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Convert function from polar to Cartesian

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112754] Convert function from polar to Cartesian
  • From: Sam Takoy <sam.takoy at yahoo.com>
  • Date: Wed, 29 Sep 2010 04:14:49 -0400 (EDT)

Hi,

I've stumbled onto an unexpected problem. I have a function in polar 
coordinate and I need to convert it to Cartesian coordinates:

fPolar[r_, a_] = r^2 Cos[2 a];
fCart[x_, y_] = fPolar[Sqrt[x^2 + y^2], ArcTan[x, y]];
fCart[x, y]

I get: (x^2 + y^2) Cos[2 ArcTan[x, y]].

But how to get rid of the ArcTan[x, y]? I've tried this:

fCart[x, y] //
  TrigExpand /. {Cos[ArcTan[x, y]] -> x/Sqrt[x^2 + y^2],
    Sin[ArcTan[x, y]] -> y/Sqrt[x^2 + y^2]}

but it does not do the substitution I need it to do. So what's the 
proper way to get rid of the ArcTan[x, y] and for the simple example 
above, to obtain x^2 - y^2?


Many thanks in advance,

Sam


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