Re: Re: Wrong behavior of CrossProduct
- To: mathgroup at smc.vnet.net
- Subject: [mg8046] Re: [mg7976] Re: [mg7958] Wrong behavior of CrossProduct
- From: seanross at worldnet.att.net
- Date: Sat, 2 Aug 1997 22:32:54 -0400
- Sender: owner-wri-mathgroup at wolfram.com
Sergio Rojas wrote: > > Hello: > > How Mathematica implement the Cross Product of two vectors? > > As far as I know the basic definition of the Cross Product between > two vectors in ANY ORTHOGONAL coordinated system is as follows: > > (a1,a2,a3)X(b1,b2,b3) = (a2*b3 - a3*b2, > a3*b1 - a1*b3, > a1*b2 - a2*b1) > > In physics this is usually illustrated by taking any three > UNIT vectors (u,u,u) with the orthogonal property: > > u[i].u[j] = Delta[i,j] where Delta[i_, j_] := If[i==j, 1, 0] > > uxu = u ; uxu = u ; uxu = u > uxu = -u ; uxu = -u ; uxu = -u > uxu = 0 ; uxu = 0 ; uxu = 0 > > Then, the above result follows by expanding: > > (a*u + a*u + a*u)x(b*u + b*u + b*u) > > and using the orthogonal property of the unit vectors. > > In my example, > > a = ( 0, 0,1) ; b = (a1,a2,0) > > axb = ( a2, -a1, 0 ) . No, you are forgetting the metric tensor, which for cartesian coordinates is the identity matrix and has non-zero diagonal elements that are a function of r and theta in spherical coordinates. Even some advanced graduate textbooks, like Arfken, don't bother to mention this. I think the failure to mention little details like this is that most physics books are written by theoreticians who rarely calculate numbers-they just generate formulas and let somebody else actually calculate the numbers. Anyway, the components of the metric tensor(usually written g) are the square of the scale factors(usually written h) for the particular component. As I recall, for cylindrical coordinates g11=1, g22=1/r^2, g33=1 and for spherical g11=1, g22=1/r^2, g33=1/r^2 sin(theta)^2, but don't quote me on that. I got sick of theoreticians and became an engineer some time ago.