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Re: Convert function from polar to Cartesian

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112783] Re: Convert function from polar to Cartesian
  • From: Peter Breitfeld <phbrf at t-online.de>
  • Date: Thu, 30 Sep 2010 04:52:15 -0400 (EDT)
  • References: <i7usgt$rdn$1@smc.vnet.net>

Sam Takoy wrote:

> 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
>

I think you can't always get rid of the ArcTan, but this works for your
case:

cartRule = {r -> Sqrt[x^2 + y^2], a -> ArcTan[x, y]};

(((fPolar[r, a] // TrigToExp) /. cartRule) // FullSimplify) // Expand

Out= x^2 - y^2

-- 
_________________________________________________________________
Peter Breitfeld, Bad Saulgau, Germany -- http://www.pBreitfeld.de


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