Saddle Surfaces

In this section we will examine surfaces in cylindrical coordinates in which z is a function of r and theta.

Question #2

The standard saddle surface in Cartesian coordinates is the graph of the equation z=x^2-y^2. Transform this equation to cylindrical coordinates by representing the surface as z=f(r,theta).
Maple does not plot functions of this sort. However, T. Murdoch of Washington and Lee University has written a Maple function called rtgraphplot that can graph cylindrical functions z=f(r,theta). The syntax is
read `/u/calcIII/calcplot.m`;
rtgraphplot(f(r,theta), r=rmin..rmax, theta=tmin..tmax);
(You only need to use the read command once per Maple session.

Question #3

Use the rtgraphplot command to plot the saddle surface in polar coordinates for r=0..1 and theta=0..2*Pi.

Question #4

Now plot the related functions

Question #5

You have seen graphs like these in a previous lab on discontinuous and nondifferentiable functions. Show that the function z= r*cos(2*theta) does not have a well-defined tangent plane at the origin. (Hint: compute the tangent planes to the surface on a circle of radius r, then show that as r approaches 0, these tangent planes do not converge.)
Up: Introduction
Previous: Surfaces in Cylindrical Coordinates
Robert E. Thurman<thurman@geom.umn.edu>
Document Created: Sat Jan 7 1995
Last modified: Tue Jan 14 13:30:39 1997