cp /math/calclab/MA1024/Pardiff_grad_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 `Pardiff_grad_start.mws`, and go through it
carefully. Then you can start working on the exercises.

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 i t comes to functions of two variables, because there are many more comp utations required to produce one graph, yet Maple performs all these com putations 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 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.

to compute the directional derivative. However, the following computation, based on the definition, is often simpler to use.

One way to think about this that can be helpful in understanding directional derivatives is to realize that is a straight line in the plane. The plane perpendicular to the plane that contains this straight line intersects the surface in a curve whose coordinate is . The derivative of at is the rate of change of at the point moving in the direction .

Maple doesn't have a simple command for computing directional
derivatives. There is a command in the `tensor` package that
can be used, but it is a little confusing unless you know something
about tensors. Fortunately, the method described above and the method
using the gradient described below are both easy to implement in
Maple. Examples are given in the `Getting Started` worksheet.

As described in the text, the gradient has several important properties, including the following.

- The gradient can be used to compute the directional derivative
as follows.

- The gradient points in the direction of maximum increase of the value of at .
- The gradient is perpendicular to the level curve of that passes through the point .
- The gradient can be easily generalized to apply to functions of three or more variables.

Maple has a fairly simple command `grad` in the `linalg`
package (which we used for curve computations). Examples of computing
gradients, using the gradient to compute directional derivatives, and
plotting the gradient field are all in the `Getting Started`
worksheet.

- Consider the following function.

- A)
- First, plot the graph of this function over the domain
and
using the
`plot3d`command. Then use the`contourplot`command to generate a contour plot of over the same domain having 20 contour lines. - B)
- What does the contour plot look like in the regions where the surface plot has a steep incline? What does it look like where the surface plot is almost flat?
- C)
- What can you say about the surface plot in a region whe re the contour plot looks like a series of nested circles?

- Consider the function from exercise 1 which looks like a valley with a mountain opposite it. Is it p
ossible to find a path from the point
to
such that the valu
e of is always between and ? You do not have to
find a formula for your path, but you must present convincing evi
dence that it exists. For example, you might want to sketch your p
ath in by hand on an appropriate countour plot.
- Consider again the function from the first exercise. Using
either method from the
`Getting Started`worksheet, compute the directional derivative of at the point , in the three directions below.- A)
- B)
- C)

- Using the method from the
`Getting Started`worksheet, plot the gradient field and the contours of on the same plot. Use the domain of and . Can you use this plot to explain the values for the directional derivatives you obtained in the previous exercises? By explaining the values, I only mean can you explain what kind of surface it is and why the values were positive, negative, or zero in terms of the contours and the gradient field?

2008-09-24