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Re: Partial Differentiation of Implicit Functions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg83196] Re: [mg83170] Partial Differentiation of Implicit Functions
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Wed, 14 Nov 2007 04:53:30 -0500 (EST)
  • References: <200711131201.HAA05102@smc.vnet.net>

On 13 Nov 2007, at 21:01, Jay wrote:

> Hi,
>
> I have some equations of the form:
> AA x^2 + BB y^2 + CC z^2 + DD x y +EE x z + FF y z + GG x + HH y +  
> II z +
> JJ== 0
>
> I want to solve for e.g. partial dy/dz = 0 and then combine the  
> result with
> the original equation to get a new implicit equation.
>
> E.g.
>
> E1 := -2 + x^2 + x z + y^2 + z^2
>
> (where I require E1 == 0)
>
> Manually performing the differentiation gives:
>
> partial dx/dz (y constant) = (x+2 z)/(2x + z) == 0
>
> I can then go back into mathematica and do
>
> Eliminate[{E1 == 0, 2*z + x == 0}, {z}]
>
> giving:
>
> 4 x y + 4 y^2 == 8 - 5 x^2
>
> This is what I want but how do I do everything in Mathematica? I  
> expected to
> be able to do something like:
>
>   Solve[E1==0,D[x,y]]
>
> but it doesn't seem to work (says "0" is not a valid variable)
>
> Thanks,
>
> Jay.
>
>
>
>
>
>
>

How about:

eq1 = -2 + x^2 + x*z + y^2 + z^2 == 0;

eq2 = Dt[eq1, z] /. {Dt[y, z] -> 0, Dt[x, z] -> 0};

Solve[{eq1, eq2}, {y, x}, {z}]
  Solve::"svars" : "Equations may not give solutions for all Solve
variables.
{{y -> (-(1/2))*Sqrt[8 - 3*x^2]},
    {y -> (1/2)*Sqrt[8 - 3*x^2]}}

or

Eliminate[{eq1, eq2}, {z}]
4*y^2 == 8 - 3*x^2

Andrzej Kozlowski


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