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Simplification and implicit function

asked 2019-09-21 09:21:08 -0600

Cyrille gravatar image

I have two question in one : 1) I have this $z= \frac{\alpha x^{\alpha-1}y^\beta y^{-\beta-1}}{\beta x^\alpha}$. There are obvious simplification since this is equal to $\frac{\alpha}{\beta} \frac{y}{x}$.

z.full_simplify() has no impact z.expand() simplify on $y$ but not on $x$

So what can I do to obtain the good result.

2) how to take the result and consider it as an implicit function to obtain the derivative de $y$. as a function of x

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answered 2019-09-21 10:08:35 -0600

Emmanuel Charpentier gravatar image

I have two question in one : 1) I have this $z=\frac{αx^{α−1}y^βy^{−β−1}}{βx^α}$. There are obvious simplification since this is equal to $\frac{α}{β}\frac{y}{x}$.

It isn't:

sage: foo=a*x^(a-1)*y^b*y^-(b+1)/(b*x^a);foo
a*x^(a - 1)*y^b*y^(-b - 1)/(b*x^a)
sage: foo.canonicalize_radical()
a/(b*x*y)

2) how to take the result and consider it as an implicit function to obtain the derivative de y. as a function of x

sage: foo.canonicalize_radical().diff(y)
-a/(b*x*y^2)

which should be obvious given the (correct) result of your simplification...

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In fact my problem was nearly similar but with your helps I have understud what was wrong in my approach. In reality I had $tms= U_x/U_y$ where the U_(x or y) where the partial dérivatives of U(x, Y) = A x^\alpha y{^\1- beta}. Even with canonicalize.radical I was unable to obtain you result since I was writing U_x/U_y.canonalize.radical() and not (U_x/U_y).canonalize.radical(). Thanks for all.

Now for 2) your result is obviously wrong since you Don't use the implicit command differenti&tion

Cyrille gravatar imageCyrille ( 2019-09-21 12:48:37 -0600 )edit

Okay.

sage: foo.canonicalize_radical()
a/(b*x*y)
sage: foo.canonicalize_radical().diff(y)
-a/(b*x*y^2)
sage: foo.diff(y)
a*x^(a - 1)*y^(b - 1)*y^(-b - 1)/x^a - a*(b + 1)*x^(a - 1)*y^b*y^(-b - 2)/(b*x^a)
sage: foo.diff(y).canonicalize_radical()
-a/(b*x*y^2)

Where is this "obviously wrong" ?

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2019-09-22 01:25:04 -0600 )edit

And, by the way:

sage: U
(x, y) |--> A*x^alpha*y^beta
sage: (U(x,y).diff(x)/U(x,y).diff(y)).canonicalize_radical()
alpha*y/(beta*x)
Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2019-09-22 01:29:46 -0600 )edit

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Asked: 2019-09-21 09:21:08 -0600

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Last updated: Sep 21