Math 21a, Fall 2008
Problem A of Week 4, Math 21a, Multivariable Calculus
Problem on parametrized Surfaces
Course head: Oliver Knill
Office: SciCtr 434

Problem A of Week 4

Problem A: Lets look at the parametrized surface
r(t,s) =  < t cos(s), t sin(s), s t > 
where s is between 0 and 6Pi and t is between 0 and 1. We call it the rose surface.
a) (2 points) If t is kept constant (like for example t=1), we obtain a grid curve on the surface. Identify these space curves algebraically and by name. Similarly, if s is kept constant (like for example s=0), we obtain grid curves on the surface. Identify these space curves by formula and by name.
b) (3 points) Find a formula for the curve obtained by intersecting the surface with the plane z=1. Sketch these planar curves in the plane. Find also the intersection of the surface with the plane y=0 as well as with the plane x=0 and sketch them in the xz-plane and yz-plane.
c) (3 points) Sketch the surface by making use of part a) and b). You are welcome to use use Mathematica if you like (*) but you can do the problem without technology.
d) (1 points) If the parameter range of the s parameter is increased from [0,6Pi] to [0,10 Pi], the shape of the rose changes. Check what applies
A: The surface gains more turns (it appears as if the rose gets more leaves)
B: The rose gets taller (the z coordinate is the "up" direction)
C: The rose gets wider (the x,y coordinates increase)
No explanations are needed in d). Just note the letters in A,B,C which apply.
e) (1 points) If the range of the t parameter is increased from [0,1] to [0,3], the shape of the rose changes. Check what applies.
A: The surface gains more turns (it appears as if the rose gets more leaves)
B: The rose gets taller (the z coordinate is the "up" direction)
C: The rose gets wider (the x,y coordinates increase)
No explanations are needed in e). Just write down the letters in A,B,C which apply.


(*) Here is an example on how to use it: to plot a cylinder in Mathematica, enter
ParametricPlot3D[ { Cos[s], Sin[s],t },{s,0,2Pi},{t,0,1}]
into a new cell, click into the cell, hold the shift key and hit return. See the lab page.
Questions and comments to knill@math.harvard.edu
Math21b | Math 21a | Fall 2008 | Department of Mathematics | Faculty of Art and Sciences | Harvard University