derivatives, and the gradient

cp ~bfarr/Pardiff_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_start.mws`, and go through it
carefully. Then you can start working on the exercises.

The Maple commands for computing partial derivatives are `D`
and `diff`. The **Getting Started** worksheet has examples
of how to use these commands to compute partial derivatives.

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 from the previous lab.

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. - Compute the two first order partial derivatives of the function
in the first exercise.
You may use either the
`D`or`diff`commands. Then evaluate them at the point , and explain the values you get in terms of the plots in the first exercise. - Consider again the function from the first exercise. Using
either method from the
`Getting Started`compute the directional derivative of at the point , in the three directions below. You may want to put an`evalf`command on the outside to get numerical values. - Using the method from the
`Getting Started`worksheet, plot the gradient field and the contours of on the same plot. Use the same 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 why the values were positive, negative, or zero in terms of the contours and the gradient field?

2003-11-30