Re: Re: Re: coordinate transformation

*To*: mathgroup at smc.vnet.net*Subject*: [mg13340] Re: [mg13308] Re: [mg13169] Re: [mg13117] coordinate transformation*From*: David Withoff <withoff>*Date*: Mon, 20 Jul 1998 02:49:58 -0400*Sender*: owner-wri-mathgroup at wolfram.com

In remarks about Calculus`VectorAnalysis` Sean Ross wrote: > This whole issue becomes painfully clear with the following exercise: > Take two cartesian vectors and their cross product. Transform the > vectors into cylindrical or spherical coordinates using the transforms > like those in the mathematica package and take their cross product in > cylindrical/spherical coordinates and transform the result back to > cartesian and compare the two. My hat off to anyone who gets the > correct answer the first two or three times trying. Here is (what appears to me to be) a straightforward way to do this: Load the Calculus`VectorAnalysis` package: In[1]:= << Calculus`VectorAnalysis` Introduce two random vectors. In[2]:= v1 = {Random[], Random[], Random[]} Out[2]= {0.0544546, 0.199285, 0.612214} In[3]:= v2 = {Random[], Random[], Random[]} Out[3]= {0.985127, 0.832121, 0.100095} Compute the cross product. By default, the vectors are assumed to be given in cartesian coordinates, and the result is in cartesian coordinates. In[4]:= CrossProduct[v1, v2] Out[4]= {-0.489489, 0.597657, -0.151008} Now do the same calculation in spherical coordinates. Convert the vectors to spherical coordinates. In[5]:= sv1 = CoordinatesFromCartesian[v1, Spherical] Out[5]= {0.646131, 0.32545, 1.30406} In[6]:= sv2 = CoordinatesFromCartesian[v2, Spherical] Out[6]= {1.29341, 1.49333, 0.7014} Compute the cross product, treating the vectors as given in spherical coordinates, and giving the result in spherical coordinates. In[7]:= sprod = CrossProduct[sv1, sv2, Spherical] Out[7]= {0.787145, 1.76384, 2.25702} Convert this result back to cartesian coordinates. In[8]:= CoordinatesToCartesian[sprod, Spherical] Out[8]= {-0.489489, 0.597657, -0.151008} Observe that Out[4] and Out[8] are the same. Is this the example that you had in mind? If so, do you have any suggestions for how this should be done instead? If not, could you offer some clarifications of your concerns? You have publicized various warnings about disagreements between Mathematica and "physical mathematics", but you haven't offered any specific examples to illustrate what Mathematica should have done instead. In the absence of any specific alternate design, it is difficult to tell if it is possible to change Mathematica so that you would be happy with it, or if your complaints are reflections of some other problem. Exactly how do you think these functions should be changed? Specific examples are desperately needed. Dave Withoff Wolfram Research