Lema:
kn^{k+(-1)} [< k^{n}
Demostració:
(n+(-1))^{k+(-1)} [< n^{k+(-1)} [< k^{n+(-1)}
Lema:
k^{n} [< (1/k)·n^{k+1}
Demostració
k^{n+1} [< n^{k+1} [< (n+1)^{k+1}
Teorema:
1+n^{k+(-1)} [< k^{n}
Demostració:
... 1+n^{k+(-1)} [< ...
... 1+( k+(-1) )·n^{k+(-1)} = 1+kn^{k+(-1)}+(-1)·n^{k+(-1)} [< ...
... kn^{k+(-1)} [< k^{n}
Teorema:
1+n^{k+1} >] k^{n}
Demostració:
... 1+n^{k+1} >] n^{k+1} >] (1/k)·n^{k+1} >] k^{n}
Teorema:
0 [< lim[ ( n^{k+(-1)}/k^{n} ) ] [< 1
Demostració:
0 [< ( n^{k+(-1)}/k^{n} ) [< ( n^{k+(-1)}/(1+n^{k+(-1)}) ) [< 1
Teorema:
0 [< lim[ ( k^{n}/n^{k+1} ) ] [< 1
Demostració:
0 [< ( k^{n}/n^{k+1} ) [< ( (1+n^{k+1})/n^{k+1} ) [< 1
( k^{oo}/k^{oo} ) = k^{oo+(-oo)} = k
( k/k )^{oo} = ( k^{0} )^{oo} = k^{0·oo} = k
( k/k )^{oo} = e^{ln( ( k/k )^{oo} )} = e^{ln( k^{0·oo} )} = e^{(0·oo)·ln(k)} = e^{ln(k)} = k
Teorema:
f(x) = x <==> ( f(x+y) = f(x)+f(y) & f(x·y) = f(x)·f(y) )
Demostració:
f(x) = f(x+0) = f(x)+f(0)
f(0) = 0
f(x) = f(x·1) = f(x)·f(1)
f(1) = 1
0 = f(0) = f(x+(-x)) = f(x)+f(-x)
f(-x) = (-1)·f(x)
1 = f(1) = f(x·(1/x)) = f(x)·f(1/x)
f(1/x) = ( 1/f(x) )
Teorema:
lim[ ( 2n+1 )/( 2^{n}+(-1) ) ] [< 2
Demostració:
lim[ ( ( 2n+1 )/( 2^{n}+(-1) ) ) ] [< ...
... lim[ ( ( 2n+1 )/( (1+n)+(-1) ) ) ] = lim[ ( ( 2n+1 )/n ) ] = 2
Teorema:
lim[ ( 3n^{2}+2n+1 )/( 3^{n}+2^{n}+1 ) ] [< 3
Demostració:
lim[ ( ( 3n^{2}+2n+1 )/( 3^{n}+2^{n}+1 ) ) ] [< ...
... lim[ ( ( 3n^{2}+2n+1 )/( (1+n^{2})+(1+n)+(-1) ) ) ] = 3
Lley:
Un coshinet de roda esta girant amb boles a dins:
(o)---o---(o)
Radi interior:
d_{t}[x(t)] = d_{t}[u(t)]·R_{1}
Radi exterior:
d_{t}[y(t)] = d_{t}[v(t)]·R_{2}
La velocitat de les boles és:
[E s(t) ][ d_{t}[z(t)] = (1/2)·( R_{1}+R_{2} )·d_{t}[s(t)] ]
La posició de les boles és:
[E s(t) ][ z(t) = (1/2)·( R_{1}+R_{2} )·s(t) ]
Deducció:
d_{t}[x(t)] = d_{t}[z(t)]+d_{t}[w(t)]·R_{0}
d_{t}[y(t)] = d_{t}[z(t)]+(-1)·d_{t}[w(t)]·R_{0}
Es defineish un < s: R ---> R & t --> s(t) > ==>
s(t) = ( ( u(t)·R_{1}+v(t)·R_{2} )/( R_{1}+R_{2} ) )
s(t)·( R_{1}+R_{2} ) = u(t)·R_{1}+v(t)·R_{2}
d_{t}[s(t)·( R_{1}+R_{2} ) ] = d_{t}[u(t)·R_{1}+v(t)·R_{2}]
d_{t}[s(t)·( R_{1}+R_{2} ) ] = d_{t}[u(t)·R_{1}]+d_{t}[v(t)·R_{2}]
( R_{1}+R_{2} )·d_{t}[s(t)] = d_{t}[u(t)]·R_{1}+d_{t}[v(t)]·R_{2}
( R_{1}+R_{2} )·d_{t}[s(t)] = d_{t}[x(t)]+d_{t}[y(t)]
(1/2)·( R_{1}+R_{2} )·d_{t}[s(t)] = (1/2)·( d_{t}[x(t)]+d_{t}[y(t)] )
(1/2)·( R_{1}+R_{2} )·d_{t}[s(t)] = d_{t}[z(t)]
int[ (1/2)·( R_{1}+R_{2} )·d_{t}[s(t)] ]d[t] = int[ d_{t}[z(t)] ]d[t] = z(t)
(1/2)·( R_{1}+R_{2} )·int[ d_{t}[s(t)] ]d[t] = z(t)
(1/2)·( R_{1}+R_{2} )·s(t) = z(t)
Lley:
Un con de altura = h:
gira sobre un disc a: d_{t}[u(t)].
rota sobre si mateish a: d_{t}[v(t)].
y l'angle entre la altura del con y el terra = a.
radi DA = h·tan(a).
radi DB = h·tan(a).
radi OD = h·cos(a)
Lley A:
d_{t}[x(t)] = h·( cos(a)·d_{t}[u(t)]+(-1)·tan(a)·d_{t}[v(t)] )
x(t) = h·( cos(a)·u(t)+(-1)·tan(a)·v(t) )
Coriolis:
cos(a)·d_{t}[u(t)] = tan(a)·d_{t}[v(t)] <==> d_{t}[s(t)] = (-0)
Lley B:
d_{t}[x(t)] = h·( cos(a)·d_{t}[u(t)]+tan(a)·d_{t}[v(t)] )
x(t) = h·( cos(a)·u(t)+tan(a)·v(t) )
Coriolis:
cos(a)·d_{t}[u(t)] = tan(a)·d_{t}[v(t)] <==> d_{t}[s(t)] = (-2)·tan(a)·d_{t}[v(t)]
Deducció:
h·d_{t}[s(t)] = (-h)·( cos(a)·d_{t}[u(t)]+(-1)·tan(a)·d_{t}[v(t)] )
d_{t}[s(t)] = (-0)
h·d_{t}[s(t)] = (-h)·( cos(a)·d_{t}[u(t)]+tan(a)·d_{t}[v(t)] )
d_{t}[s(t)] = (-2)·tan(a)·d_{t}[v(t)] en A
x(t) = int[ d_{t}[x(t)] ]d[t]
x(t) = int[ h·( cos(a)·d_{t}[u(t)]+(-1)·tan(a)·d_{t}[v(t)] ) ]d[t]
x(t) = h·int[ ( cos(a)·d_{t}[u(t)]+(-1)·tan(a)·d_{t}[v(t)] ) ]d[t]
x(t) = h·( int[ cos(a)·d_{t}[u(t)] ]d[t]+int[ (-1)·tan(a)·d_{t}[v(t)] ]d[t] )
x(t) = h·( cos(a)·int[ d_{t}[u(t)] ]d[t]+(-1)·tan(a)·int[ d_{t}[v(t)] ]d[t] )
x(t) = h·( cos(a)·u(t)+(-1)·tan(a)·v(t) )
Lley:
Un rectangle de área = L·y
se li aplica una força horitzontal = F
La tensió interna de un sub-rectangle x·y = A(x) sobre (L+(-x))·y = B(x) compleish:
T(x) = F·(x/L) <==> T(x) = F·( A(x)/(A(x)+B(x)) )
El potencial intern A(x) compleish:
A(x) = F·(x/2)·(x/L) <==> A(x) = F·(x/2)·( A(x)/(A(x)+B(x)) )
Deducció:
y = ( (A(x)+B(x))/L ) = ( A(x)/x )
(x/L) = ( A(x)/(A(x)+B(x)) )
A(x) = int[ T(x) ]d[x]
A(x) = int[ F·(x/L) ]d[x]
A(x) = (F/L)·int[ x ]d[x]
A(x) = (F/L)·(1/2)·x^{2}
A(x) = F·(x/2)·(x/L)
Lley:
Si d_{x}[q(x)] = Q·s ==> q(L) = Q·s·L
Si d_{x}[q(x)] = Q·s·2·(x/L) ==> q(L) = Q·s·L
Si d_{x}[q(x)] = Q·s·(1/e)·( 1+e^{(x/L)} ) ==> q(L) = Q·s·L
Si d_{x}[q(x)] = Q·s·(2/pi)·( 1+(-1)·(x/L)^{2} )^{(-1)·(1/2)} ==> q(L) = Q·s·L
Si d_{x}[q(x)] = Q·s·(4/pi)·( 1+(x/L)^{2} )^{(-1)} ==> q(L) = Q·s·L
Definició:
E_{e}(x,y,z) = qk·(1/r^{3})·< x,y,z > = qk·(1/r^{2})·( < x,y,z >/r )
E_{g}(x,y,z) = (-1)·qk·(1/r^{3})·< x,y,z > = (-1)·qk·(1/r^{2})·( < x,y,z >/r )
( < x,y,z >/r )o( < 1,0,0 >,< 0,1,0 >,< 0,0,1 > )o( < x,y,z >/r ) = 1
Lley:
div[ E_{e}(x,y,z) ] = 3qk·(1/r^{3})
anti-potencial[ E_{e}(x,y,z) ] = 3qk·(1/r^{3})·xyz
div[ E_{g}(x,y,z) ] = (-3)·qk·(1/r^{3})
anti-potencial[ E_{g}(x,y,z) ] = (-3)·qk·(1/r^{3})·xyz
Lley:
potencial[ E_{e}(x,y,z) ] = A_{e}(r) = (-1)·qk·(1/r)
potencial[ E_{g}(x,y,z) ] = A_{g}(r) = qk·(1/r)
Deducció:
potencial[ E_{e}(x,y,z) ] = ...
... ( int[ qk·(1/r^{3})·x ]d[x]+int[ qk·(1/r^{3})·y ]d[y]+int[ qk·(1/r^{3})·z ]d[z] ) = ...
... qk·( (-2)/r^{3} )·( (1/2)·x^{2}+(1/2)·y^{2}+(1/2)·z^{2} ) = ...
... (-1)·qk·(1/r^{3})·( x^{2}+y^{2}+z^{2} )
r^{2} = ( x^{2}+y^{2}+z^{2} ) = ...
... < x,y,z >o( < 1,0,0 >,< 0,1,0 >,< 0,0,1 > )o< x,y,z >
Lagranià eléctric:
(m/2)·d_{t}[r(t)]^{2} = (-1)·qpk·(1/r)
Lagranià graviatori:
(m/2)·d_{t}[r(t)]^{2} = qpk·(1/r)
Deducció:
2n+(-2) = (-n)
3n = 2
Solucions del camp eléctric:
r(t) = ( (3/2)·2^{(1/2)}·i·( (qpk)/m )^{(1/2)}·t )^{(2/3)}
x(t) = ( (3/2)·2^{(1/2)}·i·( (qpk)/m )^{(1/2)}·t )^{(2/3)}·cos(ut)·sin(vt)
y(t) = ( (3/2)·2^{(1/2)}·i·( (qpk)/m )^{(1/2)}·t )^{(2/3)}·sin(ut)·sin(vt)
z(t) = ( (3/2)·2^{(1/2)}·i·( (qpk)/m )^{(1/2)}·t )^{(2/3)}·cos(vt)
Solucions del camp gravitatori:
r(t) = ( (3/2)·2^{(1/2)}·( (qpk)/m )^{(1/2)}·t )^{(2/3)}
x(t) = ( (3/2)·2^{(1/2)}·( (qpk)/m )^{(1/2)}·t )^{(2/3)}·cos(ut)·sin(vt)
y(t) = ( (3/2)·2^{(1/2)}·( (qpk)/m )^{(1/2)}·t )^{(2/3)}·sin(ut)·sin(vt)
z(t) = ( (3/2)·2^{(1/2)}·( (qpk)/m )^{(1/2)}·t )^{(2/3)}·cos(vt)
Newtonià eléctric orbital:
m·d_{tt}^{2}[x(t)] = ...
... qpk·( x/r^{3} )+2·d_{t}[r(t)]·d_{t}[f(u,v)]+r(t)·d_{tt}^{2}[f(u,v)]
m·d_{tt}^{2}[y(t)] = ...
... qpk·( y/r^{3} )+2·d_{t}[r(t)]·d_{t}[g(u,v)]+r(t)·d_{tt}^{2}[g(u,v)]
m·d_{tt}^{2}[z(t)] = ...
... qpk·( z/r^{3} )+2·d_{t}[r(t)]·d_{t}[h(u,v)]+r(t)·d_{tt}^{2}[h(u,v)]
Deducció:
(2/3)+(-1)+(-1) = (-1)·(1/3)+(-1) = (-1)·(4/3)
(2/3)+(-2) = (-1)·(4/3)
d_{tt}^{2}[ f(t)·cos(ut)·sin(vt) ] = ...
... d_{t}[ d_{t}[f(t)]·cos(ut)·sin(vt)+f(x)·( (-u)·sin(ut)·sin(vt)+v·cos(ut)·cos(vt) ) ]
d_{tt}^{2}[ f(t)·sin(ut)·sin(vt) ] = ...
... d_{t}[ d_{t}[f(t)]·sin(ut)·sin(vt)+f(x)·( u·cos(ut)·sin(vt)+v·sin(ut)·cos(vt) ) ]
órbita elíptica gravitatoria:
z(t) = r
T = periode orbital de 0 a 2pi
vT = 2pi
u = (1/t)
ru·2v·( cos(ut)+(-1)·sin(ut) )+d_{t}[r]·2v·( cos(ut)+sin(ut) ) = ...
... rv^{2}+( ( (qpk)/m )·(1/r^{2}) )
... d_{t}[r]·( 4·cos(1) ) = ( r·( (2pi)/T )+( T/(2pi) )·( (qpk)/m )·(1/r^{2})
Anti-electricitat de elix positiva:
d_{t}[r]·( 4·cos(1) )+(-1)·( r·( (2pi)/T ) = ( T/(2pi) )·( (qpk_{g})/m )·(1/r^{2})
d_{t}[r]·( 4·cos(1) )+(-1)·( r·( (2pi)/T ) = (-1)·( T/(2pi) )·( (qpk_{e})/m )·(1/r^{2})
electró de gir positiu = eléctric.
protó = eléctric.
Anti-gravetat de elix negativa:
(-1)·d_{t}[r]·( 4·cos(1) )+( r·( (2pi)/T ) = ( T/(2pi) )·( ((-q)pk_{g})/m )·(1/r^{2})
(-1)·d_{t}[r]·( 4·cos(1) )+( r·( (2pi)/T ) = (-1)·( T/(2pi) )·( ((-q)pk_{e})/m )·(1/r^{2})
electró de gir negatiu = gravitatori.
neutró = gravitatori.
En física solo existen en el potencial:
int[ (1/r^{3})·x ]d[x] = ( (-2)/r^{3} )·int[ x ]d[x]
int[ (1/r^{4})·x ]d[x] = ( (-1)/r^{4} )·int[ x ]d[x]
A_{e}(r) = (-1)·qk·(1/r) || A_{e}(r) = (-1)·(1/2)·qk·(1/r^{2})
A_{g}(r) = qk·(1/r) || A_{g}(r) = (1/2)·qk·(1/r^{2})
Definició:
B_{e}(d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t) = ...
... qk·(1/r^{3})·(3/2)·< d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t >
Newtonià magnétic:
m·d_{tt}^{2}[x(t)] = p·B_{e}(d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t)
x(t) = ( (3/2)·2^{(1/2)}·i·( (qpk)/m )^{(1/2)}·t )^{(2/3)}
Lley:
div[ B_{e}(d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t) ] = 3·qk·(1/r^{3})
anti-potencial[ B_{e}(d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t) ] = ...
... 3·(27/8)·qk·(1/r^{3})·(d_{t}[x]·t)·(d_{t}[y]·t)·(d_{t}[z]·t)
Newtonià electro-magnétic:
m·d_{tt}^{2}[x(t)] = p·( E_{e}(x,y,z)+B_{e}(d_{t}[x]·t,d_{t}[y]·t,d_{t}[z]·t) )
x(t) = ( 3·i·( (qpk)/m )^{(1/2)}·t )^{(2/3)}
