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