RE: Re: RE: Re: Dot Product in Cylindrical Coordinates

*To*: mathgroup at smc.vnet.net*Subject*: [mg69362] RE: [mg69351] Re: [mg69301] RE: [mg69276] Re: Dot Product in Cylindrical Coordinates*From*: "David Park" <djmp at earthlink.net>*Date*: Sat, 9 Sep 2006 03:26:39 -0400 (EDT)

Pratik, Your example does not work for me. Needs["Calculus`VectorAnalysis`"] SetCoordinates[Cylindrical[r, theta, z]]; g = {gr[r, theta, z], gtheta[r, theta, z], gz[r, theta, z]}; f = {fr[r, theta, z], ftheta[r, theta, z], fz[r, theta, z]}; gr[r_, theta_, z_] := r gtheta[r_, theta_, z_] := theta gz[r_, theta_, z_] := z fr[r_, theta_, z_] := r^2 ftheta[r_, theta_, z_] := theta fz[r_, theta_, z_] := Cos[z] If we use CrossProduct and DotProduct along with a standard identity using Div and Curl, which should always be true, then it does not work. Div[CrossProduct[g, f]] == DotProduct[Curl[g], f] - DotProduct[Curl[f], g] // Simplify (r^2 + 3*r^4*z^2 - 5*r^3*z*Cos[z] + Sqrt[r^2*((-r)*z + Cos[z])^2] + r^2*Cos[2*z])/ (r*Sqrt[r^2*((-r)*z + Cos[z])^2]) == (theta*(-z + Cos[z]))/r However if we use the standard Dot and Cross functions the identity is verified. Div[Cross[g, f]] == Dot[Curl[g], f] - Dot[Curl[f], g] // Simplify True The point here is that the package functions DotProduct and CrossProduct are rather special and have NOTHING to do with standard vector calculus. Users should not stumble into using them where they do not apply. In standard vector calculus, in a coordinate system, an orthonormal frame is erected at each point in the space. The axes of the frame point along the coordinate directions. The components of vectors are specified in terms of this orthonormal frame. Since the frame is orthonormal, at any point we can simple use the standard Dot and Cross product for combining two vectors at that point. Curl and Div demand vector components in terms of an orthonormal frame. If businessmen can jump on computerized inventory control, lasers and barcodes and DVDs and other technology all within a decade or so of their invention, why or why can't engineering and physics schools dump the misbegotten vector calculus of over a century ago? David Park djmp at earthlink.net http://home.earthlink.net/~djmp/ From: Pratik Desai [mailto:pratikd at wolfram.com] To: mathgroup at smc.vnet.net I think it is quite imperative to note here that "There are often conflicting definitions of a particular coordinate system in the literature. When you use a coordinate system with this package, you should look at the definition given below to make sure it is what you want." --Mathematica Documentation So for cylindrical coordinate system one must define the system as: g = {g?[r, theta, z], g?[r, theta, z], gz[r, theta, z]} f = {f?[r, theta, z], f?[r, theta, z], fz[r, theta, z]} g?[r_, theta_, z_] = r g?[r_, theta_, z_] = theta gz[r_, theta_, z_] = z f?[r_, theta_, z_] = r^2 f?[r_, theta_, z_] = theta fz[r_, theta_, z_] = Cos[z] Then everything works fine: In[20]:= Div[CrossProduct[g,f]]===DotProduct[Curl[g],f]-DotProduct[Curl[f],g] Out[20]= True Hope this helps Pratik