Subsections

• Purpose
• Getting Started
• Background
• Exercises

MA 1024: 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 by going to your computer's Start menu and choose run. In the run field type:

\\filer\calclab

when you hit enter, you can then choose MA1024 and then choose the worksheet

Surf_start_B13.mw

Remember to immediately save it in your own home directory. Once you've copied and saved the worksheet, read through the background on the internet and the background of the worksheet before starting 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.

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. A third way of representing a surface $z = f(x,y)$ is through the use of level curves. The idea is that a plane $z=c$ intersects the surface in a curve. The projection of this curve on the $xy$ plane is called a level curve. A collection of such curves for different values of $c$ is a representation of the surface called a contour plot. Similar to the idea of level curves is to look at cross sections of the surface to see what two-dimensional shape is traced, not only in the $xy$ plane by letting $z$ be constant, but also in the $yz$ plane by holding $x$ constant and the $xz$ plane by holding $y$ constant.

Exercises

1. Generate a surface plot and a contour plot with 30 contours for the following funtion on the given domain:
   
   $$f(x,y) = \frac{-x}{x^2+y^2+4} ~~ -5 \leq x \leq 5, ~~ -5 \leq y \leq 5$$
   
   

   a)

   Describe the difference in proximity between the contour lines in the regions where the surface plot has a steep incline compared to where the surface plot is almost flat?

   b)

   What can you say about the surface in the region where the contour plot looks like a series of nested circles?

   c)

   Rotate the 3-d graph and give an estimate of the extrema. (Extrema are the $z$ values of the highest and lowest points on the graph.)

   d)

   Visualize the $(x,y)$ coordinate point on the contour plot where you think the extrema occur. Evaluate the function at each of these points and compare to your estimate in part c.

2. For the given equations below, plot 2 two dimensional level curves parallel to the $xy$ plane and then plot 2 two dimensional cross sections in the $xz$ plane and again, 2 two dimensional cross sections in the $yz$ plane. Identify the type or shape of the quadric surface, ie. a sphere, cylinder, cone, elliptic cone, paraboloid, elliptic parabaloid, ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, elliptic hyperboloid of one or two sheets, or a hyperbolic parabaloid (saddle). Once you have determined the shape of the surface, supply a three dimensional plot to support your conclusion.

   a)

   
   

   $$z=x^2+y^2$$

   
   

   b)

   
   

   $$z^2=1+x^2+4y^2$$

   
   

3. Consider the following function $$f(x,y) = \frac{\sin(x)}{1+y^2}$$ for $0 \leq x \leq 2 \pi$ and $-3 \leq y \leq 3$ which looks like a deep valley with a mountain opposite it. Is is possible to find a path from the point $(0,3,0)$ to $(2\pi,-3,0)$ such that the value of $z$ is always between $-0.25$ and $0.25$ ? You do not have to find a formula for your path, but you must present convincing evidence that it exists. You need to show at least 6 contours in your plot and sketch your path on an appropriate contour plot. This can be done by first clicking on the plot, then click on the Drawing option in the Maple tool bar and choose the pen.