Subsections

# MA 1024 Lab 4: Partial derivatives, directional derivatives, and the gradient

## Purpose

The purpose of this lab is to acquaint you with using Maple to compute partial derivatives, directional derivatives, and the gradient.

## Getting Started

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.

## Background

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.

### Partial derivatives

A differentiable function, , of two variables has two partial derivatives: and . 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.

### Directional derivatives

The partial derivatives and of can be thought of as the rate of change of in the direction parallel to the and axes, respectively. The directional derivative , where 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 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.

The gradient of , written , is most easily computed as

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.

## Exercises

1. 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.

2. 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.

3. 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 two directions below. You may want to put an evalf command on the outside to get numerical values.

4. Using the method from the Getting Started worksheet, plot the gradient field and the contours of on the same plot. You should prbably use smaller domain like and to help you see what is going on. 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?