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. Examples of computing
gradients, using the gradient to compute directional derivatives, and
plotting the gradient field are all in the `Getting Started`
worksheet.

- Compute the three distinct second order partial derivatives of
the following function using either the Maple
`diff`command or the`D`operator.

- Consider
. Compute the directional
derivative of at the point in the direction of the vector
using each of
the two methods demonstrated in the
`Getting Started`worksheet. By using a contour plot and/or a surface plot, explain why the value you got for the directional derivative makes sense. - Consider the simple function
. It is easy
to discover by plotting this function that it has a maximum value of 3
at the origin. Demonstrate, with plots of the gradient field and an
explanation, how the gradient field can also be used to locate
(approximately) where the maximum occurs.
- Compute the gradient of the function

and plot the gradient field over the domain and . Use your plot to find the approximate location of the maximum value of this function over this domain. - In a previous lab, you used contour plots to get information about the
behavior of the graph of a function of two variables. Use the
technique demonstrated in the
`Getting Started`worksheet to plot the gradient field and a contour plot on the same graph for the function from the previous exercise. Use the same domain of and . Which plot (contour plot or gradient field plot) do you think gives you more information about the behavior of the function? Explain your answer.

2003-12-02