Problems

Problem

Find equations in Cartesian coordinates for planes through the following set of points (in the given coordinate systems):  {1,2,2}, {2,3,5}, {3,6,3} (ParabolicCylindrical)

Problem

Graph the following surface:   u=v Cos[φ+v] in Paraboloidal coordinates

Problem

Find the equation for the tangent plane to the above surface at the point {,,} in {u,v,φ}.  Graph that plane and the surface on the same coordinate axes (as I did in the example), to make sure you aren't hallucinating.

Problem

Graph the following curve:

{t Sin[t],t Cos[t],t}in Paraboloidal and Toroidal coordinates for 0.5≤t≤π.   (Assume the parameter a=1 in the definition.  Warning: you may have to make slight changes in your domain for this to work (hint: in Toroidal coordinates, don't let v go all the way in to 0).  :  Mathematica seems a little confused about the order of the coordinates in Toroidal; in the Help System, they are listed in the order {u,v,φ}, but if you do Coordinates[Toroidal], it lists them as {v,u,φ}.  For our purposes, let's use the ordering given in the program, {v,u,φ}.)

Find the arc-length of this curve over this interval in each coordinate system.  
Should the two arc-lengths be the same?  Why or why not?  (Hint: use NIntegrate to work the integrals.  Trying to work them exactly is a really bad idea...)

Created by Mathematica  (April 3, 2007)