RE: Re: Dot Product in Cylindrical Coordinates

*To*: mathgroup at smc.vnet.net*Subject*: [mg69300] RE: [mg69276] Re: Dot Product in Cylindrical Coordinates*From*: "David Park" <djmp at earthlink.net>*Date*: Wed, 6 Sep 2006 04:28:10 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Paul, I still want to keep this alive. Perhaps the terminology is ambiguous? In vector calculus it is usually conventional to erect an orthonormal frame at each point in space. So for cylindrical coordinates at the point {r, theta, z}, we would have an orthonormal frame with unit axes pointing in the r, theta and z directions. Then if someone specifies a vector, its components are in terms of the orthonormal frame. This is not the same as a set of cylindrical coordinates with respect to the origin. When I use Sergio's vectors in that sense I obtain 2 as the answer. In fact, if we have an orthonormal frame field the dot product is just the ordinary dot product. If we used a coordinate basis, which is ortho but not normal, and specified components in terms of it, then the result would depend on the point of application, specifically on r in this case. So it is much more convenient to use an orthonormal frame field, and that is what is usually done. I don't think that is a quixotic interpretation. So I think the VectorAnalysis DotProduct used directly on vectors is a little trap that people can fall into. David Park djmp at earthlink.net http://home.earthlink.net/~djmp/ From: Paul Abbott [mailto:paul at physics.uwa.edu.au] To: mathgroup at smc.vnet.net In article <edadqd$pgc$1 at smc.vnet.net>, Sergio Miguel Terrazas Porras <sterraza at uacj.mx> wrote: > When I calculate the dot product of vectors {1,Pi/4,0} and {2,0,1} in > Cylindrical Coordinates Mathematica 5.1 returns the result Sqrt[2], when the > result should be 2. Notwithstanding several of the other responses, the result _is_ Sqrt[2]. When you write {1,Pi/4,0}, surely you mean {rho, phi, z} == {1, Pi/4, 0} and _not_ that {x, y, z} == {1, Pi/4, 0} ? After loading Needs["Calculus`VectorAnalysis`"] and selecting Cylindrical coordinates, SetCoordinates[Cylindrical]; then in cartesian coordinates, this point is p1 = CoordinatesToCartesian[{1, Pi/4, 0}] {1/Sqrt[2], 1/Sqrt[2], 0} Similarly, p2 = CoordinatesToCartesian[{2, 0, 1}] {2, 0, 1} Hence the dot product of the coordinate vectors (relative to the origin {0,0,0}), computed in cartesian coordinates, is p1 . p2 Sqrt[2] This is the same result that you got, presumably using, DotProduct[ {1, Pi/4, 0}, {2, 0, 1} ] Sqrt[2] Of course, if you really mean {x, y, z} == {1, Pi/4, 0} then there is no need to load Calculus`VectorAnalysis`: the dot product is just {1, Pi/4, 0} . {2, 0, 1} 2 Note that Dot is _not_ modified when this package is loaded so Jean-Marc's response, Needs["Calculus`VectorAnalysis`"] SetCoordinates[Cylindrical]; {1, Pi/4, 0} . {2, 0, 1} is bogus -- the first two lines have no effect on the third. Modifying Andrzej's code, we have Simplify[(JacobianMatrix[] . p1) . (JacobianMatrix[] . p2)] Sqrt[2] Cheers, Paul _______________________________________________________________________ Paul Abbott Phone: 61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) AUSTRALIA http://physics.uwa.edu.au/~paul