Problems

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

a. {2, π/3, π/6}, {1, π/12, π/3}, {3, π/4, π/4} (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 {(-7π)/12, (7π)/6, π/6} in {u, v, ϕ}.

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.)

Graph the following surface:

{u Cos[v], u Sin[v], u^2} in Paraboloidal coordinates and in Toroidal coordinates

Graph the following curve:

{t Sin[t], t Cos[t], t}in Paraboloidal and Toroidal coordinates for FormBox[RowBox[{0.5, ≤, t, ≤, π}], TraditionalForm].  Find the arc-lenth of this curve over this interval in each coordinate system.  Should they be the same?  Why or why not?  (Hint: use NIntegrate to work the integrals.)

Created by Mathematica  (October 18, 2004)