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The purpose of this lab is to acquaint you with using Maple to compute
partial derivatives, directional derivatives, and the gradient.
To assist you, there is a worksheet associated with this lab that
contains examples. You can
copy that worksheet to your home directory with the following command,
which must be run in a terminal window, not in Maple.
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.
For a function of a single real variable, the derivative
gives information on whether the graph of is increasing or
decreasing. Finding where the derivative is zero was important in
finding extreme values. For a function of two (or more)
variables, the situation is more complicated.
A differentiable function, , of two variables has two partial
. As you have learned in class, computing partial derivatives is
very much like computing regular derivatives. The main difference is
that when you are computing
, you must treat
the variable as if it was a constant and vice-versa when computing
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.
The partial derivatives
of can be thought of as the rate of change of in
the direction parallel to the and axes, respectively. The
is a unit vector, is the rate of change of in the
direction . There are several different ways that the
directional derivative can be computed. The method most often used
for hand calculation relies on the gradient, which will be described
below. It is also possible to simply use the definition
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
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.
The gradient of , written , is most easily computed as
As described in the text, the gradient has several important
properties, including the following.
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
- Compute the three distinct second order partial derivatives of
the following function using either the Maple diff command or
the D operator.
. 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
. 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
. Which plot (contour plot
or gradient field plot) do you think gives you more information about
the behavior of the function? Explain your answer.
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William W. Farr