Subsections

# 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_C12.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  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  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 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. 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 plane by letting be constant, but also in the plane by holding constant and the plane by holding constant.

## Exercises

1. Use the command plot3d to generate a surface plot with the option ,axes=boxed and the command contourplot to generate a contour plot with 30 contours for the following funtion on the given domain:

a)
Describe the difference in proximity between the contour lines in 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 regions 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 values of the highest and lowest points on the graph.)
d)
Use an approximate coordinate location from the contour plot where you think the extrema occur and evaluate the function at each of these points to see if your estimations in part c were close.
2. Consider the following function

which represents the deviation, in inches, of last year's rainfall from the average annual rainfall in a certain area.
a)
Use the plot3d command to graph the surface corresponding to the function . Use axes that are boxed.
b)
What are (approximately using values from the plot) the maximum and minimum values of the rainfall deviation?
c)
Use a contour plot and shade in the region on your printout of this domain in which the deviation was between inches and inches.
3. For the given equations below, plot 2 two dimensional level curves parallel to the plane and then plot 2 two dimensional cross sections in the plane and again, 2 two dimensional cross sections in the 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)

b)