Maple #6 is due Thursday 31 Oct 2002

For FULL credit STAPLE your sheets together.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Actually unstapled assignments will not be accepted.

You are to plot the five functions below. Each plot must be rotated to 
a "nice" position and "look good (smooth)" . Each plot must have include
axes and the title must include your name and what was plotted. Note
that the expressions below are not necessarily in a form that Maple
likes. 


To plot the surface <x(u, v), y(u, v), z(u, v)> a <= u <= b, c <= v <= d
with Maple use plot3d([x(u, v), y(u, v), z(u, v)], u=a..b, v=c..d)

1&2 Plot the following surfaces from u=0..2, v=0..4 (from Test 3 Fall 98)
Each plot is separate.
  1a. <cos v, sin v, u>
  1b. <(2+u)cos 2v, (2+u)sin 2v, v>
  2a. <u cos v, u sin v, u>
  2b. <sin u cos v, sin u sin v, cos u>

3.  Line integrals. Use maple to draw  curtain plots like in the
maple worksheet of Oct 24 (or see below).
Eventually each line integral over a curve C reduces to integrating a
function like int(g(t),t=a..b). What we want is a `curtain' of height
g(t) over the curve C. If C is given by <x(t), y(t)>, t=a..b then
we want to plot the surface <x(t), y(t), s*g(t)> t=a..b, s=0..1.
We can do this with plot3d as follows
plot3d([x(t),y(t),s*g(t)],t=a..b,s=0..1,title=`line curtain`,axes=BOXED);
Draw the curtain for C the half the unit circle from (1,0) to (-1,0)
counterclockwise and F the vector field <y,0>

4. Plot a torus as a surface. It is the circle in the xz-plane
given by (x-2)^2+z^2=1 rotated about the z-axis. Be sure to give
the parametric equations of the surface.

5.  Plot a Klein bottle. The following should help. The handle and
bulb `fit together' to make one smooth surface.
c:=.6;a:=.2;
plot1 the handle:
Range u=Pi/2..5*Pi/2,v=Pi/4..5*Pi/4
A:=c*(cos(v)*sin(v) -0.5 + a*sin(u)*sin(v)/sqrt(sin(v)^2+cos(2*v)^2));
B:=a*c*cos(u);
C:=cos(v)+a*c*sin(u)*cos(2*v)/sqrt(sin(v)^2+cos(2*v)^2);
parametric equations of this part of the surface [A, B, C]

plot2 the bulb: v=5*Pi/4..9*Pi/4
Range u=Pi/2..5*Pi/2,v=5*Pi/4..9*Pi/4
r:=sin(v)*cos(v) - (a+1/2);
A:=c* sin(u) * r;
B:= -c * cos(u) * r;
C:=cos(v);
parametric equations of this part of the surface [A, B, C]