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MathGroup Archive 1998

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Re: Re: coordinate transformation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg13252] Re: [mg13169] Re: [mg13117] coordinate transformation
  • From: David Withoff <withoff>
  • Date: Fri, 17 Jul 1998 03:17:47 -0400
  • Sender: owner-wri-mathgroup at wolfram.com

> S.-Svante Wellershoff wrote:
> >
> > Can someone explain me, how to use mathematica for transformations
> > between
> > different coordinate systems?
> >
> > Like (x,y,z) in carthesian equals (r sin t cos p, r sin t sin p, r cos
> > t) in spherical system.
> >
> > Thanks, Svante
> >
> > p.s.: hope its no faq!
> >
> > ---------------------------------------------------------------------
> >   S.-Svante Wellershoff      svante.wellershoff at physik.fu-berlin.de
> >                              http://www.physik.fu-berlin.de/~welle/
> > ---------------------------------------------------------------------
> >   Institut fuer Experimentalphysik
> >   Freie Universitaet Berlin         phone +49-(0)30-838-6234 (-6057)
> >   Arnimallee 14                     fax   +49-(0)30-838-6059
> >   14195 Berlin - Germany
> > ---------------------------------------------------------------------
>
> There is an add-on package which does this called
> Calculus`VectorAnalysis`.  Please be careful.  Your definition above
> and the ones built in to mathematica and the ones quoted in nearly
> every physics book I have ever seen are valid only for vectors of
> infinitesimal spatial extent(field vectors, not displacement vectors)
> centered at the origin.  You must also transform the unit vectors along
> with the components.  If you do any subsequent vector differential
> operations, you must also take into account the coordinate metric which
> depends on the unit system and the location of the vectors.

Yes, there are functions in Calculus`VectorAnalysis` to change between
common coordinate systems.  For example:

In[1]:= << Calculus`VectorAnalysis`

In[2]:= ?CoordinatesToCartesian
CoordinatesToCartesian[pt] gives the Cartesian coordinates of the point
pt
   given in the default coordinate system. CoordinatesToCartesian[pt,
   coordsys] gives the Cartesian coordinates of the point given in the
   coordinate system coordsys.

In[3]:= CoordinatesToCartesian[{r, t, p}, Spherical]

Out[3]= {r Cos[p] Sin[t], r Sin[p] Sin[t], r Cos[t]}

The documentation for this and other functions in this package can be
found in the Standard Add-On Packages guide, which is included in the
on-line documentation.  I expect that that is what you want.

The remark about transformations that are "valid only for vectors of
infinitesimal spatial extent centered at the origin" is confusing, but
may refer to the fact that, when working with a vector field, the field
vectors are typically described using a locally cartesian coordinate
system derived from the coordinate system that is used for the host
space. You can get into all sorts of trouble if you get the host space
(or the coordinate system that is used to describe it) mixed up with
the spaces (or their coordinate systems) used for the field vectors. 
If you aren't working with vector fields, then none of that is
relevant, of course, and it isn't relevant to the
Calculus`VectorAnalysis` package in any case, since the functions in
that package only deal with one space at a time, and are not intended
for transformations between, say, one locally cartesian space and
another.

Dave Withoff
Wolfram Research


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