Afinitats:
El Ker de l'afinitat és un espai afí.
< (-c),c >+( < a,(-a) >,< (-a),a > )o< x,y > = < 0,0 >
x = y+(c/a)
< (-c),c >+( < a,(-b) >,< (-a),b > )o< x,y > = < 0,0 >
x = (b/a)·y+(c/a)
< c,c >+( < a,0 >,< 0,a > )o< x,y > = < 0,0 >
x = (1/a)·(-c)
y = (1/a)·(-c)
< c,d >+( < a,0 >,< 0,b > )o< x,y > = < 0,0 >
x = (1/a)·(-c)
y = (1/b)·(-d)
Espai afí:
ax+by = c
x = ( (-b)/a )·y+(c/a)
Base afí:
p+E = < (c/a),0 >+k·< (-b),a >
Base afí:
p+E = < c,0 >+k·< (-b),1 >
Espai afí:
x+by = c
Base afí:
p+E = < c,0 >+k·< (-b),a >
Espai afí:
ax+by = ac
p+E = < c,d >+k·< 1,1 >
q+F = < a,b >+i·< 1,0 >+j·< 0,1 >
(p+E) [&] (q+F) = < c,d >+k·< 1,1 > = < a,b >+(k+(-a)+c)·< 1,0 >+(k+(-b)+d)·< 0,1 >
i = k+(-a)+c
j = k+(-b)+d
Sigui det(A) != 0 ==>
< (-u),(-v) >+A o < x,y > = 0
Sigui C = < u,v > ==>
A o < x,y > = C
ax+by = u
cx+dy = v
(1/det(A))·( a( (-1)·bv+du )+b( av+(-1)·cu ) ) = u
(1/det(A))·( c( (-1)·bv+du )+d( av+(-1)·cu ) ) = v
x = (-1)·(det(Y|C)/det(A))
y = det(X|C)/det(A)
métricas bi-lineales:
< x,y >o( < a,0 >,< 0,a > )o< x,y > = 0
x = s·i
y = s
< x,y >o( < a,0 >,< 0,a > )o< x,y > = 1
x = (1/a)^{(1/2)}·cos(s) = (1/a)^{(1/2)}·( 1+(-1)·t^{2} )^{(1/2)}
y = (1/a)^{(1/2)}·sin(s) = (1/a)^{(1/2)}·t
x [&] y = { s : s = (pi/4)+n·pi }
< x,y >o( < a,0 >,< 0,(-a) > )o< x,y > = 0
x = s
y = s
< x,y >o( < a,0 >,< 0,(-a) > )o< x,y > = 1
x = (1/a)^{(1/2)}·cos(s)
y = i·(1/a)^{(1/2)}·sin(s)
x [&] y = { s : s = arctan(-i) }
< x,y >o( < a,a >,< a,a > )o< x,y > = 0
x = (1/a)^{(1/2)}·s
y = (1/a)^{(1/2)}·(-s)
< x,y >o( < a,a >,< a,a > )o< x,y > = 1
x = (1/a)^{(1/2)}·s
y = (1/a)^{(1/2)}·((-s)+1)
x [&] y = { s : s = (1/2) }
métricas bi-lineales diferenciales:
Primera forma-métrica fonamental:
< d_{u}[f(u,v)],d_{v}[f(u,v)] >o...
... ( < d[u]d[u],d[v]d[u] >,< d[u]d[v],d[v]d[v] > )o...
... < d_{u}[f(u,v)],d_{v}[f(u,v)] > = d[S(u,v)]d[S(u,v)]
Segona forma-métrica fonamental:
< d_{u}[f(u,v)],d_{v}[f(u,v)] >o...
... ( ...
... < (1/2)·u^{2}·d_{uu}^{2}[f(u,v)]·d[u]d[u],vu·d_{vu}^{2}[f(u,v)]·d[v]d[u] >, ...
... < uv·d_{uv}^{2}[f(u,v)]·d[u]d[v],(1/2)·v^{2}·d_{vv}^{2}[f(u,v)]·d[v]d[v] > ...
... )o...
... < d_{u}[f(u,v)],d_{v}[f(u,v)] > = d[S(u,v)]d[S(u,v)]
f(u,v) = e^{iu·t}+e^{iv·t}
d_{u}[f(u,v)] = it·e^{iu·t}
d_{v}[f(u,v)] = it·e^{iv·t}
d_{uu}^{2}[f(u,v)] = (-1)·t^{2}e^{iu·t}
d_{vv}^{2}[f(u,v)] = (-1)·t^{2}e^{iv·t}
(1/2)·( S(u) )^{2} = int-int[ t^{4}·(1/2)·u^{2}·e^{3iu·t} ]d[u]d[u] = ...
... t^{4}·(1/2)·u^{2} [o( (1/2)·u^{2} )o] ...
... (1/4!)·u^{4} [o( (1/2)·u^{2} )o] (-1)·( 1/(3t) )^{2}·e^{3iu·t}
(1/2)·( S(v) )^{2} = int-int[ t^{4}·(1/2)·v^{2}·e^{3iv·t} ]d[v]d[v] = ...
... t^{4}·(1/2)·v^{2} [o( (1/2)·v^{2} )o] ...
... (1/4!)·v^{4} [o( (1/2)·v^{2} )o] (-1)·( 1/(3t) )^{2}·e^{3iv·t}
( S(u,v) )^{2} = ( S(u) )^{2}+( S(v) )^{2}
d[S(u,v)]d[S(u,v)] = d[S(u)]d[S(u)]+d[S(v)]d[S(v)]
Unitats a la menys 1 amb producte integral
Sigui d_{t}[x(t)] = p·t·se^{st} ==>
x(t) = (p/m)·t [o(t)o] (1/2)·t^{2} [o(t)o] e^{st}
d_{tt}^{2}[x(t)] = (p/m)·se^{st}+(p/m)·t·s^{2}·e^{st}
Sigui d_{t}[x(t)] = (F/m)·t^{2}·se^{st} ==>
x(t) = (F/m)·t [o(t)o] (1/3!)·t^{3} [o(t)o] e^{st}
d_{tt}^{2}[x(t)] = (F/m)·2t·se^{st}+(F/m)·t^{2}·s^{2}·e^{st}
Sigui m·d_{t}[x(t)] = p·t^{n}·s_{n}e^{s_{n}·t^{n}} ==>
x(t) = (p/m)·t [o(t)o] ( 1/(n+1) )·t^{n+1} [o(t)o] ...
... ( 1/((-n)+2) )·t^{(-n)+2} [o(t)o] e^{s_{n}·t^{n}}
d_{tt}[x(t)] = (p/m)·nt^{n+(-1)}·s_{n}e^{s_{n}·t^{n}}+...
... (p/m)·t^{n}·(s_{n})^{2}·nt^{n+(-1)}·e^{s_{n}·t^{n}}
Sigui m·d_{t}[x(t)] = F·t^{n+1}·s_{n}e^{s_{n}·t^{n}} ==>
x(t) = (F/m)·t [o(t)o] ( 1/(n+2) )·t^{n+2} [o(t)o] ...
... ( 1/((-n)+2) )·t^{(-n)+2} [o(t)o] e^{s_{n}·t^{n}}
d_{tt}[x(t)] = (F/m)·(n+1)·t^{n}·s_{n}e^{s_{n}·t^{n}}+...
... (F/m)·t^{n+1}·(s_{n})^{2}·nt^{n+(-1)}·e^{s_{n}·t^{n}}
Álgebra-jjeko lineal-dokitx vectorial-dokitx.
Geometri-jjeko lineal-dokitx afini-dokitx.
q-esteike zapati-jjeko,
sere-po-mitxli mio.
q-eseike zapati-jjeko,
no sere-po-mitxli mio.
esteike zapati-jjoika,
sere-proika mio.
eseike zapati-jjoika,
no sere-proika mio.
