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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.
To assist you, there is a worksheet associated with this lab that
contains examples to get you aquainted with the lab. You can copy that worksheet to your home directory with the following command, which must be run in a terminal window for example, not in Maple.
cp /math/calclab/MA1024/Surf_start.mws My_Documents
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.
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.
- For the given equations below, plot two dimensional level curves parallel to the plane and then plot two dimensional cross sections in the plane and 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.
- Create a contour plot for the function
for the values using two different methods; first using cross sections and then using Maple's contourplot command.
- Consider the following function
which looks like a deep valley with a mountain opposite it. Is is possible to find a path from the point to 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 want to sketch your path in by hand on an appropriate countour plot. You need to show at least 6 contours in your plot.
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