Preliminares:
Principio:
[Ev][ d_{t}[x] = v ]
Ley:
x(t) = vt+h
Deducción:
Principio:
[Eg][ d_{tt}^{2}[x] = g ]
Ley:
Si g = 0 ==> [Ev][ d_{t}[x] = v ]
Ley:
d_{t}[x] = gt+v
x(t) = g·(1/2)·t^{2}+vt+h
Deducción:
Procedimiento en coordenadas cartesianas:
---------------------------------------------------
Principio: [ de Fuerza ]
[EF_{k}][ m·d_{tt}^{2}[x] = sum[k = 1]-[n][ F_{k} ] ]
Ley: [ de Impulsión ]
[Ep_{k}][ m·d_{t}[x] = sum[k = 1]-[n][ p_{k} ] ]
Deducción:
int[ m·d_{tt}^{2}[x] ]d[t] = m·int[ d_{tt}^{2}[x] ]d[t] = m·d_{t}[x]
int[ sum[k = 1]-[n][ F_{k} ] ]d[t] = sum[k = 1]-[n][ int[ F_{k} ]d[t] ] = sum[k = 1]-[n][ p_{k} ]
Ley: [ de Energía ]
[EU_{k}][ (m/2)·d_{t}[x]^{2} = sum[k = 1]-[n][ U_{k} ] ]
Deducción:
int[ m·d_{tt}^{2}[x] ]d[x] = int[ m·d_{tt}^{2}[x]·d_{t}[x] ]d[t] = (m/2)·d_{t}[x]^{2}
int[ sum[k = 1]-[n][ F_{k} ] ]d[x] = sum[k = 1]-[n][ int[ F_{k} ]d[x] ] = sum[k = 1]-[n][ U_{k} ]
Ley: [ de Potencia ]
[EN_{k}][ (m/u)·d_{t}[x]^{[o(ut)o] 2} = sum[k = 1]-[n][ N_{k} ] ]
Deducción:
int[ m·d_{tt}^{2}[x] ]d[ d_{t}[x] ] = int[ (m/u)·d_{tt}^{2}[x]^{2} ]d[ut] = (m/u)·d_{t}[x]^{[o(ut)o] 2}
int[ sum[k = 1]-[n][ F_{k} ] ]d[ d_{t}[x] ] = sum[k = 1]-[n][ int[ F_{k} ]d[ d_{t}[x] ] ] = ...
... sum[k = 1]-[n][ N_{k} ]
Fuerza constante:
Ley:
Sea F(t) = F ==>
d_{t}[x] = (F/m)·t
x(t) = (F/m)·(1/2)·t^{2}
Deducción:
Sea F(t) = F ==>
U(x) = Fx
(m/2)·d_{t}[x]^{2} = U(x)
Deducción:
U(x) = int[ F ]d[x] = F·int[ d[x] ] = Fx
(m/2)·d_{t}[x]^{2} = (m/2)·( (F/m)·t )^{2} = F·( (F/m)·(1/2)·t^{2} ) = Fx = U(x)
Ley:
Sea F(t) = F ==>
N(d_{t}[x]) = F·d_{t}[x]
(m/u)·d_{t}[x]^{[o(ut)o] 2} = N(d_{t}[x])
Deducción:
N(d_{t}[x]) = int[ F ]d[ d_{t}[x] ] = F·int[ d[ d_{t}[x] ] ] = F·d_{t}[x]
(m/u)·d_{t}[x]^{[o(ut)o] 2} = (m/u)·( (F/m)·t )^{[o(ut)o] 2} = int[ (m/u)·(F/m)^{2} ]d[ut] = ...
... F·( (F/m)·t ) = F·d_{t}[x] = N(d_{t}[x])
Fuerza lineal de carga variable:
Ley:
Sea F(t) = Itg ==>
d_{t}[y] = (1/m)·Ig·(1/2)·t^{2}
y(t) = (1/m)·Ig·(1/6)·t^{3}
Deducción:
U(y) = (1/m)·(Ig)^{2}·(1/8)·t^{4}
(m/2)·d_{t}[y]^{2} = U(y)
Deducción:
U(y) = int[ Itg ]d[y] = int[ Itg d_{t}[y] ]d[t] = int[ (1/m)·( Ig )^{2}·(1/2)·t^{3}· ]d[t] = ...
... (1/m)·( Ig )^{2}·(1/8)·t^{4}
(m/2)·d_{t}[y]^{2} = (m/2)·( ( (1/m)·Ig )·(1/2)·t^{2} )^{2} = (1/m)·( Ig )^{2}·(1/8)·t^{4} = U(y)
N(d_{t}[y]) = (1/m)·( Ig )^{2}·(1/3)·t^{3}
(m/u)·d_{t}[y]^{[o(ut)o] 2} = N(d_{t}[y])
Deducción:
N(d_{t}[y]) = int[ Itg ]d[ d_{t}[y] ] = int[ (1/m)·Itg d_{tt}^{2}[y] ]d[t] = ...
... int[ (1/m)·( Itg )^{2} ]d[t] = (1/m)·( Ig )^{2}·(1/3)·t^{3}
(m/u)·d_{t}[y]^{[o(ut)o] 2} = (m/u)·( ( (1/m)·Ig )·(1/2)·t^{2} )^{[o(ut)o] 2} = ...
... int[ (1/m)·( Ig )^{2}·t^{2} ]d[t] = (1/m)·( Ig )^{2}·(1/3)·t^{3} = N(d_{t}[y])
Fuerzas de amortiguación y de resistencia de fluido:
Horizontal:
Ley:
m·d_{tt}^{2}[x] = (-k)·x
x(t) = re^{(k/m)^{(1/2)}·it}
Deducción:
m·d_{tt}^{2}[x] = m·d_{t}[ d_{t}[x] ] = m·d_{t}[ d_{t}[ re^{(k/m)^{(1/2)}·it} ] ] = ...
... m·d_{t}[ r·d_{t}[ e^{(k/m)^{(1/2)}·it} ] ] = m·d_{t}[ re^{(k/m)^{(1/2)}·it}·(k/m)^{(1/2)}·i ] = ...
... mr·(k/m)^{(1/2)}·i·d_{t}[ e^{(k/m)^{(1/2)}·it} ] = ...
... mr·( (k/m)^{(1/2)}·i )^{2} e^{(k/m)^{(1/2)}·it} = ...
... (-k)·re^{(k/m)^{(1/2)}·it} = (-k)·x
Ley
m·d_{tt}^{2}[x] = (-b)·d_{t}[x]
d_{t}[x] = ve^{(-1)·(b/m)·t}
Deducción:
m·d_{tt}^{2}[x] = m·d_{t}[ d_{t}[x] ] = m·d_{t}[ ve^{(-1)·(b/m)·t} ] = ...
... mv·d_{t}[ e^{(-1)·(b/m)·t} ] = mv·e^{(-1)·(b/m)·t}·(-1)·(b/m) = ...
... (-b)·ve^{(-1)·(b/m)·t} = (-b)·d_{t}[x]
U(x) = (-k)·(1/2)·x^{2}
(m/2)·d_{t}[x]^{2} = U(x)
Deducción:
U(x) = int[ (-k)·x ]d[x] = (-k)·int[ x ]d[x] = (-k)·(1/2)·x^{2}
(m/2)·d_{t}[x]^{2} = (m/2)·( i·(k/m)^{(1/2)}·re^{(k/m)^{(1/2)}·it} )^{2} = ...
... (-k)·(1/2)·( r^{2}·e^{(k/m)^{(1/2)}·2it} ) = (-k)·(1/2)·x^{2} = U(x)
N(d_{t}[x]) = (-b)·(1/2)·d_{t}[x]^{2}
(m/u)·d_{t}[x]^{[o(ut)o] 2} = N(d_{t}[x])
Deducción:
N(d_{t}[x]) = int[ (-b)·d_{t}[x] ]d[ d_{t}[x] ] = ...
... (-b)·int[ d_{t}[x] ]d[ d_{t}[x] ] = (-b)·(1/2)·d_{t}[x]^{2}
(m/u)·d_{t}[x]^{[o(ut)o] 2} = (m/u)·( ve^{(-1)·(b/m)·t} )^{[o(ut)o] 2} = ...
... (m/u)·int[ ( (-1)·(b/m)·v )^{2}·e^{(-2)·(b/m)·t} ]d[ut] = ...
... (-b)·(1/2)·( v^{2}·e^{(-2)·(b/m)·t} ) = (-b)·(1/2)·d_{t}[x]^{2} = N(d_{t}[x])
Fuerzas de amortiguación y de resistencia de fluido:
Vertical:
Ley:
m·d_{tt}^{2}[y] = (-k)·y+qg
y(t) = re^{(k/m)^{(1/2)}·it}+(1/k)·qg
Deducción:
m·d_{tt}^{2}[y] = m·d_{t}[ d_{t}[y] ] = m·d_{t}[ d_{t}[ re^{(k/m)^{(1/2)}·it}+(1/k)·qg ] ] = ...
... m·d_{t}[ d_{t}[ re^{(k/m)^{(1/2)}·it} ]+d_{t}[ (1/k)·qg ] ] = ...
... m·d_{t}[ d_{t}[ re^{(k/m)^{(1/2)}·it} ]+0 ] = m·d_{t}[ d_{t}[ re^{(k/m)^{(1/2)}·it} ] = ...
... m·d_{tt}^{2}[ re^{(k/m)^{(1/2)}·it} ] = (-k)·re^{(k/m)^{(1/2)}·it} = ...
... (-k)·re^{(k/m)^{(1/2)}·it}+(-1)·qg+qg = (-k)·( re^{(k/m)^{(1/2)}·it}+(1/k)·qg )+qg = (-k)·y+qg
Ley:
m·d_{tt}^{2}[y] = (-b)·d_{t}[y]+qg
d_{t}[y] = ve^{(-1)·(b/m)·t}+(1/b)·qg
Deducción:
m·d_{tt}^{2}[y] = m·d_{t}[ d_{t}[y] ] = m·d_{t}[ ve^{(-1)·(b/m)·t}+(1/b)·qg ] = ...
... m·( d_{t}[ ve^{(-1)·(b/m)·t} ]+d_{t}[ (1/b)·qg ] ) = ...
... m·( d_{t}[ ve^{(-1)·(b/m)·t} ]+0 ) = m·d_{t}[ ve^{(-1)·(b/m)·t} ] = (-b)·ve^{(-1)·(b/m)·t} = ...
... (-b)·ve^{(-1)·(b/m)·t}+(-1)·qg+qg = (-b)·( ve^{(-1)·(b/m)·t}+(1/b)·qg )+qg = (-b)·d_{t}[y]+qg
Obertura de hombros y caderas robótica:
Por amortiguador de retorno con empuje de obertura de fluido.
Ley:
m·d_{tt}^{2}[y] = (-k)·y+( sin(w)·qg+sin(s)·pg )
y(t) = re^{(k/m)^{(1/2)}·it}+(1/k)·( sin(w)·qg+sin(s)·pg )
Ley:
m·d_{tt}^{2}[y] = (-b)·d_{t}[y]+( sin(w)·qg+sin(s)·pg )
d_{t}[y] = ve^{(-1)·(b/m)·t}+(1/b)·( sin(w)·qg+sin(s)·pg )
Estiramiento de rodillas y codos robótica:
Por amortiguador de retorno con empuje de obertura de fluido.
Ley:
m·d_{tt}^{2}[y] = (-k)·y+( F+(-1)·qg )
y(t) = re^{(k/m)^{(1/2)}·it}+(1/k)·( F+(-1)·qg )
Ley:
m·d_{tt}^{2}[y] = (-b)·d_{t}[y]+( F+(-1)·qg )
d_{t}[y] = ve^{(-1)·(b/m)·t}+(1/b)·( F+(-1)·qg )
Motores robóticos que aumentan la fuerza según la carga:
Ley:
m·d_{tt}^{2}[y] = (-k)·y+( Itg+qg )
y(t) = re^{(k/m)^{(1/2)}·it}+(1/k)·( Itg+qg )
Ley:
m·d_{tt}^{2}[y] = (-b)·d_{t}[y]+( Itg+qg )
d_{t}[y] = ve^{(-1)·(b/m)·t}+(1/b)·( Itg+qg )+(-1)·(1/b)^{2}·Ig
----------------------------------------------
Procedimiento en coordenadas polares:
----------------------------------------------
Principio: [ de Inercia angular ]
[EI_{ck}][ mdx = sum[k = 1]-[n][ I_{ck} ] ]
[ d ] = ( metro / ( radian )^{2} )
Ley: [ de Impulsión angular ]
[EL_{k}][ md·d_{t}[x] = sum[k = 1]-[n][ L_{k} ] ]
Deducción:
d_{t}[ mdx ] = md·d_{t}[x] = md·d_{t}[x]
d_{t}[ sum[k = 1]-[n][ I_{ck} ] ] = sum[k = 1]-[n][ d_{t}[ I_{ck} ] ] = ...
... sum[k = 1]-[n][ L_{k} ]
Ley: [ de Fuerza angular ]
[EH_{k}][ md·d_{tt}^{2}[x] = sum[k = 1]-[n][ H_{k} ] ]
Deducción:
d_{t}[ md·d_{t}[x] ] = md·d_{t}[ d_{t}[x] ] = md·d_{tt}^{2}[x]
d_{t}[ sum[k = 1]-[n][ L_{k} ] ] = sum[k = 1]-[n][ d_{t}[ L_{k} ] ] = ...
... sum[k = 1]-[n][ H_{k} ]
Principio: [ de Energía angular ]
[EU_{k}][ I_{c}·(1/2)·d_{t}[w]^{2} = sum[k = 1]-[n][ U_{k} ] ]
Ley:
Sea U(w) = U ==>
Si I_{c} = M·(r/s)^{2} ==>
x(t) = (M/m)·(1/d)·(r/s)^{2}
w(t) = ( (2/M)·U )^{(1/2)}·(s/r)·t
Deducción:
mdx = M·(r/s)^{2}
I_{c}·(1/2)·d_{t}[w]^{2} = M·(r/s)^{2}·(1/2)·d_{t}[ ( (2/M)·U )^{(1/2)}·(s/r)·t ]^{2} = ...
... M·(r/s)^{2}·(1/2)·( ( (2/M)·U )^{(1/2)}·(s/r)·d_{t}[ t ] )^{2} = ...
... M·(r/s)^{2}·(1/2)·( ( (2/M)·U )^{(1/2)}·(s/r) )^{2} = U
Principio: [ Fundamental de la dinámica angular ]
[EM_{k}][ d_{w}[ U(w) ] = sum[k = 1]-[n][ M_{k} ] ]
[ M_{k} ] = ( Joule / Radian )
Ley:
L(t)·(1/2)·d_{t}[w]+I_{c}·d_{tt}^{2}[w] = sum[k = 1]-[n][ M_{k} ]
Deducción:
d_{w}[ I_{c}·(1/2)·d_{t}[w]^{2} ] = (1/d_{t}[w])·d_{t}[ I_{c}·(1/2)·d_{t}[w]^{2} ] ...
... (1/d_{t}[w])·( d_{t}[ I_{c} ]·(1/2)·d_{t}[w]^{2}+I_{c}·d_{t}[ (1/2)·d_{t}[w]^{2} ] ) = ...
... L(t)·(1/2)·d_{t}[w]+I_{c}·d_{tt}^{2}[w]
Ley: [ de Momento de Fuerza ]
Si d_{t}[ I_{c} ] = 0 ==>
[EM_{k}][ I_{c}·d_{tt}^{2}[w] = sum[k = 1]-[n][ M_{k} ] ]
Ley: [ de Momento de Impulsión ]
Si d_{t}[ I_{c} ] = 0 ==>
[EK_{k}][ I_{c}·d_{t}[w] = sum[k = 1]-[n][ K_{k} ] ]
Deducción:
int[ I_{c}·d_{tt}^{2}[w] ]d[t] = I_{c}·int[ d_{tt}^{2}[x] ]d[t] = I_{c}·d_{t}[w]
int[ sum[k = 1]-[n][ M_{k} ] ]d[t] = sum[k = 1]-[n][ int[ M_{k} ]d[t] ] = ...
... sum[k = 1]-[n][ K_{k} ]
Ley: [ de Potencia angular ]
Si d_{t}[ I_{c} ] = 0 ==>
[EN_{k}][ (I_{c}/u)·d_{t}[w]^{[o(ut)o] 2} = sum[k = 1]-[n][ N_{k} ] ]
Deducción:
int[ I_{c}·d_{tt}^{2}[w] ]d[ d_{t}[w] ] = int[ (I_{c}/u)·d_{tt}^{2}[w]^{2} ]d[ut] = ...
.. (I_{c}/u)·d_{t}[w]^{[o(ut)o] 2}
int[ sum[k = 1]-[n][ M_{k} ] ]d[ d_{t}[w] ] = sum[k = 1]-[n][ int[ M_{k} ]d[ d_{t}[w] ] ] = ...
... sum[k = 1]-[n][ N_{k} ]
Ley:
Si ( d_{t}[ I_{c} ] = 0 & M(w) = F·(x/s) ) ==>
d_{t}[w] = (1/I_{c})·F·(x/s)·t
w(t) = (1/I_{c})·F·(x/s)·(1/2)·t^{2}
Deducción:
Problema.
Ley:
Si ( d_{t}[ I_{c} ] = 0 & M(w) = F·(x/s) ) ==>
U(w) = F·(x/s)·w
I_{c}·d_{t}[w]^{2} = U(w)
Deducción:
U(w) = int[ M(w) ]d[w] = int[ F·(x/s) ]d[w] = F·(x/s)·int[ d[w] ] = F·(x/s)·w
(I_{c}/2)·d_{t}[w]^{2} = (I_{c}/2)·( (1/I_{c})·F·(x/s)·t )^{2} = ...
... F·(x/s)·( (1/I_{c})·F·(x/s)·(1/2)·t^{2} ) = F·(x/s)·w = U(w)
Ley:
Si ( d_{t}[ I_{c} ] = 0 & M(w) = F·(x/s) ) ==>
N(d_{t}[w]) = F·(x/s)·d_{t}[w]
(I_{c}/u)·d_{t}[w]^{[o(ut)o] 2} = N(d_{t}[w])
Deducción:
N(d_{t}[w]) = int[ M(w) ]d[ d_{t}[w] ] = int[ F·(x/s)·d_{tt}^{2}[w] ]d[t] = ...
... int[ (1/I_{c})·( F·(x/s) )^{2} ]d[t] = (1/I_{c})·( F·(x/s) )^{2}·int[ d[t] ] = ...
... (1/I_{c})·( F·(x/s) )^{2}·t = F·(x/s)·d_{t}[w]
(I_{c}/u)·d_{t}[w]^{[o(u)o] 2} = (I_{c}/u)·( (1/I_{c})·F·(x/s)·t )^{[o(ut)o] 2} = ...
... (I_{c}/u)·int[ ( (1/I_{c})·F·(x/s) )^{2} ]d[ut] = ...
...(1/I_{c})·( F·(x/s) )^{2}·int[ d[t] ] = (1/I_{c})·( F·(x/s) )^{2}·t = ...
... F·(x/s)·d_{t}[w] = N(d_{t}[w])
Inercias angulares constantes:
Principio:
[Ef][ I_{c} = int[ ( (r·f(n))/s )^{2}·d_{n}[m(n)] ]d[n] ]
Ley:
Si f(n) = n^{0} ==> I_{c} = int[ (r/s)^{2}·d_{n}[m(n)] ]d[n]
Si f(n) = (n/r) ==> I_{c} = int[ (n/s)^{2}·d_{n}[m(n)] ]d[n]
Principio:
[Ef][Eg][ I_{c} = int-int[ (1/2)·( ( (r·f(p))/s )^{2}+( (r·g(q))/s )^{2} )·d_{pq}^{2}[m(p,q)] ]d[p]d[q] ]
Ley:
Si ( f(p) = p^{0} & g(q) = q^{0} ) ==> I_{c} = int-int[ (r/s)^{2}·d_{pq}^{2}[m(p,q)] ]d[p]d[q]
Si ( f(p) = (p/r) & g(q) = (q/r) ) ==> ...
... I_{c} = int-int[ (1/2)·( (p/s)^{2}+(q/s)^{2} )·d_{pq}^{2}[m(p,q)] ]d[p]d[q]
Ley:
Sea ( U(w) = U & F(x) = int[ f(x) ]d[x] ) ==>
Si I_{c} = int[ (r/s)^{2}·Ma·f(an) ]d[n] ==>
I_{c} = (r/s)^{2}·M·F(an)
x(t) = (1/(md))·(r/s)^{2}·M·F(an)
w(t) = ( 2U )^{(1/2)}·(s/r)·( 1/(M·F(an)) )^{(1/2)}·t
K(t) = ( 2U )^{(1/2)}·(r/s)·( M·F(an) )^{(1/2)}
Ley:
Sea U(w) = U ==>
Si I_{c} = int[ (n/s)^{2}·Ma ]d[n] ==>
I_{c} = (1/3)·(n/s)^{2}·Man
x(t) = (1/(md))·(1/3)·(n/s)^{2}·Man
w(t) = ( 6U )^{(1/2)}·(s/n)·( 1/(Man) )^{(1/2)}·t
K(t) = ( 6U )^{(1/2)}·(1/3)·(n/s)·( Man )^{(1/2)}
Ley:
Sea ( U(w) = U & F(x) = int[ f(x) ]d[x] & G(x) = int[ g(x) ]d[x] ) ==>
Si I_{c} = int-int[ (r/s)^{2}·Ma^{2}·f(ap)·g(aq) ]d[p]d[q] ==>
I_{c} = (r/s)^{2}·M·F(ap)·G(aq)
x(t) = (1/(md))·(r/s)^{2}·M·F(ap)·G(aq)
w(t) = ( 2U )^{(1/2)}·(s/r)·( 1/(M·F(ap)·G(aq)) )^{(1/2)}·t
K(t) = ( 2U )^{(1/2)}·(r/s)·( M·F(an)·G(aq) )^{(1/2)}
Deducción:
I_{c} = int-int[ (r/s)^{2}·M·f(ap)·g(aq) ]d[ap]d[aq] = (r/s)^{2}·M·int-int[ f(ap)·g(aq) ]d[ap]d[aq] = ...
... (r/s)^{2}·M·int[ g(aq)·int[ f(ap) ]d[ap] ]d[aq] = (r/s)^{2}·M·int[ g(aq)·F(ap) ]d[aq] = ...
... (r/s)^{2}·M·F(ap)·int[ g(aq) ]d[aq] = (r/s)^{2}·M·F(ap)·G(aq)
Ley:
Sea ( U(w) = U & F(x) = int[ f(x) ]d[x] & G(x) = int[ g(x) ]d[x] ) ==>
Si I_{c} = int-int[ (r/s)^{2}·Ma^{2}·( f(ap)+g(aq) ) ]d[p]d[q] ==>
I_{c} = (r/s)^{2}·M·( F(ap)·aq+ap·G(aq) )
x(t) = (1/(md))·(r/s)^{2}·M·( F(ap)·aq+ap·G(aq) )
w(t) = ( 2U )^{(1/2)}·(s/r)·( 1/(M·( F(ap)·aq+ap·G(aq) )) )^{(1/2)}·t
K(t) = ( 2U )^{(1/2)}·(r/s)·( M·( F(ap)·aq+ap·G(aq) ) )^{(1/2)}
Ley:
Sea ( U(w) = U & p = r·sin(nw) & q = r·cos(nw) ) ==>
Si I_{c} = int-int[ (1/2)·( (p/s)^{2}+(q/s)^{2} )·Ma^{2} ]d[p]d[q] ==>
I_{c} = (r/s)^{2}·M·(1/4)·npi·(ar)^{2}
x(t) = (1/(md))·(r/s)^{2}·M·(1/4)·npi·(ar)^{2}
w(t) = ( (8/(npi))·U )^{(1/2)}·(s/r)·(1/(ar))·(1/M)^{(1/2)}·t
K(t) = ( (8/(npi))·U )^{(1/2)}·(r/s)·(ar)·M^{(1/2)}
Deducción:
d[p] = nr·cos(nw)·d[w] & d[q] = nr·(-1)·sin(nw)·d[w]
Sea ( U(w) = U & p = (r/i)·sinh(nw) & q = r·cosh(nw) ) ==>
Si I_{c} = int-int[ (1/2)·( (p/s)^{2}+(q/s)^{2} )·Ma^{2} ]d[p]d[q] ==>
I_{c} = (r/s)^{2}·M·(1/4)·npi·(ar)^{2}
Motores de rotación.
Ley:
Sea U(w) = U ==>
Si d[I_{c}] = (1/s)^{2}·Mrv·d[t] ==>
x(t) = (1/(md))·(1/s)^{2}·Mrvt
w(t) = ( (8/M)·(1/(rv))·U )^{(1/2)}·st^{(1/2)}
Ley:
Sea U(w) = U ==>
Si d[I_{c}] = (1/s)^{2}·Mrgt·d[t] ==>
x(t) = (1/(md))·(1/s)^{2}·Mrg·(1/2)·t^{2}
w(t) = ( (4/M)·(1/(rg))·U )^{(1/2)}·s·ln(ut)
Articulaciones robóticas y de vehículo.
Ley:
Sea U(w) = U ==>
Si d[L(t)] = (1/s)^{2}·Mv·d[x] ==>
x(t) = re^{(M/m)·(v/d)·(1/s)^{2}·t}
w(t) = ( (8/(mdr))·U )^{(1/2)}·(-1)·(m/M)·(d/v)·s^{2}·e^{(-1)·(1/2)·(M/m)·(v/d)·(1/s)^{2}·t}
Ley:
Sea U(w) = U ==>
Si d[L(t)] = (1/s)^{2}·Mg·d[tx] ==>
x(t) = re^{(M/m)·(g/d)·(1/s)^{2}·(1/2)·t^{2}}
w(t) = ( (8/(mdr))·U )^{(1/2)}·...
... ( (-1)·(m/M)·(d/g)·s^{2}·ln(ut) ) [o(t)o] e^{(-1)·(1/2)·(M/m)·(g/d)·(1/s)^{2}·(1/2)·t^{2}}
Ley:
Sea U(w) = U ==>
Si d[L(t)] = (1/s)^{2}·Mav·2x·d[x] ==>
x(t) = ( (-1)·(M/m)·(v/d)·(1/s)^{2}·at )^{(-1)}
w(t) = ( (2/(md))·U )^{(1/2)}·( (-1)·(M/m)·(v/d)·(1/s)^{2}·a )^{(1/2)}·(2/3)·t^{(3/2)}
Deducción:
d_{x}[ x^{2} ] = 2x
d_{x}[ x^{2} ]·d[x] = 2x·d[x]
d[ x^{2} ] = 2x·d[x]
L(t) = int[ d[L(t)] ] = int[ (1/s)^{2}·Mav·2x·d[x] ] = ...
... int[ (1/s)^{2}·Mav ]d[x^{2}] = (1/s)^{2}·Mav·int[ d[x^{2}] ] = (1/s)^{2}·Mavx^{2}
Ley:
Sea U(w) = U ==>
Si d[L(t)] = (1/s)^{2}·Mag·( d[t]·x^{2}+t·2x·d[x] ) ==>
x(t) = ( (-1)·(M/m)·(g/d)·(1/s)^{2}·a·(1/2)·t^{2} )^{(-1)}
w(t) = ( (2/(md))·U )^{(1/2)}·( (-1)·(M/m)·(g/d)·(1/s)^{2}·a·(1/2) )^{(1/2)}·(1/2)·t^{2}
Deducción:
d_{t}[ tx^{2} ] = d_{t}[t]·x^{2}+t·2x·d_{t}[x]
d_{t}[ tx^{2} ]·d[t] = ( d_{t}[t]·x^{2}+t·2x·d_{t}[x] )·d[t]
d[ tx^{2} ] = d[t]·x^{2}+t·2x·d[x]
L(t) = int[ d[L(t)] ] = int[ (1/s)^{2}·Mag·( d[t]·x^{2}+t·2x·d[x] ) ] = ...
... int[ (1/s)^{2}·Mag ]d[tx^{2}] = (1/s)^{2}·Mag·int[ d[tx^{2}] ] = (1/s)^{2}·Magtx^{2}
Ley: [ de rezo al Mal ]
Los hombres no están atacando,
a los xtraterrestres.
Los xtraterrestres no están atacando,
a los hombres.
Ley:
Se matan entre ellos en su mundo.
Cometen adulterio entre ellos en su mundo.
Principio:
U(x,y,z) = Potencial[ Q(x,y,z) ]
U(yz,zx,xy) = Anti-Potencial[ Q(yz,zx,xy) ]
Principio:
div-exp[ U(x,y,z) ] = sum[k = 1]-[3][ d_{xyz}^{3}[ e^{U_{k}(x,y,z)} ]
Si div-exp[ U(x,y,z) ] = 0 ==>
div-exp[ U(x,y,z) ] = d_{xyz}^{3}[ e^{sum[k = 1]-[3][ U_{k}(x,y,z) ]} ]
Principio:
Anti-div-exp[ U(yz,zx,xy) ] = sum[k = 1]-[3][ d_{kij}^{2}[ e^{U_{k}(yz,zx,xy)} ] ]
Si Anti-div-exp[ U(yz,zx,xy) ] = 0 ==>
Anti-div-exp[ U(yz,zx,xy) ] = d_{kij}^{2}[ e^{sum[k = 1]-[3][ U_{k}(yz,zx,xy) ]} ]
Ley:
Si Q(x,y,z) = U·< (1/x),(1/y),(1/z) > ==>
U(x,y,z) = U·( ln(ax)+ln(ay)+ln(az) )
div-exp[ U(x,y,z) ] = Ua^{3}
F(z) = int-int[ div-exp[ U(x,y,z) ] ]d[x]d[x]+int-int[ div-exp[ U(x,y,z) ] ]d[y]d[y] = ...
... Ua^{3}·(1/2)·( x^{2}+y^{2} )
Ley:
Si Q(yz,zx,xy) = U·< (1/(yz)),(1/(zx)),(1/(xy)) > ==>
U(yz,zx,xy) = U·( ln(byz)+ln(bzx)+ln(bxy) )
Anti-div-exp[ U(yz,zx,xy) ] = Ub^{3}·4xyz
F(z) = int-int[ Anti-div-exp[ U(yz,zx,xy) ] ]d[x]d[y]+int-int[ Anti-div-exp[ U(yz,zx,xy) ] ]d[y]d[x] = ...
... Ub^{3}·2z·(xy)^{2}
Ley:
Si Q(x,y,z) = aU·< ((y+z)/x),((z+x)/y),((x+y)/z) > ==>
U(x,y,z) = U·( (ay+az)·ln(ax)+(az+ax)·ln(ay)+(ax+ay)·ln(az) )
div-exp[ U(x,y,z) ] = Ua^{3}·( ...
... (ax)^{ay+az+(-1)}·ln(ax)·( 2+(ay+az)·ln(ax) )+...
... (ay)^{az+ax+(-1)}·ln(ay)·( 2+(az+ax)·ln(ay) )+...
... (az)^{ax+ay+(-1)}·ln(az)·( 2+(ax+ay)·ln(az) ) )
Ley:
E(x_{k}) = int-int[ div-exp[ U_{k}(x,y,z) ] ]d[(1/a)^{2}·(i+j)] = ...
... U·(ak)^{ai+aj+(-1)}·[o(ai+aj)o] ( 2+(1/2)·(ai+aj)·ln(ak) )·(ai+aj)
x_{k}(t) = ...
... (1/a)·Anti-[ ( s /o(s)o/ ...
... int[ (as)^{ai+aj+(-1)}·[o(ai+aj)o] ( 2+(1/2)·(ai+aj)·ln(as) )·(ai+aj) ]d[s] )^{[o(s)o] (1/2)} ]-( ...
... ( (2/m)·U )^{(1/2)}·at )
Deducción:
d_{ax}[ (ax)^{ay+az} ] = (ay+az)·(ax)^{ay+az+(-1)}
d_{ay}[ (ay+az)·(ax)^{ay+az+(-1)} ] = (ax)^{ay+az+(-1)}·( 1+ay·ln(ax)+az·ln(ax) )
d_{az}[ (ax)^{ay+az+(-1)}·( 1+(ay+az)·ln(ax) ) ] = (ax)^{ay+az+(-1)}·ln(ax)·( 2+(ay+az)·ln(ax) )
Ley:
Si Q(yz,zx,xy) = bU·< ((zx+xy)/(yz)),((xy+yz)/(zx)),((yz+zx)/(xy)) > ==>
U(yz,zx,xy) = U·( (bzx+bxy)·ln(byz)+(bxy+byz)·ln(bzx)+(byz+bzx)·ln(bxy) )
Anti-div-exp[ U(yz,zx,xy) ] = Ub·( ...
... (byz)^{bzx+bxy+(-1)}·(bz+by)·( 1+ln(byz) )+...
... (bzx)^{bxy+byz+(-1)}·(bx+bz)·( 1+ln(bzx) )+...
... (bxy)^{byz+bzx+(-1)}·(by+bx)·( 1+ln(bxy) ) )
Ley:
E(x_{k}) = int[ Anti-div-exp[ U_{k}(yz,zx,xy) ] ]d[(1/b)·k] = U·(bij)^{bik+bkj+(-1)}·( ( 1/ln(bij) )+1 )
x_{k}(t) = (a/b)·Anti-[ ( s /o(s)o/ int[ (bij)^{ais+asj+(-1)}·( ( 1/ln(bij) )+1 ) ]d[s] )^{[o(s)o] (1/2)} ]-( ...
... ( (2/m)·U )^{(1/2)}·(b/a)·t )
Deducción:
d_{byz}[ (byz)^{bzx+bxy} ] = (bzx+bxy)·(byz)^{bzx+bxy+(-1)}
d_{x}[ (bzx+bxy)·(byz)^{bzx+bxy+(-1)} ] = (byz)^{bzx+bxy+(-1)}·(bz+by)·( 1+ln(byz) )
Áreas y Volumenes:
Teorema:
x = r·cos(w)
y = r·sin(w)
d[x]d[y] = (1/2)·( d[x]d[y]+d[x]d[y] ) = ...
... (1/2)·( d_{r}[x]·d_{w}[y]+(-1)·d_{w}[x]·d_{r}d[y] )·d[r]d[w]
Teorema:
Área de un círculo:
A(r) = int[w = 0]-[2pi][ inr[r = 0]-[r][ r ]d[r] ]d[w] = ...
... int[w = 0]-[2pi][ (1/2)·r^{2} ]d[w] = (1/2)·r^{2}·int[w = 0]-[2pi][ d[w] ] = pi·r^{2}
Perímetro de un círculo:
B(r) = d_{r}[ A(r) ] = d_{r}[ pi·r^{2} ] = 2pi·r
Teorema:
Área de un sector circular:
A(r,w) = int[w = 0]-[w][ inr[r = 0]-[r][ r ]d[r] ]d[w] = ...
... int[w = 0]-[w][ (1/2)·r^{2} ]d[w] = (1/2)·r^{2}·int[w = 0]-[w][ d[w] ] = (1/2)·wr^{2}
Perímetro de un sector circular:
B(r,w) = d_{r}[ A(r,w) ] = d_{r}[ (1/2)·wr^{2} ] = wr
Teorema:
d[z] = r·sin(s)·d[s]
x = r·cos(2w)
y = r·sin(2w)
d[x]d[y]d[z] = (1/2)·( d[x]d[y]d[z]+d[x]d[y]d[z] ) = ...
... (1/2)·( d_{r}[x]·d_{w}[y]+(-1)·d_{w}[x]·d_{r}d[y] )·r·sin(s)·d[r]d[w]d[s] =
Teorema:
Volumen de una esfera:
A(r) = int[s = 0]-[pi][ int[w = 0]-[2pi][ int[r = 0]-[r][ r^{2}·sin(s) ]d[r] ]d[w] ]d[s] = ...
... int[s = 0]-[pi][ int[w = 0]-[2pi][ (1/3)·r^{3}·sin(s) ]d[w] ]d[s] = ...
... (1/3)·r^{3}·int[s = 0]-[pi][ sin(s)·int[w = 0]-[2pi][ d[w] ] ]d[s] = ...
... (1/3)·r^{3}·int[s = 0]-[pi][ sin(s)·2pi ]d[s] = (2/3)·pi·r^{3}·int[s = 0]-[pi][ sin(s) ]d[s] = ...
... (2/3)·pi·r^{3}·(-1)·( cos(pi)+(-1)·cos(0) ) = (4/3)·pi·r^{3}
Superficie de una esfera:
B(r) = d_{r}[ A(r) ] = d_{r}[ (4/3)·pi·r^{3} ] = 4pi·r^{2}
Teorema:
Volumen de un hemisferio:
A(r,w) = int[s = 0]-[pi][ int[w = 0]-[w][ int[r = 0]-[r][ r^{2}·sin(s) ]d[r] ]d[w] ]d[s] = ...
... int[s = 0]-[pi][ int[w = 0]-[w][ (1/3)·r^{3}·sin(s) ]d[w] ]d[s] = ...
... (1/3)·r^{3}·int[s = 0]-[pi][ sin(s)·int[w = 0]-[w][ d[w] ] ]d[s] = ...
... (1/3)·r^{3}·int[s = 0]-[pi][ sin(s)·w ]d[s] = (2/3)·wr^{3}·int[s = 0]-[pi][ sin(s) ]d[s] = ...
... (2/3)·wr^{3}·(-1)·( cos(pi)+(-1)·cos(0) ) = (2/3)·wr^{3}
Superficie de un hemisferio:
B(r,w) = d_{r}[ A(r,w) ] = d_{r}[ (2/3)·wr^{3} ] = 2wr^{2}
--------------------------------
Mecánica Teórica y Ondas:
--------------------------------
Teorema:
Sea d_{x}[F(x)] = f(x) ==>
d_{x}[ Anti-[F(s)]-(x) ] = ( 1/f( Anti-[F(s)]-(x) ) )
Demostración:
d_{y}[ Anti-[F(s)]-( F(y) ) ] = d_{y}[y] = 1
d[ Anti-[F(s)]-( F(y) ) ] = d[y]
d_{F(y)}[ Anti-[F(s)]-( F(y) ) ] = d_{F(y)}[y] = ( 1/d_{y}[F(y)]) = ( 1/f(y) )
Sea y = Anti-[F(s)]-(x) ==>
d_{F( Anti-[F(s)]-(x) )}[ Anti-[F(s)]-( F( Anti-[F(s)]-(x) ) ) ] = ( 1/f( Anti-[F(s)]-(x) ) )
Teorema:
Sea d_{x}[F(x)] = f(x) ==>
d_{x}[ Anti-[( s /o(s)o/ F(s) )]-(x) ] = f( Anti-[( s /o(s)o/ F(s) )]-(x) )
Demostración:
d_{x}[ Anti-[( s /o(s)o/ F(s) )]-(x) ] = ( 1/( 1/f( Anti-[( s /o(s)o/ F(s) )]-(x) ) ) ) = ...
... f( Anti-[( s /o(s)o/ F(s) )]-(x) )
Teorema:
d_{x}[ arc-sin(x) ] = ( 1/(1+(-1)·x^{2})^{(1/2)} )
Demostración:
d_{x}[ arc-sin(x) ] = ( 1/cos( arc-sin(x) ) ) = ( 1/(1+(-1)·(sin( arc-sin(x) ))^{2})^{(1/2)} )
Teorema:
d_{x}[ arc-cos(x) ] = (-1)·( 1/(1+(-1)·x^{2})^{(1/2)} )
Demostración:
d_{x}[ arc-cos(x) ] = (-1)·( 1/sin( arc-cos(x) ) ) = ( 1/(1+(-1)·(cos( arc-cos(x) ))^{2})^{(1/2)} )
Racionamiento antiguo:
Teorema:
d_{x}[ e^{x} ] = e^{x}
Demostración:
d_{x}[ e^{x} ] = ( 1/(1/y) ) = y = e^{x}
Teorema:
d_{x}[ ln(x) ] = (1/x)
Demostración:
d_{x}[ ln(x) ] = ( 1/e^{y} ) = ( 1/e^{ln(x)} ) = (1/x)
Teorema:
Sea x(t) = Anti-[ ( s /o(s)o/ int[ F(s) ]d[s] )^{[o(s)o] (1/2)} ]-( 2^{(1/2)}·ut ) ] ==>
(1/2)·d_{t}[x(t)]^{2} = F(s)·u^{2}
d_{tt}^{2}[x(t)] = (1/2)·( F(s) )^{(-1)·(1/2)}·f(s)·( F(s) )^{(1/2)}·2u^{2} = f(s)·u^{2}
Demostración:
d_{t}[ F(s) ] = d_{s}[F(s)]·d_{t}[s]
Teorema:
Sea x(t) = Anti-[ ( s /o(s)o/ int[ H(ut) [o(ut)o] F(s) ]d[s] )^{[o(s)o] (1/2)} ]-( 2^{(1/2)}·ut ) ] ==>
(1/2)·d_{t}[x(t)]^{2} = ( H(ut) [o(ut)o] F(s) )·u^{2}
d_{tt}^{2}[x(t)] = ...
... (1/2)·( H(ut) [o(ut)o] F(s) )^{(-1)·(1/2)}·h(ut)·f(s)·( H(ut) [o(ut)o] F(s) )^{(1/2)}·2u^{2} = ...
... h(ut)·f(s)·u^{2}
Demostración:
u·d_{ut}[ H(ut) [o(ut)o] F(s) ] = u·d_{ut}[H(ut)]·d_{ut}[F(s)] = ...
... u·h(ut)·d_{s}[F(s)]·d_{ut}[s] = h(ut)·d_{s}[F(s)]·d_{t}[s]
Teorema:
Sea x(t) = Anti-[ ( s /o(s)o/ int[ (ut) [o(ut)o] F(s) ]d[s] )^{[o(s)o] (1/2)} ]-( 2^{(1/2)}·ut ) ] ==>
(1/2)·d_{t}[x(t)]^{2} = ( (ut) [o(ut)o] F(s) )·u^{2}
d_{tt}^{2}[x(t)] = ...
... (1/2)·( (ut) [o(ut)o] F(s) )^{(-1)·(1/2)}·f(s)·( (ut) [o(ut)o] F(s) )^{(1/2)}·2u^{2} = f(s)·u^{2}
Ley:
d[H(t)] = (1/pi)^{2}·MgI·d[tx]
x(t) = (1/a)·Anti-[ ( s /o(s)o/ int[ (1/2)·(ut)^{2} [o(ut)o] (1/2)·s^{2} ]d[s] )^{[o(s)o] (1/2)} ]-( ...
... ( 2·(M/m)·gI·(1/u)·(1/d) )^{(1/2)}·(1/pi)·t )
Deducción:
H(t) = int[ d[H(t)] ] = int[ (1/pi)^{2}·MgI ]d[tx] = (1/pi)^{2}·MgI·int[ d[tx] ] = (1/pi)^{2}·MgItx
md·d_{tt}^{2}[x] = (1/pi)^{2}·MgItx
md·d_{tt}^{2}[x]·d[x] = (1/pi)^{2}·MgI·(1/u)·(ut)·x·d_{ut}[x]·d[ut]
md·(1/2)·d_{t}[ax]^{2} = (1/pi)^{2}·MgI·(1/u)·(1/2)·(ut)^{2} [o(ut)o] (1/2)·(ax)^{2}
Ley:
d[H(t)] = (1/pi)^{2}·Mv·d[(1/t)·x]
x(t) = (1/a)·Anti-[ ( s /o(s)o/ int[ ln(ut) [o(ut)o] (1/2)·s^{2} ]d[s] )^{[o(s)o] (1/2)} ]-( ...
... ( 2·(M/m)·(vu)·(1/d) )^{(1/2)}·(1/pi)·t )
Deducción:
H(t) = int[ d[H(t)] ] = int[ (1/pi)^{2}·Mv ]d[(1/t)·x] = (1/pi)^{2}·Mv·int[ d[(1/t)·x] ] = ...
... (1/pi)^{2}·Mv·(1/t)·x
md·d_{tt}^{2}[x] = (1/pi)^{2}·Mv·(1/t)·x
md·d_{tt}^{2}[x]·d[x] = (1/pi)^{2}·Mvu·(1/(ut))·x·d_{ut}[x]·d[ut]
md·(1/2)·d_{t}[ax]^{2} = (1/pi)^{2}·Mvu·ln(ut) [o(ut)o] (1/2)·(ax)^{2}
Ley:
Le tenéis que decir a Esquerra Republicana,
que queréis el título de la universidad de Stroniken como el mío,
enseñando el testimonio de uno mismo con Dios,
escrito con vuestra letra:
Jûan Garriga Peralta-Peraltotzak:
Filósofo de la ciencia matemática y de la ciencia lógica,
por la universidad de Stroniken.
Y que vos lo envíen.
Ley: [ de onda electro-magnética plana de superficie ]
Lap[ E_{e}(x,y,t) ] = (-2)·(1/c)^{2}·d_{tt}^{2}[ int[ B_{e}(x,y,t) ]d[t] ]
Deducción:
E_{e}(x,y,t)+int[ B_{e}(x,y,t) ]d[t] = 0 = m·d_{tt}^{2}[ < x,y > ]
x(t) = ct·cos(w)
y(t) = ct·sin(w)
Lap[ int[ B_{e}(x,y,t) ]d[t] ] = ...
... ( 1/(d[x]^{2}+d[y]^{2}) )·(d[x]^{2}+d[y]^{2}) [o] Lap[ int[ B_{e}(x,y,t) ]d[t] ]
Lap[ E_{e}(x,y,t) ]+Lap[ int[ B_{e}(x,y,t) ]d[t] ]= 0^{3}
Lap[ E_{e}(x,y,t) ]+2·(1/c)^{2}·d_{tt}^{2}[ int[ B_{e}(x,y,t) ]d[t] ] = 0^{3}
Lap[ E_{e}(x,y,t) ] = (-2)·(1/c)^{2}·d_{tt}^{2}[ int[ B_{e}(x,y,t) ]d[t] ]
Ley: [ de onda gravito-magnética plana de superficie ]
Lap[ int[ B_{g}(x,y,t) ]d[t] ] = (-2)·(1/c)^{2}·d_{tt}^{2}[ E_{g}(x,y,t) ]
Deducción:
int[ B_{g}(x,y,t) ]d[t]+E_{g}(x,y,t) = 0 = m·d_{tt}^{2}[ < x,y > ]
x(t) = ct·cos(w)
y(t) = ct·sin(w)
Lap[ E_{g}(x,y,t) ] = ( 1/(d[x]^{2}+d[y]^{2}) )·(d[x]^{2}+d[y]^{2}) [o] Lap[ E_{g}(x,y,t) ]
Lap[ int[ B_{g}(x,y,t) ]d[t] ]+Lap[ E_{g}(x,y,t) ] = 0^{3}
Lap[ int[ B_{g}(x,y,t) ]d[t] ]+2·(1/c)^{2}·d_{tt}^{2}[ E_{g}(x,y,t) ] = 0^{3}
Lap[ int[ B_{g}(x,y,t) ]d[t] ] = (-2)·(1/c)^{2}·d_{tt}^{2}[ E_{g}(x,y,t) ]
Ecuaciones de ondas elípticas planas de superficie:
Teorema:
d_{xx}^{2}[u(x,y,t)]+d_{yy}^{2}[u(x,y,t)] = (-2)·(1/c)^{2}·d_{tt}^{2}[u(x,y,t)]
u(x,y,t) = (1/2)·( ...
... e^{ax+ay+acit || ln( H(ax,ay) )+act}+...
... e^{ax+ay+acit || ln( H(ax,ay) )+(-1)·act} )
u(x,y,0) = H(ax,ay)
d_{t}[ u(x,y,0) ] = 0
Teorema:
d_{xx}^{2}[u(x,y,t)]+d_{yy}^{2}[u(x,y,t)] = (-2)·(1/c)^{2}·d_{tt}^{2}[u(x,y,t)]
u(x,y,t) = sum[k = 1]-[oo][ ...
... int[h(ax,ay)+(-1)·(1/2)·act·0 || (2t)^{(1/2)}]-[h(ax,ay)+(1/2)·act·0 || (2t)^{(1/2)}][ w ]d[w] ]·...
... e^{ax+ay+acit || 0}
u(x,y,0) = 0
d_{t}[ u(x,y,0) ] = ac·h(ax,ay)
Ecuaciones de ondas hiperbólicas planas de superficie:
Teorema:
d_{xx}^{2}[u(x,y,t)]+d_{yy}^{2}[u(x,y,t)] = 2·(1/c)^{2}·d_{tt}^{2}[u(x,y,t)]
u(x,y,t) = (1/2)·( ...
... e^{ax+ay+act || ln( H(ax,ay) )+act}+...
... e^{ax+ay+act || ln( H(ax,ay) )+(-1)·act} )
u(x,y,0) = H(ax,ay)
d_{t}[ u(x,y,0) ] = 0
Teorema:
d_{xx}^{2}[u(x,y,t)]+d_{yy}^{2}[u(x,y,t)] = 2·(1/c)^{2}·d_{tt}^{2}[u(x,y,t)]
u(x,y,t) = sum[k = 1]-[oo][ ...
... int[h(ax,ay)+(-1)·(1/2)·act·0 || (2t)^{(1/2)}]-[h(ax,ay)+(1/2)·act·0 || (2t)^{(1/2)}][ w ]d[w] ]·...
... e^{ax+ay+act || 0}
u(x,y,0) = 0
d_{t}[ u(x,y,0) ] = ac·h(ax,ay)
Ley:
d_{xx}^{2}[u(x,y,t)]+d_{yy}^{2}[u(x,y,t)] = (-2)·(1/c)^{2}·d_{tt}^{2}[u(x,y,t)]
u(x,y,t) = E_{e}·(1/2)·( ...
... e^{ax+ay+acit || ln( (ax)^{2}+(ay)^{2} )+act}+...
... e^{ax+ay+acit || ln( (ax)^{2}+(ay)^{2} )+(-1)·act} )
u(x,y,0) = (ax)^{2}+(ay)^{2}
d_{t}[ u(x,y,0) ] = 0
Ley:
d_{xx}^{2}[u(x,y,t)]+d_{yy}^{2}[u(x,y,t)] = (-2)·(1/c)^{2}·d_{tt}^{2}[u(x,y,t)]
u(x,y,t) = E_{e}·sum[k = 1]-[oo][ ...
... int[(ax+ay)+(-1)·act·0 || (2t)^{(1/2)}]-[(ax+ay)+act·0 || (2t)^{(1/2)}][ w ]d[w] ]·...
... e^{ax+ay+acit || 0}
u(x,y,0) = 0
d_{t}[ u(x,y,0) ] = 2ac·(ax+ay)
