succesions y canvi de variable de integral múltiple

Constructor:

Si ( [An][ a_{n+1} [< a_{n}+a_{1} ] & a_{1} [< 0 ) ==> ...

... ( b_{n} = (a_{n}·n) es decreishent || b_{n} = (a_{n}·n) es creishent estricte ).

a_{n+1} [< (n+1)·a_{1} [< 0

a_{n+1}(n+1) [< a_{n+1}·n [< (a_{n}+a_{1})·n = a_{n}·n+a_{1}n [< a_{n}·n

a_{n+1}(n+1) [< a_{n}·n

Si ( [An][ a_{n+1} >] a_{n}+a_{1} ] & a_{1} >] 0 ) ==> ...

... ( b_{n} = (a_{n}·n) es creishent || b_{n} = (a_{n}·n) es decreishent estricte ).

a_{n+1} >] (n+1)·a_{1} >] 0

a_{n+1}(n+1) >] a_{n+1}·n >] (a_{n}+a_{1})·n = a_{n}·n+a_{1}n >] a_{n}·n

a_{n+1}·(n+1) >] a_{n}·n

Destructor:

Si ( [An][ a_{n+1} [< a_{n}+a_{1} ] & a_{1} [< 0 ) ==> ...

... ( b_{n} = (a_{n}/n) es decreishent || b_{n} = (a_{n}/n) es creishent estricte ).

a_{n+1} [< (n+1)·a_{1} [< 0

( a_{n+1}/(n+1) ) >] ( a_{n+1}/n ) > ...

... ( (a_{n}+a_{1})/n ) = (a_{n}/n)+(a_{1}/n) > (a_{n}/n)

( a_{n+1}/(n+1) ) > (a_{n}/n) || ( a_{n+1}/(n+1) ) [< (a_{n}/n)

Si ( [An][ a_{n+1} >] a_{n}+a_{1} ] & a_{1} >] 0 ) ==> ...

... ( b_{n} = (a_{n}/n) es creishent || b_{n} = (a_{n}/n) es decreishent estricte ).

a_{n+1} >] (n+1)·a_{1} >] 0

( a_{n+1}/(n+1) ) [< ( a_{n+1}/n ) < ...

... ( (a_{n}+a_{1})/n ) = (a_{n}/n)+(a_{1}/n) < (a_{n}/n)

( a_{n+1}/(n+1) ) < (a_{n}/n) || ( a_{n+1}/(n+1) ) >] (a_{n}/n)

Constructor:

Si ( [An][ a_{n+1} >] a_{n}+a_{1} ] & a_{1} >] 0 ) ==> a_{n} es creishent.

Si ( [An][ a_{n+1} [< a_{n}+a_{1} ] & a_{1} [< 0 ) ==> a_{n} es decreishent.

a_{n+1} >] a_{n}+a_{1} >] a_{n}

a_{n+1} [< a_{n}+a_{1} [< a_{n}

Destructor:

Si ( [An][ a_{n+1} >] a_{n}+a_{1} ] & a_{1} < 0 ) ==> a_{n} es decreishent.

a_{n+1} >] a_{n}+a_{1} >] a_{n}

lim[a_{n}] = (-1)

a_{n+1} >] (n+1)·a_{1}

lim[a_{n}] = sup{ M : a_{n} >] M }

Si ( [An][ a_{n+1} [< a_{n}+a_{1} ] & a_{1} > 0 ) ==> a_{n} es creishent.

a_{n+1} [< a_{n}+a_{1} [< a_{n}

lim[a_{n}] = 1

a_{n+1} [< (n+1)·a_{1}

lim[a_{n}] = inf{ M : a_{n} [< M }

d[x]d[y] = (1/2)·( d_{r}[x]d_{s}[y]+d_{s}[x]d_{r}[y] )d[r]d[s]

x(r,s) = r·cos(s)

y(r,s) = r·sin(s)

d[x(r+h,s)] = cos(s)·d[r]

d[y(r,s+h)] = r·cos(s)·d[s]

d[y(r+h,s)] = sin(s)·d[r]

d[x(r,s+h)] = (-r)·sin(s)·d[s]

d[x]d[y] = (1/2)·( r·( cos(s) )^{2}+(-1)·r·( sin(s) )^{2} )d[r]d[s]

int-int[f(x^{2}+y^{2})]d[x]d[y] = ...

... (1/4)·int-int[f(r^{2})·2r·cos(2s)]d[r]d[s] = ...

... (1/4)·int[ int[f(r^{2})]d[r^{2}]·cos(2s) ]d[s] = ...

... (1/8)·sin(2s)·int[f(r^{2})]d[r^{2}] = ...

... (1/8)·sin(2·arc-tan(y/x))·int[f(x^{2}+y^{2})]d[x^{2}+y^{2}]

x(r,s) = r·( cos(s) )^{2}

y(r,s) = r·( sin(s) )^{2}

d[x(r+h,s)] = ( cos(s) )^{2}·d[r]

d[y(r,s+h)] = r·2·sin(s)·cos(s)·d[s]

d[y(r+h,s)] = ( sin(s) )^{2}·d[r]

d[x(r,s+h)] = (-r)·2·cos(s)·sin(s)·d[s]

d[x]d[y] = sin(s)·cos(s)·( r·( cos(s) )^{2}+(-1)·r·( sin(s) )^{2} )d[r]d[s]

int-int[f(x+y)]d[x]d[y] = ...

... int-int[f(r)·r·( sin(s)·cos(s) )·cos(2s)]d[r]d[s] = ...

... (1/2)·int-int[f(r)·r·sin(2s)·cos(2s)]d[r]d[s] = ...

... (1/2)·int[( int[f(r)]d[r]·r+(-1)·int-int[f(r)]d[r]d[r] )·sin(2s)·cos(2s)]d[s] = ...

... (1/8)·( int[f(r)]d[r]·r+(-1)·int-int[f(r)]d[r]d[r] )·( sin(2s) )^{2}

x(r,s) = r^{n+1}·( cos(s) )^{2n+2}

y(r,s) = r^{n+1}·( sin(s) )^{2n+2}

d[x(r+h,s)] = (n+1)·r^{n}·( cos(s) )^{2n+2}·d[r]

d[y(r,s+h)] = r^{n+1}·(2n+2)·( sin(s) )^{2n+1}·cos(s)·d[s]

d[y(r+h,s)] = (n+1)·r^{n}·( sin(s) )^{2n+2}·d[r]

d[x(r,s+h)] = (-1)·r^{n+1}·(2n+2)·( cos(s) )^{2n+1}·sin(s)·d[s]

d[x]d[y] = ...

... (n+1)^{2}·( sin(s)·cos(s) )^{2n+1}·...

... ( r^{2n+1}·( cos(s) )^{2}+(-1)·r^{2n+1}·( sin(s) )^{2} )d[r]d[s]

int-int[ f( x^{(1/(n+1))}+y^{(1/(n+1))} ) ]d[x]d[y] = ...

... int-int[f(r)·r^{2n+1}·(n+1)^{2}·( sin(s)cos(s) )^{2n+1}·cos(2s)]d[r]d[s] = ...

... ( int[f(r)]d[r]·[o(r)o](1/2)·r^{(2n+2)} )·(1/2^{2n+3})·( sin(2s) )^{(2n+2)} = ...

... sum[ k = 0 ---> (2n+1) ][ ...

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

... int-[k+1]-int[f(r)]d[r]...(k+1) d[r] ...

... ]·...

... (1/2^{2n+3})·( sin(2s) )^{(2n+2)}

d[x]d[y]d[z] = ...

... (1/3)·( d_{r}[x]d_{u}[y]d_{v}[z]+...

... d_{u}[x]d_{v}[y]d_{r}[z]+d_{v}[x]d_{r}[y]d_{u}[z] )d[r]d[u]d[v]

x(r,u,v) = r·cos(u)·cos(v)

y(r,u,v) = r·sin(u)·cos(v)

z(r,u,v) = r·sin(v)

d[x(r+h,u,v)] = cos(u)·cos(v)·d[r]

d[y(r,u+h,v)] = r·cos(u)·cos(v)·d[u]

d[z(r,u,v+h)] = r·sin(v)·d[v]

d[x(r,u,v+h)] = (-r)·cos(u)·sin(v)·d[v]

d[y(r+h,u,v)] = sin(u)·cos(v)·d[r]

d[z(r,u+h,v)] = 0·d[u]

d[x(r,u+h,v)] = (-r)·sin(u)·cos(v)·d[u]

d[y(r,u,v+h)] = (-r)·sin(u)·sin(v)·d[v]

d[z(r+h,u,v)] = cos(v)·d[r]

d[x]d[y]d[z] = ...

... (1/3)·( cos(v) )^{2}·sin(v)·( r^{2}·( cos(u) )^{2}+r^{2}·( sin(u) )^{2} )d[r]d[u]d[v]

int-int-int[f(x^{2}+y^{2}+z^{2})]d[x]d[y]d[z] = ...

... (1/3)·int-int-int[f(r^{2})·r^{2}·( cos(v) )^{2}·sin(v)]d[r]d[u]d[v] = ...

... (1/6)·int-int-int[f(r^{2})·2r·r·( cos(v) )^{2}·sin(v)]d[r]d[u]d[v] = ...

... (1/6)·( int[f(r^{2})]d[r]·r+(-1)·int-int[f(r^{2})]d[r]d[r] )·(-1)·(1/3)·( cos(v) )^{3}·u

... (1/6)·( ...

... (1/2)·int[f(r^{2})]d[r^{2}]+...

... (-1)·(1/4)·int-int[f(r^{2})·(1/r^{2})]d[r^{2}]d[r^{2}] ...

... )·...

... (-1)·(1/3)·( cos(v) )^{3}·u

x(u,v) = u+v

y(u,v) = u+(-v)

d_{u}[x(u+h,v)] = 1+v

d_{v}[y(u,v+h)] = u+(-1)

d_{u}[y(u+h,v)] = 1+(-v)

d_{v}[x(u,v+h)] = u+1

d[x]d[y] = ( u+(-v) )·d[u]d[v]

int-int[f(xy)]d[x]d[y] = ...

... int-int[ f(u^{2}+(-1)·v^{2})·(u+(-v)) ]d[u]d[v] = ...

... (1/2)·int-int[ f(u^{2}+(-1)·v^{2})·(2u+(-2)·v) ]d[u]d[v] = ...

... (1/2)·( int[ int[ f(u^{2}+(-1)·v^{2}) ]d[u^{2}+(-1)·v^{2}] ]d[u^{2}+(-1)·v^{2}] [o(v)o] ...

... ( v/o(v)o/((-1)·v^{2}) )+...

... int[ int[ f(u^{2}+(-1)·v^{2}) ]d[u^{2}+(-1)·v^{2}] ]d[u^{2}+(-1)·v^{2}] [o(u)o] ...

... ( u/o(u)o/(u^{2}) ) )

d_{x}[( f(x) )^{g(y)}] = g(y)·( f(x) )^{g(y)+(-1)}·d_{x}[f(x)]

d_{yx}^{2}[( f(x) )^{g(y)}] = ...

... g(y)·( f(x) )^{g(y)+(-1)}d_{x}[f(x)]d_{y}[ln(g(y))+ln(d_{x}[f(x)])+(g(y)+(-1))·ln(f(x))]

d_{y}[( f(x) )^{g(y)}] = ( f(x) )^{g(y)}d_{y}[g(y)·ln(f(x))] = ...

... ( f(x) )^{g(y)}( d_{y}[g(y)]·ln(f(x)) )

d_{xy}^{2}[( f(x) )^{g(y)}] = ...

... g(y)·( f(x) )^{g(y)+(-1)}d_{x}[f(x)]( d_{y}[g(y)]/(g(y))+d_{y}[g(y)]ln(f(x)) )

d_{xy}^{2}[f(x,y)] = d_{yx}^{2}[f(x,y)]

d_{xx}^{2}[f(x,y)]·d_{x}[x]d_{y}[x] = d_{xx}^{2}[f(x,y)]·d_{y}[x]d_{x}[x]

d_{yy}^{2}[f(x,y)]·d_{x}[y]d_{y}[y] = d_{yy}^{2}[f(x,y)]·d_{y}[y]d_{x}[y]

int-[n]-int[ d^{n}[f(x_{1},...,x_{n})] ] = f(x_{1},...,x_{n})

int-[n]-int[ d_{x_{1}...x_{n}}^{n}[f(x_{1},...,x_{n})] ] d[x_{1}]...d[x_{n}] = ...

... f(x_{1},...,x_{n})

economía y camps vectorials escalars

Impuestos:

Sueldo:

100 euros.

1000 euros.

d_{x}[y]+ny = (1/p)·x

y(x) = e^{(-n)·x}·int[(1/p)·x·e^{nx}]d[x] = ...

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

p = ( (n+1)/n^{2} ) & x = 1

Si n = 10 ==> p = (0.11)

d_{x}[y]+ny = (1/p)·x^{2}

y(x) = e^{(-n)·x}·int[(1/p)·x^{2}·e^{nx}]d[x] = ...

... (1/p)·( (1/n)·x^{2}+(-1)·(1/n)^{2}·2x+2·(1/n)^{3} ) = 1

p = ( (n^{2}+(-2)·n+2)/n^{3} ) & x = 1

Si n = 10 ==> p = (0.082)

sueldo:

1800€

k = 1000·( (0.082)+( (0.88)/10 ) ) = 170€

< a·d_{x}[ ],b·d_{y}[ ] > [o] < s·f(x),s·g(y) > = ...

... s·( < a·d_{x}[ ],b·d_{y}[ ] > [o] < f(x),g(y) > )

a·d_{x}[s·f(x)]+b·d_{y}[s·g(y)] = s·( a·d_{x}[f(x)]+b·d_{y}[g(y)] )

< a·d_{x}[ ],b·d_{y}[ ] > [o] < f(x)+F(x),g(y)+G(y) > = ...

... ( < a·d_{x}[ ],b·d_{y}[ ] > [o] < f(x),g(y) > )+...

... ( < a·d_{x}[ ],b·d_{y}[ ] > [o] < F(x),G(y) > )

a·d_{x}[f(x)+F(x)]+b·d_{y}[g(y)+G(y)] = ...

... ( a·d_{x}[f(x)]+b·d_{y}[g(y)] )+( a·d_{x}[F(x)]+b·d_{y}[G(y)] )

< a·d_{x}[ ],b·d_{y}[ ] > [o] < (1/a)·x,(-1)·(1/b)·y > = 0

< a·d_{x}[ ],b·d_{y}[ ] > [o] < (-1)·(1/a)·x,(1/b)·y > = 0

< a·d_{x}[ ],b·d_{y}[ ] > [o] < (1/a)·( x^{2}+2yx ),(-1)·(1/b)·( y^{2}+2xy ) > = 0

< a·d_{x}[ ],b·d_{y}[ ] > [o] < (-1)·(1/a)·( x^{2}+2yx ),(1/b)·( y^{2}+2xy ) > = 0

< a·d_{x}[ ],b·d_{y}[ ] > [o] ...

... < (1/a)·( f(x)+d_{y}[f(y)]·x ),(-1)·(1/b)·( f(y)+d_{x}[f(x)]·y ) > = 0

< a·d_{x}[ ],b·d_{y}[ ] > [o] ...

... < (-1)·(1/a)·( f(x)+d_{y}[f(y)]·x ),(1/b)·( f(y)+d_{x}[f(x)]·y ) > = 0

a·d_{x}[ (1/a)·( f(x)+d_{y}[f(y)]·x ) ]+b·d_{y}[ (-1)·(1/b)·( f(y)+d_{x}[f(x)]·y ) ] = ...

... d_{x}[ f(x)+d_{y}[f(y)]·x ]+(-1)·d_{y}[ f(y)+d_{x}[f(x)]·y ] = ...

... ( d_{x}[f(x)]+d_{y}[f(y)] )+(-1)·( d_{y}[f(y)]+d_{x}[f(x)] ) = 0

a·d_{x}[ (-1)·(1/a)·( f(x)+d_{y}[f(y)]·x ) ]+b·d_{y}[ (1/b)·( f(y)+d_{x}[f(x)]·y ) ] = ...

... (-1)·d_{x}[ f(x)+d_{y}[f(y)]·x ]+d_{y}[ f(y)+d_{x}[f(x)]·y ] = ...

... (-1)·( d_{x}[f(x)]+d_{y}[f(y)] )+( d_{y}[f(y)]+d_{x}[f(x)] ) = 0

 

< a·int[ ]d[x],b·int[ ]d[y] > [o] < s·f(x),s·g(y) > = ...

... s·( < a·int[ ]d[x],b·int[ ]d[y] > [o] < f(x),g(y) > )

a·int[s·f(x)]d[x]+b·int[s·g(y)]d[y] = s·( a·int[f(x)]d[x]+b·int[g(y)]d[y] )

< a·int[ ]d[x],b·int[ ]d[y] > [o] < f(x)+F(x),g(y)+G(y) > = ...

... ( < a·int[ ]d[x],b·int[ ]d[y] > [o] < f(x),g(y) > )+...

... ( < a·int[ ]d[x],b·int[ ]d[y] > [o] < F(x),G(y) > )

a·int[f(x)+F(x)]d[x]+b·int[g(y)+G(y)]d[y] = ...

... ( a·int[f(x)]d[x]+b·int[g(y)]d[y] )+( a·int[F(x)]d[x]+b·int[G(y)]d[y] )

< a·int[ ]d[x],b·int[ ]d[y] > [o] ...

... < (1/a)·( f(x)+int[f(y)]d[x]·d_{x}[1] ),(-1)·(1/b)·( f(y)+int[f(x)]d[x]·d_{y}[1] ) > = 0

< a·int[ ]d[x],b·int[ ]d[y] > [o] ...

... < (-1)·(1/a)·( f(x)+int[f(y)]d[x]·d_{x}[1] ),(1/b)·( f(y)+int[f(x)]d[x]·d_{y}[1] ) > = 0

Derivada direccional:

Direccions unitaries ortogonals al gradient.

F(x,y) = sin(x)+cos(y)

< cos(x), (-1)·sin(y) > [o] < sin(y),cos(x) > = 0

< cos(x), (-1)·sin(y) > [o] < (-1)·sin(y),(-1)·cos(x) > = 0

F(x,y) = x^{n+1}+y^{n+1}

< (n+1)·x^{n}, (n+1)·y^{n} > [o] ...

... ( 1/(y^{2n}+x^{2n})^{(1/2)} )·< y^{n},(-1)·x^{n} > = 0

< (n+1)·x^{n}, (n+1)·y^{n} > [o] ...

... ( 1/(y^{2n}+x^{2n})^{(1/2)} )·< (-1)·y^{n},x^{n} > = 0

F(x,y) = f(x)+f(y)

< d_{x}[f(x)], d_{y}[f(y)] > [o] ...

... ( 1/(d_{y}[f(y)]^{2}+d_{x}[f(x)]^{2})^{(1/2)} )·< d_{y}[f(y)],(-1)·d_{x}[f(x)] > = 0

< d_{x}[f(x)], d_{y}[f(y)] > [o] ...

... ( 1/(d_{y}[f(y)]^{2}+d_{x}[f(x)]^{2})^{(1/2)} )·< (-1)·d_{y}[f(y)],d_{x}[f(x)] > = 0

F(x,y) = f(x)·y+f(y)·x

< d_{x}[f(x)]y+f(y), d_{y}[f(y)]x+f(x) > [o] ...

... ( 1/((d_{y}[f(y)]x+f(x))^{2}+(d_{x}[f(x)]y+f(y))^{2})^{(1/2)} )·...

... < d_{y}[f(y)]x+f(x),(-1)·(d_{x}[f(x)]y+f(y)) > = 0

< d_{x}[f(x)]y+f(y), d_{y}[f(y)]x+f(x) > [o] ...

... ( 1/((d_{y}[f(y)]x+f(x))^{2}+(d_{x}[f(x)]y+f(y))^{2})^{(1/2)} )·...

... < (-1)·(d_{y}[f(y)]x+f(x)),d_{x}[f(x)]y+f(y) > = 0

temari de matemàtiques

Grau en Matemàtiques:

Primer Any:

Semestre I:

Àlgebra I

Estructures algebràiques.

Teorema de Pitágoras.

Trigonometría

Identitats trigonométriques

Binomi

Ecuació de primer grau

Ecuació de segon grau

Ecuació de tercer grau.

Ecuació de cuart grau.

Sistemes d'ecuacións algebràiques en dues variables.

Equacions polinomiques de simbols de Potch-Hammer.

Semestre II:

Análisis Matemàtic I

Definició de cos ordenat.

Desigualtats en un cos ordenat.

Valor absolut positiu y negatiu.

Fórmules de sumació triangulars.

Límits de funcions.

Succesions de número e y derivació de logaritme.

Succesions convergents.

Succesions monótones.

Succesions no acotades.

Continuitat.

Segon Any:

Semestre I:

Àlgebra lineal:

Espais vectorials.

Subespais vectorials.

Combinacions lineals.

Determinants.

Sistemes de ecuacions lineals.

Aplicacions lineals.

Diagonalització.

Formes canóniques.

Producte escalar.

Àlgebra tensorial.

Semestre II:

Càlcul Diferencial I:

Definició de derivada.

Derivades de monomis.

Serie exponencial.

Series trigonométriques elíptiques.

Series trigonométriques hiperbóliques.

Derivada de la exponencial.

Derivada de les trigonométriques elíptiques.

Derivada de les trigonométriques hiperbóliques.

Regla de la cadena.

Derivacio del logaritme.

Linealitat de la derivada.

Formula del producte y quocient.

Derivació logarítmica.

Funcions monótones.

Teorema del valor mitx.

Hôpital comprovació de limits.

Series de Taylor polinómics

Series de Taylor exponencials

Derivació de la funció inversa.

Derivació de arc-funcions trigonométriques elíptiques.

Derivació de arc-funcions trigonométriques hiperbóliques.

Funcions pow[n]:

Àlgebra y Derivació.

Funcions sum[n]

Àlgebra y Derivació.

Recta tangent.

Tercer any:

Semestre I:

Càlcul integral I:

Linealitat de la integral.

Integrals inmediates.

Integració per parts.

Integració de funcions racionals.

Integració de funcions trigonométriques.

Producte integral.

Fórmules de sumació triangular racionals.

Integral definida per sumes.

Semestre II:

Análisis Matemàtic II:

Series convergents

Series divergents.

Series telescópiques.

Polinomis de Bernoulli.

Series de Fourier.

Series harmóniques

Cuart any:

Primer Semestre:

Càlcul diferencial y integral II

Derivada parcial.

Derivada direcional.

Pla tangent

LaGrange

Camps vectorials diferencials y integrals escalars.

Integral en varies variables

Integral de camí.

Integral de superficie.

Semestre II:

ecuacions diferencials

Ecuació lineal de primer ordre

Ecuació lineal de segon ordre.

métode del exponent k+(-1) = nk en la homogenia.

métode de xu y xu^{n+1}

operador [[k]]

ecuacions diferencials y series.

Master en matemàtiques:

Primer Any:

Semestre I:

Teoria de conjunts:

Àlgebra de conjunts.

Relacions.

Relacions connectives.

Relacions de equivalencia.

Relacions de ordre.

Intervals.

Funcions expansives y contractives.

Ordinals y Cardinals.

Semestre II:

Topología:

Espais topologics

Bases Topologiques.

Homotopia.

Cadenes de complexos Homologia

Teoría de Wiles.

Categoríes

Especies combinatories

Segon Any:

Semestre I:

Análisis complex y ecuacions en derivades parcials:

Derivada imperial.

Residus.

Integrals circulars.

Ecuacions en derivades parcials.

Semestre II:

Probabilitats y geometría diferencial:

Distribucions.

Distribucions condicionades.

Esperança matemática.

Primera forma Fonamental.

Segona forma fonamental.

Master en Física:

Primer Any:

Semestre I:

Mecànica, mecànica industrial y circuits eléctrics:

Semestre II

Termodinamica y tecnología industrial:

Segon Any:

Semestre I:

Electro-Gravito-Magnetisme y relativitat general:

Semestre II:

Mecànica cuántica, Gauge y Teoría de cordes:

ecuació geométrica y derivació de inversa

(-1)·( cosh[1:n+1](x) )^{n+1}+sinh[1:n+1](x) = (-1)

d_{x}[y] = ( y^{(n+1)}+(-1) )/( y+(-m) )

(y+(-m))·d_{x}[y] = y^{(n+1)}+(-1)

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

sinh[1:n+1]( f(x) )+(-m)·( f(x) ) ) = x

sinh[1:n+1]-sum[(-m)]( f(x) ) = x

f(x) = anti-sinh[1:n+1]-sum[(-m)](x)

y(x) = cosh[1:n+1]( anti-sinh[1:n+1]-sum[(-m)](x) )

d_{x}[ anti-sinh[1:n+1]-sum[(-m)](x) ] = ...

... ( 1/( cosh[1:n+1]( anti-sinh[1:n+1]-sum[(-m)](x) )+(-m) ) )

f^{o(-1)}(x) = y

d_{x}[f^{o(-1)}(x)] = ( 1/d_{y}[f(y)] )

y = ln(x)

d_{x}[ln(x)] = ( 1/d_{y}[e^{y}] ) = (1/e^{y}) = (1/x)

y = arc-sin(x)

d_{x}[arc-sin(x)] = ( 1/d_{y}[sin(y)}] ) = (1/cos(y)) = ...

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

d_{x}[ anti-ln-pow[n](x) ] = ...

... ( y/(n·y^{n}ln(y)+y^{n}) ) = ( y/(n·ln-pow[n](y)+y^{n}) )

... ( anti-ln-pow[n](x)/(nx+( anti-ln-pow[n](x) )^{n}) )

d_{x}[ anti-e-pow[n](x) ] = ...

... ( y/(n·y^{n}e^{y}+y^{n}e^{y}y) ) = ( y/(n·e-pow[n](y)+e-pow[n](y)y) )

... ( 1/e-pow[n](y) )·( y/(n+y) ) = (1/x)·( anti-e-pow[n](x)/(n+anti-e-pow[n](x)) )

d_{x}[ anti-ln-[+]-sum[n](x) ] = ...

... ( 1/((1/y)+n) ) = ( y/(1+ny) )

... ( anti-ln-[+]-sum[n](x)/(1+n·anti-ln-[+]-sum[n](x)) )

d_{x}[ anti-e-[+]-sum[n](x) ] = ...

... ( 1/(e^{y}+n) ) = ( 1/(e^{y}+ny+n·(1+(-y)) ) = ( 1/( e-[+]-sum[n](y)+n·(1+(-y)) ) )

... ( 1/( x+n·(1+(-1)·anti-e-[+]-sum[n](x)) ) )

ln(x)+nx = c

ln-[+]-sum[n](x) = c

x = anti-ln-[+]-sum[n](c)

e^{x}+nx = c

e-[+]-sum[n](x) = c

x = anti-e-[+]-sum[n](c)

ln(x)+nx^{m+1} = c

x^{m}·( ln(x)/x^{m}+nx ) = c

x^{m}·( ln-pow[(-m)](x)+nx ) = c

x^{m}·( ln-pow[(-m)]-[+]-sum[n](x) ) = c

( ln-pow[(-m)]-pow[m]-[+]-sum[n]-pow[m](x) ) = c

( ln-[+]-sum[n]-pow[m](x) ) = c

x = anti-ln-[+]-sum[n]-pow[m](c)

e^{x}+nx^{m+1} = c

x = anti-e-[+]-sum[n]-pow[m](c)

x^{k}·ln(x)+nx^{m+1} = c

x = anti-ln-pow[k]-[+]-sum[n]-pow[m](c)

x^{k}·e^{x}+nx^{m+1} = c

x = anti-e-pow[k]-[+]-sum[n]-pow[m](c)

françé y ecuació diferencial

sacboir [o] kacboir

sé-pont [o] ké-pont

saps-pont [o] kaps-pont

sap-pont [o] kap-pont

sacboms [o] kacboms

sacboz [o] kacboz

sacben-puá [o] kacben-puá

bacboir [o] dacboir

bé-pont [o] dé-pont

baps-pont [o] daps-pont

bap-pont [o] dap-pont

bacboms [o] dacboms

bacboz [o] dacboz

bacben-puá [o] dacben-puá

Il sap-pont de-le-com vack ser bacboire-dom de la Font.

Ila sap-pont de-le-com vack ser bacboire-dom de la Font.

vuloir

ye vule ye-de-muá <==> vull-de-puá

tú vule tú-de-tuá <==> vols-de-puá

vule pont-de-suá <==> vol-de-puá

vuloms

vuloz

vulen-puá

fatzoir [o] detzir

ye fatze ye-de-muá [o] ye ditze ye-de-muá <==> fetx-kû [o] ditx-kû

tú fatze tú-de-tuá [o] tú ditze tú-de-tuá <==> fetx-kes [o] ditx-kes

fatze pont-de-suá [o] ditze pont-de-suá <==> fetx-ka [o] ditx-ka

fatzems [o] detzims <==> fem [o] diem

fatzez [o] detziz <==> feu [o] dieu

fatzen-puá [o] ditzen-puá <==> fetx-ken [o] ditx-ken

Il vule pont-de-suá fatzoire-dom un café avec ila-de-suá.

Ila vule pont-de-suá fatzoire-dom un café avec il-de-suá.

ye fatze ye-de-muá un café avec tú-de-tuá,

si tú vule tú-de-tuá.

tú fatze tú-de-tuá un café avec ye-de-muá,

si ye vule ye-de-muá.

vatxnar [o] datxnar

vaitx-pont [o] daitx-pont

vas-pont [o] das-pont

vack-pont [o] dack-pont

vatxnoms [o] datxnoms

vatxnoz [o] datxnoz

van-pont [o] dan-pont

tenoir [o] venir

ye tine ye-de-muá [o] ye vine ye-de-muá

tú tine tú-de-tuá [o] tú vine tú-de-tuá

tine pont-de-suá [o] vine pont-de-suá

tenems [o] venims

tenez [o] veniz

tenen-puá [o] venen-puá

nus venims de le nort y vatxnoms cap a le sur.

nus venims de le sur y vatxnoms cap a le nort.

d_{x}[y] = f(x)·(y/x)^{( n/(n+1) )}

y = xu^{n+1}

u+(n+1)·x·d_{x}[u] = f(x)

u = e^{(-1)·(1/(n+1))·ln(x)}·int[(1/(n+1))·(1/x)·f(x)·e^{(1/(n+1))·ln(x)}]d[x]

y = x·( e^{(-1)·(1/(n+1))·ln(x)}·int[(1/(n+1))·(1/x)·f(x)·e^{(1/(n+1))·ln(x)}]d[x] )^{(n+1)}

y = x·e^{(-1)·ln(x)}·( int[(1/(n+1))·(1/x)·f(x)·e^{(1/(n+1))·ln(x)}]d[x] )^{(n+1)}

Teorema de Pitágoras:

(a+b)^{2} = h^{2}+4·(1/2)·ab

[ah]+[hb] = [ab] = (pi/2)

[ah]+[hh]+[hb] = pi

[hh] = (pi/2)

Identitat Pitagórica: 

a^{2}+b^{2} = h^{2}

(a^{2}/h^{2})+(b^{2}/h^{2}) = (h^{2}/h^{2})

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

( cos(x) )^{2}+( sin(x) )^{2} = 1

Si lo mundo vos odia,

pensad que ya no conocen al que me envió,

porque no son de esta especie,

y miente su alma.

Si lo mundo no vos odia,

pensad que aun conocen al que me envió,

porque son de esta especie,

y no miente su alma.

Ye estare ye-de-muá fatzointu-dom un café avec tú-de-tuá

Tú estare tú-de-tuá fatzointu-dom un café avec ye-de-muá

Ye havere ye-de-muá fatzoitu-dom un café avec tú-de-tuá

Tú havere tú-de-tuá fatzoitu-dom un café avec ye-de-muá

Françé-de-le-Patuá-y-Occitán-de-le-Pamuá:

Tú vols-de-puá [ fatzoire-dom ]-[ fatzoir ] un café avec [ ye-de-muá ]-[ ye-de-mi ]

Ye vull-de-puá [ fatzoire-dom ]-[ fatzoir ] un café avec [ tú-de-tuá ]-[ tú-de-ti ]

ye tine ye-de-muá anai-dom-otza-duá,

perque fatze pont-de-suá otzaté.

ye tine ye-de-muá anai-dom-bero-duá,

perque fatze pont-de-suá beroté.

Métode:

y = xu^{n+1}

d_{x}[y] = f(x)·(y/x)^{( n/(n+1) )}+(y/x)

(n+1)·x·d_{x}[u] = f(x)

y(x) = x·( (1/(n+1))·int[f(x)]d[x] [o(x)o] ln(x) )^{(n+1)}

d_{x}[y] = f(x)·(y/x)^{( n/(n+1) )}

u+(n+1)·x·d_{x}[u] = f(x)

y(x) = x·e^{(-1)·ln(x)}·( int[(1/(n+1))·(1/x)·f(x)·e^{(1/(n+1))·ln(x)}]d[x] )^{(n+1)}

Teoría:

d_{x}[ sin-[f(x)]-d[n:1]( h(x) ) ] = ...

... ( ( sin-[f(x)]-d[n:1]( h(x) ) )·( cos-[f(x)]-d[n:1]( h(x) ) )+(-1)·f(x) )·d_{x}[h(x)]

( cos-[f(x)]-d[n:1]( h(x) ) )+( sin-[f(x)]-d[n:1]( h(x) ) )^{n} = 1

Métode:

y = xu^{n+1}

d_{x}[y] = f(x)·(y/x)^{( n/(n+1) )}+(y/x)^{n}

u^{n(n+1)+(-n)} = u^{n^{2}}

u·( 1+(-1)·u^{n^{2}+(-1)} )+(n+1)·x·d_{x}[u] = f(x)

y(x) = x·( sin-[f(x)]-d[n^{2}+(-1):1]( (-1)·(1/(n+1))·ln(x) ) )^{n+1}

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

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

y(x) = x·( sin-[f(x)]-d[m(n+1)+(-n)+(-1):1]( (-1)·(1/(n+1))·ln(x) ) )^{n+1}

Teoría:

d_{x}[ sin-[f(x)]-d[n:1]( h(x) )·F(x) ] = ...

... ( ( sin-[f(x)]-d[n:1]( h(x) )·F(x) )·( cos-[f(x)]-d[n:1]( h(x) )·F(x) )+...

... (-1)·f(x)·( F(x) )^{2} )·d_{x}[h(x)]+...

... ( sin-[f(x)]-d[n:1]( h(x) )·F(x) )·d_{x}[F(x)]

( cos-[f(x)]-d[n:1]( h(x) )·F(x) )+( sin-[f(x)]-d[n:1]( h(x) )·F(x) )^{n} = 1

Si F(x) = 1 ==> d_{x}[F(x)] = 0

Si F(x) = k ==>

sin-[f(x)]-d[n:1]( h(x) )·k = ( sin-[k^{2}·f(x)]-d[n:1]( h(x) ) )

cos-[f(x)]-d[n:1]( h(x) )·k = ( cos-[k^{2}·f(x)]-d[n:1]( h(x) ) )

d_{x}[ sin-[k^{2}·f(x)]-d[n:1]( h(x) )/sin-[f(x)]-d[n:1]( h(x) ) ] = 0

d_{x}[ cos-[k^{2}·f(x)]-d[n:1]( h(x) )/cos-[f(x)]-d[n:1]( h(x) ) ] = 0

d_{x}[ sin-[f(x)]-d[n:1]( h(x) )·(F(x)+G(x)) ] = ...

... ( ( sin-[f(x)]-d[n:1]( h(x) )·(F(x)+G(x)) )·( cos-[f(x)]-d[n:1]( h(x) )·(F(x)+G(x)) )+...

... (-1)·f(x)·(F(x)+G(x))^{2} )·d_{x}[h(x)]+...

... ( sin-[f(x)]-d[n:1]( h(x) )·(F(x)+G(x)) )·d_{x}[F(x)+G(x)]

d_{x}[ sin-[f(x)]-d[n:1]( h(x) )·(F(x)+G(x)) ] = ...

... d_{x}[ sin-[f(x)]-d[n:1]( h(x) )·F(x) ]+...

... d_{x}[ sin-[f(x)]-d[n:1]( h(x) )·(2·F(x)·G(x))^{(1/2)} ]+...

... d_{x}[ sin-[f(x)]-d[n:1]( h(x) )·G(x) ]

sin-[f(x)]-d[n:1]( h(x) )·(2·F(x)·G(x))^{(1/2)}·d_{x}[(2·F(x)·G(x))^{(1/2)}] = ...

... sin-[f(x)]-d[n:1]( h(x) )·( F(x)d_{x}[G(x)]+d_{x}[F(x)]G(x) )

métode:

y = xu^{n+1}

d_{x}[y] = f(x)·(y/x)^{( n/(n+1) )}+(y/x)^{n}+g(x)·(y/x)

u·( 1+(-1)·u^{n^{2}+(-1)} )+(-1)·g(x)·u+(n+1)·x·d_{x}[u] = f(x)

y(x) = ...

... x·( sin-[( f(x)/( int[(1/(n+1))·(1/x)·g(x)]d[x] )^{2} )]-...

... d[n^{2}+(-1):1]( (-1)·(1/(n+1))·ln(x) )·...

... int[(1/(n+1))·(1/x)·g(x)]d[x] )^{n+1}

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

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

y(x) = ...

... x·( sin-[( f(x)/( int[(1/(n+1))·(1/x)·g(x)]d[x] )^{2} )]-...

... d[m(n+1)+(-n)+(-1):1]( (-1)·(1/(n+1))·ln(x) )·...

... int[(1/(n+1))·(1/x)·g(x)]d[x] )^{n+1}

ecuacions diferencials

d_{x}[ arc-sin-up-[1]-pow[n](x) ] = ...

... ( arc-sin-up-[1]-pow[n](x) )^{n}·( 1+(-1)·( arc-sin-up-[1]-pow[n](x) )^{2} )^{(1/2)}

d_{x}[y(x)] = ( sin(y) )^{n}

d_{x}[ arc-sin-up-[k]-pow[n](x) ] = ...

... ( arc-sin-up-[k]-pow[n](x) )^{n}·...

... ( 1+(-1)·( arc-sin-up-[k]-pow[n](x) )^{k+1} )^{(1/(k+1))}

d_{x}[y(x)] = ( sin[k](y) )^{n}

anti-arc-sin-dawn-[1]-pow[(-n)](x) = ...

... (1/((-n)+1))·x^{(-n)+1} [o(x)o] (-1)·( 1+(-1)·x^{2} )^{(1/2)} [o(x)o] ln(x)

d_{x}[y(x)] = x^{(-n)}·(1+(-1)·x^{2})^{(-1)·(1/2)}

anti-arc-sin-dawn-[k]-pow[(-n)](x) = ...

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

... (-1)·(1/k)·( 1+(-1)·x^{k+1} )^{(k/(k+1))} [o(x)o] (1/((-k)+1))·x^{(-k)+1}

d_{x}[y(x)] = x^{(-n)}·(1+(-1)·x^{k+1})^{(-1)·(1/(k+1))}

stowed stehed

crash [o] scratch

clash [o] sclatch

prash [o] spratch

plash [o] splatch

brash [o] sbratch

blash [o] sblatch

grash [o] sgratch

glash [o] sglatch