cp ~bfarr/Surf_start.mws ~

You can copy the worksheet now, but you should read through the lab
before you load it into Maple. Once you have read to the exercises,
start up Maple, load
the worksheet `Surf_start.mws`, and go through it
carefully. Then you can start working on the exercises.

One of the most valuable services provided by computer software such as Maple is that it allows us to produce intricate graphs with a minimum of effort on our part. This becomes especially apparent when it comes to functions of two variables, because there are many more computations required to produce one graph, yet Maple performs all these computations with only a little guidance from the user.

The simplest way of describing a surface in Cartesian coordinates is as the graph of a function over a domain, e.g. a set of points in the plane. The domain can have any shape, but a rectangular one is the easiest to deal with. Another common, but more difficult way of describing a surface is as the graph of an equation , where is a constant. In this case, we say the surface is defined implicitly. A third way of representing a surface is through the use of level curves. The idea is that a plane intersects the surface in a curve. The projection of this curve on the plane is called a level curve. A collection of such curves for different values of is a representation of the surface called a contour plot.

Some surfaces are difficult to describe in Cartesian coordinates, but easy to describe using either cylindrical or spherical coordinates. The obvious examples are cylinders and spheres, but there are many other situations where these coordinate systems are useful.

- Generate a surface plot and contour plot for the
following function on the given domain:

for and .- What does the contour plot look like in the regions where the surface plot has a steep incline? What does it look like where the surface plot is almost flat?
- What can you say about the surface plot in a region where the contour plot looks like a series of nested circles?

- Generate a surface plot for the following functions on the domains
given.
- for and . Use cylindrical coordinates.
- for and . Use spherical coordinates.

- Consider the function for and , which looks like a deep valley with a mountain opposite it. Is it possible to find a path on the surface from the point to the point such that the value of is always between and ? You do not have to find a formula for your path, but you must present convincing evidence that it exists. For example, you might sketch your path in by hand on an appropriate contour plot.

2003-04-01