derivatives, and the gradient

cp /math/calclab/MA1024/Gradient_start_D09.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 `Gradient_start_D09.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.

- Using method 2 from the
`Getting Started`worksheet, compute the directional derivative of at the point in each of the directions below. Explain your results in terms of being positive, negative or zero and what that tells about the surface at that point in the given direction. - Now, find the directional derivative at each of the points and in the direction in
as well as another direction perpendicular to this. (Be sure to use unit direction vectors each time.) What do your results suggest about the surface at these points? What is the difference between a zero answer in this exercise compared to the zero answer from exercise 1?
- Using the method from the
`Getting Started`worksheet, plot the gradient field and the contours of on the same plot over the intervals and . Use 30 contours, a grid and`fieldstrength=fixed`for the gradient plot. Describe the surface of at the points and using both the gradient field and the countour plot in your explanation.

2009-11-09