+(-1)( [z+(-t)]+t ) ]+[z+(-t)]+t}


[(-1)( [(-1)[0]]+[0] )] = ln(4/3)


e^{[(-1)( [(-1)[0]]+[0] )]}=(1/e^{[(-1)[0]]+[0]})+1 = (2/3)·(1/2)+1 = (1/3)+1 = (4/3)

ecuació exponencial II

e^{x}+e^{y}+e^{z} = e^{x}+e^{[y+(-z)]+z}


e^{x}+e^{y}+e^{z} = e^{[x+(-1)( [y+(-z)]+z ) ]+[y+(-z)]+z}


e^{x}+e^{y}+1 = e^{[x+(-1)[y] ]+[y]}


e^{x}+1+e^{z} = e^{[x+(-1)( [(-z)]+z ) ]+[(-z)]+z}
e^{x}+( (1/e^{z})+1 )·e^{z} = e^{[x+(-1)( [(-z)]+z ) ]+[(-z)]+z}


1+e^{y}+e^{z} = e^{[(-1)( [y+(-z)]+z ) ]+[y+(-z)]+z}
1+e^{[y+(-z)]+z} = e^{[(-1)( [y+(-z)]+z ) ]+[y+(-z)]+z}


[(-1)[0]] = ln(3/2)


e^{[(-1)[0]]}=(1/e^{[0]})+1 = (1/2)+1 = (3/2)

ecuació exponencial

e^{x}+e^{y} = ( e^{x+(-y)}+1 )·e^{y}
e^{x}+e^{y} = e^{[ x+(-y) ]}·e^{y}
e^{x}+e^{y} = e^{[ x+(-y) ]+y}


[0] = ln(2)
e^{[0]} = e^{0}+1 = 2


e^{x} = (1/n!)·x^{n} <==> x = e[( 1/n! )]


e^{z} = k·(1/n!)·x^{n} <==> ...
... z = ln(k)+e[( 1/n! )] ...
... x = e[( 1/n! )]


e^{z} = (1/n)·x^{n} <==> ...
... z = ln( (n+(-1))! )+e[( 1/n! )] ...
... x = e[( 1/n! )]


e^{z} = (n+1)·x^{n} <==> ...
... z = ln( (n+1)! )+e[( 1/n! )] ...
... x = e[( 1/n! )]


e^{z} = (1/n!)·x^{n} + a <==> ...
... z = [ e[( 1/n! )] + (-1)·ln(a) ] + ln(a) ...
... x = e[( 1/n! )]


e^{z} = (1/p!)·x^{p} + (1/q!)·y^{q} <==> ...
... z = [ e[( 1/p! )] + (-1)·e[( 1/q! )] ] + e[( 1/q! )] ...
... x = e[( 1/p! )] ...
... y = e[( 1/q! )]


e^{z} = x^{p} + y^{q} <==> ...
... z = [ ( ln(p!)+e[( 1/p! )] ) + (-1)·( ln(q!)+e[( 1/q! )] ) ] + ( ln(q!)+e[( 1/q! )] ) ...
... x = e[( 1/p! )] ...
... y = e[( 1/q! )]


e^{z} = ax^{p} + by^{q} <==> ...
... z = [ ( ln(a)+ln(p!)+e[( 1/p! )] ) + (-1)·( ln(b)+ln(q!)+e[( 1/q! )] ) ] + ( ln(b)+ln(q!)+e[( 1/q! )] ) ...
... x = e[( 1/p! )] ...
... y = e[( 1/q! )]

derivades ecuació potencial amb exponencial

Si ( ( f(x) )^{m}+e^{x} = ( f(x) )^{n} & f(x) >] 0 & d_{x}[f(x)] > 0 ) ==> m( f(x) )^{m} [< n( f(x) )^{n}


m( f(x) )^{(m+(-1))}·d_{x}[f(x)]+e^{x} = n( f(x) )^{(n+(-1))}·d_{x}[f(x)]


m( f(x) )^{(m+(-1))}·d_{x}[f(x)] [< n( f(x) )^{(n+(-1))}·d_{x}[f(x)]


m( f(x) )^{(m+(-1))} [< n( f(x) )^{(n+(-1))}

derivades ecuació potencial

Si ( ( f(x) )^{m} + x = ( f(x) )^{n} & f(x) >] 0 & d_{x}[f(x)] > 0 ) ==> m( f(x) )^{m} [< n( f(x) )^{n}


m( f(x) )^{(m+(-1))}·d_{x}[f(x)] + 1 = n( f(x) )^{(n+(-1))}·d_{x}[f(x)]


m( f(x) )^{(m+(-1))}·d_{x}[f(x)] [< n( f(x) )^{(n+(-1))}·d_{x}[f(x)]


m( f(x) )^{(m+(-1))} [< n( f(x) )^{(n+(-1))}

biótica: gatos

color
{
gato-negro
}
{
gato-negro-y-blanco
}
{
gato-blanco
}


color
{
gato-negro-persa
}
{
gato-gris-siamés
}
{
gato-blanco-persa
}


color
{
gato-granate-naranja
}
{
gato-naranja
}
{
gato-ocre-naranja
}


color
{
gato-rojo-marrón
}
{
gato-marrón
}
{
gato-amarillo-marrón
}

biótica: cerdo y jabalí

color
{
cerdo-negro-y-lila
}
{
cerdo-gris
}
{
cerdo-blanco-y-lila
}


color
{
jabalí-blanco-y-marrón
}
{
jabalí-gris
}
{
jabalí-negro-y-marrón
}

biótica: cabra y oveja

color
{
oveja-blanca
}
{
oveja-gris
}
{
oveja-negra
}


color
{
cabra-negra
}
{
cabra-gris
}
{
cabra-blanca
}

biótica: hipopótamo y rinoceronte

color
{
rinoceronte-blanco-con-dos-cuernos
}
{
rinoceronte-gris-con-dos-cuernos
}
{
rinoceronte-negro-con-dos-cuernos
}


color
{
hipopótamo-negro-con-dos-colmillos
}
{
hipopótamo-gris-con-dos-colmillos
}
{
hipopótamo-blanco-con-dos-colmillos
}

biótica: elefante

color
{
mamut-blanco-de-orejas-pequeñas
}
{
elefante-gris-de-orejas-grandes
}
{
mamut-negro-de-orejas-pequeñas
}


color
{
mastodonte-blanco-de-orejas-grandes
}
{
elefante-gris-de-orejas-pequeñas
}
{
mastodonte-negro-de-orejas-grandes
}

biótica: toro y buey

color
{
toro-blanco
}
{
buey-blanco-y-negro
}
{
toro-negro
}


color
{
buey-rojo-marrón
}
{
toro-marrón
}
{
buey-amarillo-marrón
}

biótica: cavallo, cebra y jirafa

color
{
caballo-blanco
}
{
cebra-blanca-y-negra
}
{
caballo-negro
}


color
{
jirafa-roja-marrón
}
{
caballo-marrón
}
{
jirafa-amarilla-marrón
}

biótica: ranas y sapos

color
{
rana-roja
}
{
rana-marrón
}
{
rana amarilla
}


color
{
sapo-verdoso
}
{
sapo-azul
}
{
sapo-lila
}

credits de biótica

Dual-bolets: 20 credits
Dual-fruites: 20 credits
Dual-animals: 30 credits
Dual-fulles: 20 credits
Dual-flors: 20 credits


Total: 110 credits

química: equilibri químic de la oxidació del nitrur de hidrógen

( [ 2·NH_{3} ]·[ O_{2} ] )/( [ N_{2}O_{2} ]·[ 3·H_{2}] ) = (48/60) = (8/10)  = (4/5)


5·[ 2·NH_{3} ]·[ O_{2} ] <==> 4·[ N_{2}O_{2} ]·[ 3·H_{2}]


E = 240

química: equilibri químic de la combustió del metà

( [ CH_{4} ]·[ O_{2} ] )/( [ CO_{2} ]·[ 2·H_{2} ] ) = 32/32 = 1


[ CH_{4} ]·[ O_{2} ] <==> [ CO_{2} ]·[ 2·H_{2} ]


E = 32

química: equilibri químic del aigua

( [ 2·H_{2} ]·[ O_{2} ] )/[ 2·H_{2}O ] = 16/8 = 2


[ 2·H_{2} ]·[ O_{2} ] <==> 2·[ 2·H_{2}O ]


E = 16

credits de economia

Computació-de-vectors-y-matrius: 20 credits
Álgebra: 20 credits
Álgebra-lineal-y-geometría-lineal: 20 credits
Análisis-matemàtic: 40 credits
Ecuacions-diferencials: 10 credits


micro-economia: 30 credits
micro-economía-de-geometría: 20 credits
simetría-de-consum: 30 credits
models-de-preus-de-ecuacions-diferencials: 30 credits


total: 220 credits

credits de química

Reaccions-y-entalpies: 60 credits
Molecules-poligonals-constructors-y-destructors: 40 credits
Molecules-poligonals-para-constructors-y-para-destructors: 40 credits
Configuracions-electróniques: 20 credits
equilibri-químic: 60 credits


total: 220 credits

óptica: reflexió atravesant

x_{1} = R·cos(s_{1})
x_{2} = R·cos(s_{1})


y_{1} = R·sin(s_{1})
y_{2} = (-1)·R·sin(s_{1})

óptica: reflexió rebotant

x_{1} = R·cos(s_{1})
x_{2} = (-1)·R·cos(s_{1})


y_{1} = R·sin(s_{1})
y_{2} = R·sin(s_{1})

óptica: refracció

x_{1} = R· cos(s_{1})
x_{2} = (-1)·R·sin(s_{1}) )


y_{1} = R·sin(s_{1})
y_{2} = (-1)·R·cos(s_{1})

óptica: reflexió de mirall

x_{1} = R·cos(s_{1})
x_{2} = R·cos(s_{1})


y_{1} = R·sin(s_{1})
y_{2} = R·sin(s_{1})

credits de física

Computació-de-vectors-y-matrius: 20 credits
Álgebra: 20 credits
Álgebra-lineal-y-geometría-lineal: 20 credits
Análisis-matemàtic: 40 credits
Ecuacions-diferencials: 10 credits


Mecánica-classica: 20 credits
Termodinámica: 10 credits
Electromagnetisme-y-gravitomagnetisme: 10 credits
Mecanisme-de-gauge: 10 credits
Mecanica-tensorial: 10 credits
Teoría-de-cordes: 10 credits
Mecanica-cuántica: 10 credits
Óptica: 10 credits
Circuits-eléctrics: 10 credits
Integrals-de-càrrega: 10 credits


Total: 220 credits

credits de matemàtiques

Computació-de-vectors-y-matrius: 20 credits
Álgebra: 20 credits
Álgebra-lineal-y-geometría-lineal: 20 credits
Análisis-matemàtic: 40 credits
Ecuacions-diferencials: 10 credits


Càlcul-integral: 10 credits
Topologia: 10 credits
Números-figurats y particions: 20 credits
Teoría-de-conjunts: 10 credits
Teoría-de-números: 10 credits
Probabilitats: 10 credits
Series-y-Sumes: 10 credits
Geometría-diferencial: 10 credits
Análisis-complexa: 10 credits
Análisis-funcional: 10 credits


Total: 220 credits

números hexagonals

n=2
0110
1111
0110


n=3
0011100
0111110
1111111
0111110
0011100


P(n) = ( n+2·(n+(-1)) )·( 1+2·(n+(-1)) )+(-2)·n(n+(-1))
P(n) = ( n+(n+1)·2·(n+(-1))+4·(n+(-1))^{2} )+(-2)·n(n+(-1))
P(n) = ( n+2·(n^{2}+(-1))+4·(n^{2}+(-2)n+1) )+(-2)·n(n+(-1))
P(n) = 4n^{2}+(-5)n+2


P(2) = 8
P(3) = 23

números triangulars

n=1
10


n=2
100
110


n=3
1000
1100
1110


P(n) = (1/2)·(n+1)·n

números mitx-hexagonals

n=2
0110
1111


n=3
0011100
0111110
1111111


n=4
0001111000
0011111100
0111111110
1111111111


n=5
0000111110000
0001111111000
0011111111100
0111111111110
1111111111111


P(n) = ( n+2(n+(-1)) )·n+(-1)(n+(-1))·n
P(n) = n( 2n+(-1) ) =  2n^{2}+(-n)


P(0) = 0
P(1/2) = 0


P(2) = 6
P(3) = 15
P(4) = 28
P(5) = 45

integrals exponencials

∫ [e^{( f(x) )^{n}}] d[x] = e^{( f(x) )^{n}} [o(x)o] ( ( f(x) )^{n} )^{[o(x)o](-1)}


∫ [e^{( ax )^{n}}] d[x] = ...
...e^{( ax )^{n}} [o(x)o] (1/a^{2})·(1/n)·( 1/((-n)+2) )·( ( ax )^{(-n)+2} )


∫ [e^{( ax^{2}+bx )^{n}}] d[x] = ...
...e^{( ax^{2}+bx )^{n}} [o(x)o]...
...(1/n)·( 1/((-n)+2) )·( ( ax^{2}+bx )^{(-n)+2} ) [o(x)o] (1/2a)·(-1)·( 2ax+b )^{(-1)}


∫ [e^{( ax^{3}+bx^{2}+cx )^{n}}] d[x] = ...
...e^{( ax^{3}+bx^{2}+cx )^{n}} [o(x)o]...
...(1/n)·( 1/((-n)+2) )·( ( ax^{3}+bx^{2}+cx )^{(-n)+2} ) [o(x)o] ...
...(-1)·( 3ax^{2}+2bx+c )^{(-1)} [o(x)o] (1/6a)·ln(6ax+b)


∫ [e^{( ax^{4}+bx^{3}+cx^{2}+dx )^{n}}] d[x] = ...
...e^{( ax^{4}+bx^{3}+cx^{2}+dx )^{n}} [o(x)o]...
...(1/n)·( 1/((-n)+2) )·( ( ax^{4}+bx^{3}+cx^{2}+dx )^{(-n)+2} ) [o(x)o] ...
...(-1)·( 4ax^{3}+3bx^{2}+2cx+d )^{(-1)} [o(x)o] ln(12ax^{2}+6bx+2c) [o(x)o] (1/24a)·ln(24ax+6b)

integrals potencials

∫ [( f(x) )^{n}] d[x] = (1/(n+1))( f(x) )^{(n+1)} [o(x)o] ( f(x) )^{[o(x)o](-1)}


∫ [ ( ax^{2}+bx )^{n} ] d[x] = (1/(n+1))( ax^{2}+bx )^{(n+1)} [o(x)o] (1/2a)·ln(2ax+b)


∫ [ ( ax^{3}+bx^{2}+cx )^{n} ] d[x] = ...
...(1/(n+1))( ax^{3}+bx^{2}+cx )^{(n+1)} [o(x)o] ( ln(3ax^{2}+2bx+c) [o(x)o] (1/6a)·ln(6ax+2b) )


∫ [ ( ax^{4}+bx^{3}+cx^{2}+dx )^{n} ] d[x] = ...
...(1/(n+1))( ax^{4}+bx^{3}+cx^{2}+dx )^{(n+1)} [o(x)o] ...
...( ln(4ax^{3}+3bx^{2}+2cx+d) [o(x)o] ln(12ax^{2}+6bx+2c) [o(x)o] (1/24a)·ln(24ax+6b) )

integral de serie geométrica

∫ [ e^{x}·( (x^{(n+1)}+(-1))/(x+(-1)) ) ] d[x] = e^{x}+∑ ( x^{(k+1)}·er_{m;k+1}(x) )


∫ [ e^{x}·( (x^{(n+1)}+(-1))/(x+(-1)) ) ] d[x]= ∫ [ e^{x}+∑ ( e^{x}·x^{k} ) ] d[x]


∫ [ e^{(-x)}·( (x^{(n+1)}+(-1))/(x+(-1)) ) ] d[x] = (-1)·( e^{(-x)}+∑ ( (-x)^{(k+1)}·er_{m;k+1}(-x) ) )

índex de física

índex de etiquetes de física

álgebra: índex-algebràic de un grup normal

A={1,3,5}
B={2,4,6}
C={3,5,7}


A+{1}={0}+B
A+{0}={(-1)}+B


A+{2}={0}+C
A+{0}={(-2)}+C


B+{1}={0}+C
B+{0}={(-1)}+C


E={1,3,5}
F={5,15,25}
G={2,6,10}


E·{5}={1}·F
E·{1}={(1/5)}·F


E·{2}={1}·G
E·{1}={(1/2)}·G


F·{(2/5)}={1}·G
F·{1}={(5/2)}·G

álgebra: sistema cuadrat

x^{2}+y^{2} =  p
x+y = q


(x+y)^{2}+(-2)xy = p


q^{2}+(-p) = 2xy


x^{2}+(q+(-x))^{2} =  p
(q+(-y))^{2}+y^{2} =  p


2x^{2}+(-2)qx+q^{2} =  p
2y^{2}+(-2)qy+q^{2} =  p


x^{2}+(-q)x+( (q^{2}+(-p))/2 ) =  0
y^{2}+(-q)y+( (q^{2}+(-p))/2 ) =  0


x = (1/2)( q+( 2p+(-1)q^{2} )^{(1/2)} )
y = (1/2)( q+(-1)( 2p+(-1)q^{2} )^{(1/2)} )

álgebra: exponent directe cuadrat

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


1+4 = 5 = (1/2)( ( 1+2 )^{2}+( 2+(-1) )^{2} )
4+4 = 8 = (1/2)( ( 2+2 )^{2}+( 2+(-2) )^{2} )


1+9 = 10 = (1/2)( ( 1+3 )^{2}+( 3+(-1) )^{2} )
4+9 = 13 = (1/2)( ( 2+3 )^{2}+( 3+(-2) )^{2} )
9+9 = 18 = (1/2)( ( 3+3 )^{2}+( 3+(-3) )^{2} )


1+16 = 17 = (1/2)( ( 1+4 )^{2}+( 4+(-1) )^{2} )
4+16 = 20 = (1/2)( ( 2+4 )^{2}+( 4+(-2) )^{2} )
9+16 = 25 = (1/2)( ( 3+4 )^{2}+( 4+(-3) )^{2} )
16+16 = 32 = (1/2)( ( 4+4 )^{2}+( 4+(-4) )^{2} )

índex de matemàtiques

index de etiquetes matemàtiques

índex de etiquetes

index de etiquetes.