On Jan 16, 2008 8:14 PM, Ondrej Certik <[EMAIL PROTECTED]> wrote:
> On Jan 16, 2008 7:00 PM, David Joyner <[EMAIL PROTECTED]> wrote:
> >
> > Hi:
> >
> > I don't remember this topic coming up before but at some point
> > it must be faced since it arises in Calculus 1. AFAIK, SAGE has no routines
> > for
> > (1) computing the "implicit derivative" y' if y is defined implicitly
> > by F(x,y)=0,
> > (2) plotting (x,y) subject to F(x,y)=0.
> >
> > (1): This is easy for SAGE to compute: y' = - F_x(x,y)/F_y(x,y).
> > The problem is that it just isn't implemented yet to my knowledge.
> > How should this be implemented? implicit_derivative(F(x,y),x) or something??
> > Suggestions?
>
> Is there a need for a special function? I would suggest to use regular
> differentiation means.
>
> For inspiration, I just tried this in sympy:
>
> In [1]: g = Function("g")
>
> In [2]: f(g(x), x)
> Out[2]: f(g(x), x)
>
> In [3]: e=f(g(x), x)
>
> In [4]: e
> Out[4]: f(g(x), x)
>
> In [7]: e.args
> Out[7]: (g(x), x)
>
> In [8]: e.diff(x)
> Out[8]:
> d d d
> ─────(f(g(x), x))*──(g(x)) + ──(f(g(x), x))
> dg(x) dx dx
>
> In [9]: Basic.set_repr_level(1)
> Out[9]: 2
>
> In [10]: e.diff(x)
> Out[10]: D(f(g(x), x), g(x))*D(g(x), x) + D(f(g(x), x), x)
>
>
> (you need to use fixed width fonts to see [8], otherwise you'll just see a
> mess)
> But I am not satisfied with the output, especially with
>
> D(f(g(x), x), x)
>
> since it is not clear if it means the total derivative, or just with
> respect to the second parameter. Or do you mean something like this:
>
> In [4]: e=f(x)**2+f(x)*x**3-3*f(x)
>
> In [5]: e.diff(x)
> Out[5]:
> d 3 d d 2
> - 3*──(f(x)) + x *──(f(x)) + 2*f(x)*──(f(x)) + 3*x *f(x)
> dx dx dx
>
> In [6]: Basic.set_repr_level(1)
> Out[6]: 2
>
> In [7]: e.diff(x)
> Out[7]: -3*D(f(x), x) + x**3*D(f(x), x) + 2*f(x)*D(f(x), x) + 3*x**2*f(x)
>
> In [9]: solve(e.diff(x).subs(f(x).diff(x), y) == 0, [y])
> Out[9]: [3*x**2/(3 - x**3 - 2*f(x))*f(x)]
>
> in [7] I differentiate the implicit definition of f(x), in [9] I solve
> for the D(f(x), x) by substituting it for "y" and solving for y.
>
> I cannot now figure out how to do this in Sage, but I am not sure what
> the best interface to all of this is. But I think the less functions
> (idioms) one needs to remember to do any calculus stuff in Sage, the
> better.
To make myself more clear, I'd suggest:
def implicit_derivative(eq, f)
x = f.args[0]
return solve(eq.diff(x), f)
or something like this. At least the goal of the calculus in Sage
should be to support this imho.
Ondrej
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