   Next: About this document ... Up: lab_template Previous: lab_template

Subsections

# MA 1024: 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. On your Maple screen go to File - Open then type the following in the white rectangle:
\\filer\calclab\MA1024\Directional_Derivatives_and_Gradients_Start.mw


You can copy the worksheet now, but you should read through the lab before. Once you have read to the exercises,go to Maple and go through the worksheet carefully. Then 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

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

For the function ,
1. 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.
1. 2. 3. 2. 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?

3. 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.   Next: About this document ... Up: lab_template Previous: lab_template
Jane E Bouchard
2013-09-11