Converting between Spherical and Cartesian coordinates

*To*: mathgroup at smc.vnet.net*Subject*: [mg53296] Converting between Spherical and Cartesian coordinates*From*: Alain Cochard <alain at geophysik.uni-muenchen.de>*Date*: Wed, 5 Jan 2005 01:21:09 -0500 (EST)*Reply-to*: alain at geophysik.uni-muenchen.de*Sender*: owner-wri-mathgroup at wolfram.com

I use spherical coordinates: In[1]:= << Calculus`VectorAnalysis` ; In[2]:= SetCoordinates[Spherical] Out[2]= Spherical[Rr, Ttheta, Pphi] and I convert: In[3]:= CoordinatesFromCartesian[{x,y,z}] 2 2 2 z Out[3]= {Sqrt[x + y + z ], ArcCos[------------------], ArcTan[x, y]} 2 2 2 Sqrt[x + y + z ] Now, if I use the expressions for x, y, and z, I expect to get back to Rr, Ttheta, Pphi, but: In[4]:= FullSimplify[%3/.{ x->Rr Sin[Ttheta]Cos[Pphi], y->Rr Sin[Ttheta]Sin[Pphi], z->Rr Cos[Ttheta]}, Rr>=0 && 0 <= Ttheta <= Pi && -Pi < Pphi <= Pi] Out[4]= {Rr, Ttheta, ArcTan[Rr Cos[Pphi] Sin[Ttheta], > Rr Sin[Pphi] Sin[Ttheta]]} Even if I remove the equalities from the assumptions, I am not quite there: In[5]:= FullSimplify[%3/.{ x->Rr Sin[Ttheta]Cos[Pphi], y->Rr Sin[Ttheta]Sin[Pphi], z->Rr Cos[Ttheta]}, Rr>0 && 0 < Ttheta < Pi && -Pi < Pphi < Pi] Out[5]= {Rr, Ttheta, ArcTan[Cos[Pphi], Sin[Pphi]]} But I would have thought that for a given Pphi in (-Pi,Pi), there is a unique value for ArcTan[Cos[Pphi], Sin[Pphi]], which is... Pphi. And this seems to me consistent with the fact that a given (single) point is unique, or, in other words, that to a given (perhaps not-too-specific) {x,y,z} corresponds a unique {Rr, Ttheta, Pphi}, and vice versa. What am I missing here? Thanks in advance, AC