Subsections

• Purpose
• Getting Started
• Background
• Exercises

MA 1024A Lab 3: Surfaces in Cartesian Coordinates

Purpose

The purpose of this lab is to introduce you to some of the Maple commands that can be used to plot surfaces in three dimensions.

Getting Started

To assist you, there is a worksheet associated with this lab that contains examples and even solutions to some of the exercises. You can copy that worksheet to your home directory with the following command, which must be run in a terminal window, not in Maple.

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.

Background

The graph of a function of a single real variable is a set of points $(x,f(x))$ in the plane. Typically, the graph of such a function is a curve. For functions of two variables in Cartesian coordinates, the graph is a set of points $(x,y,f(x,y))$ in three-dimensional space. For this reason, visualizing functions of two variables is usually more difficult. For students, it is usually even more difficult if the surface is described in terms of polar or spherical coordinates.

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 $z = f(x,y)$ over a domain, e.g. a set of points in the $xy$ 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 $F(x,y,z) = C$, where $C$ is a constant. In this case, we say the surface is defined implicitly.

Exercises

1. Plot the graphs of the following functions over the given domains. Use the plot3d command.
   1. $f(x,y) = (2x^2+y^2)\exp(1-x^2-y^2)$, for $-2\leq x \leq 2$ and $-2\leq y\leq 2$.
   2. $g(x,y) = \cos(x)\sin(y)$, for $0\leq x\leq 2\pi$ and $0\leq y\leq 2\pi$.
   3. $h(x,y) = x^3-3xy^2$ over the interior of the ellipse $x^2/4+y^2/9 = 1$.
2. Use the implicitplot3d command to plot the graphs of the following equations. You should come up with plot ranges that show the surfaces clearly.
   1. The cylinder $x^2+y^2 = 4$.
   2. $x^2+\exp(-y)+z^2 = 4$. Can you explain why the surface only exists for $y \geq - \ln(4)$?
3. Consider the equation
   
   $$2x^2+4y^2+z^2-2xy+xz+yz + 4x -12 = 0$$
   
   
   In the book, it says that such an equation can be reduced by rotation and translation to one of the two forms
   
   $$Ax^2+By^2+Cz^2+J = 0$$
   
   
   or
   
   $$Ax^2+By^2+Iz = 0$$
   
   
   Use the implicitplot3d command to graph the surface corresponding to the equation given above. You should be able to identify the graph as one of the types of graphs, i.e. paraboloid, hyperboloid, or ellipsoid, shown in the text. Can you use your graph to determine which of the two forms shown above the equation can be transformed into? Do not try to do the transformation - you don't need to to answer the question.