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Partial Differentiation of Implicit Functions

• To: mathgroup at smc.vnet.net
• Subject: [mg83170] Partial Differentiation of Implicit Functions
• From: "Jay" <jay at aleka.freeserve.co.uk>
• Date: Tue, 13 Nov 2007 07:01:08 -0500 (EST)
• Organization: Helsinki Television

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.

• Follow-Ups:
  • Re: Partial Differentiation of Implicit Functions
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>