Subsections

• Purpose
• Getting Started
• Background
• Exercises

MA 1024 Lab 2: Surfaces

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/Surfaces_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 Surfaces_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, 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.

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.

Two common ways of representing the graph of a function of two variables are the surface plot and the contour plot. The first is simply a representation of the graph in three-dimensional space. The second draws the level curves $f(x,y)=C$ for several values of $C$ in the $x,y$ plane. We will explore how to produce these kinds of graphs in Maple, and how to use the graphs to study the functions.

Exercises

1. Generate a surface plot and contour plot for the following functions on the given domains.
   1. $f(x,y)= 2x^2+xy+3y^2$ for $-3\leq x \leq 3$ and $-3 \leq y\leq 3$.
   2. $g(x,y) = 2x^2+4xy+y^2$ for $-3\leq x \leq 3$ and $-3 \leq y\leq 3$.
   What do the contour and surface plots tell you about the difference between these two functions? For example, which function has contours that look like nested ellipses? How is this the graph of this function different from the graph of the other function?

2. A group of oceanographers is mapping the ocean floor to assist in the recovery of a sunken ship. Using sonar, they develop the model
   
   $$D = \frac{-4x}{x^2+y^2+1}\;\;\;\;\;-5 \leq x \leq 0\,,\;\;-5 \leq y \leq 5$$
   
   
   where $D$ is the depth and $x$ and $y$ are the distances in kilometers.
   1. Graph the surface corresponding to the function $D$ over the given domain.

   2. Plot at least $5$ contours of the function $D$ and note the values of $D$ at these contours.

3. Can you find a path on the ocean floor that connects the point $(3,2,-6/7)$ to the point $(3,-2,-6/7)$ so that the depth is always between $-1$ and $-0.8$? (Hint - use a contour plot, and sketch your path by hand directly on the contour plot.)