Re: Re: Re: Wrong behavior of CrossProduct

*To*: mathgroup at smc.vnet.net*Subject*: [mg8088] Re: [mg8044] Re: [mg7996] Re: [mg7958] Wrong behavior of CrossProduct*From*: seanross at worldnet.att.net*Date*: Tue, 5 Aug 1997 03:22:58 -0400*Sender*: owner-wri-mathgroup at wolfram.com

Mark Evans wrote: > > > > > You raise a good point and put your finger on a subtlety that escapes > > most people. The vector v={0,b,c} in spherical coordinates represents a > > vector of zero length for displacement vectors. Consider,however, the > > case of a gradient field, such as an electric field. Certainly we could > > conceive of an electric field that, at some point in space, had no > > radial component, but only a theta or phi component. The magnitude of > > the field would not be zero because the radial component was zero. Most > > mechanics and electodynamics textbooks pass over this subtlety because > > physical cross products that occur in nature don't involve displacement > > vectors, they involve field vectors and vector differential operators > > which occur at a local point in space. > > > > Sean, > > Mathematica's vector math is implemented in a package. Do you think > there should be two packages, one for displacement vectors, another for > field vectors? What do you think Wolfram Research should do to resolve > these subtleties? How would you go about resolving them? > > Mark I wouldn't resolve them. Anyone who actually needs to calculate numbers does it in Cartesian coordinates after doing symbolic math in whatever other system. If I ever got lazy and wanted Mma to do my symbolic cross products, I would write it out myself in tensor notation: g[i,j] epsilon[i,j,k] A[i] B[j]. The last three factors are the same as the cartesian CrossProduct[A,B] command, so I would only have to put in the correct metric tensor myself. It would take a decent package to correctly include all the subleties of covariant, contravariant and displacement vectors and include all the metrics for all the possible combinations. If I were to "fix" Mma, I would eliminate the non-cartesian cross-products altogether and let people construct their own with matrix multiplication.

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**Re: Re: Re: Wrong behavior of CrossProduct**

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