Problems

Find equations in Cartesian coordinates for planes through the following sets of points (in the given coordinate systems):

a. {2,,}, {1,,}, {3,,} (spherical)
b. {1,2,2}, {2,3,5}, {3,6,3} (ParabolicCylindrical)

Graph the following surface:

u=v Cos[φ+v] in Paraboloidal coordinates

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).

Graph the same equation as above, but this time in Toroidal coordinates.  (Assume the parameter a=1 in the definition.  Warning: you may have to make slight changes in your domain for this to work.  :  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 Help System.)

Graph the following surface:

{s Cos[t],s Sin[t],} in Paraboloidal coordinates and in Toroidal coordinates (i.e., using the same formulas, graph it once interpreting them as (u,v,φ} in Paraboloidal and a second time interpreting them as {u,v,φ} in Toroidal).

Graph the following curve:

{t Sin[t],t Cos[t],t}in Paraboloidal and Toroidal coordinates for 0.5≤t≤π (see the previous problem for an explanation).  

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  (March 29, 2006)