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})