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Peer Review

Project LINKS Module: The Gradient



Overall Numeric Rating:

4 stars
Content Quality: 4 stars
Effectiveness: 4 stars
Ease of Use: 4 stars
Reviewed: Jun 17, 2002 by Mathematics
Overview: This module provides a summary and motivation for the various properties of the gradient such as being the direction of steepest ascent and being the normal vector for level curves and surfaces. Since the module was not complete at the time of reviewing, it was reviewed up to the part ?Other coordinate systems?. The remainder appears to be in various stages of construction. The ratings in this review refer only to the part of the module actually reviewed.

Please also see the reviews of the following individual Project LINKS

as well as the review of the Project
LINKS Modules Gateway site
Type of Material: Should run in any browser.
Technical Requirements: Should run in any browser.

The modules are intended to be used in a studio classroom setting for a day or two, with a professor and TA available. Thus many questions are open-ended. The user should be aware of this underlying idea even though, unlike most Project LINKS modules, this one explicitly states that it may be useful for study in or out of the classroom or laboratory.
Also see the ASSUMPTIONS
page on the Project LINKS
Modules Gateway site
Identify Major Learning Goals: To show the various uses of the gradient vector in applications and in
mathematics. From the module objectives:

?You will:

1. Understand that the gradient is a vector that points in the direction
of the greatest rate of increase of a function;

2. Visualize and sketch gradient vectors;

3. Calculate the gradient of functions of two and three variables in
several coordinate systems;

4. Know properties of the gradient that make it useful in physical
and mathematical applications, and in recreational activities, such as
hiking, skiing, and mountain climbing.?
Target Student Population: Students taking physics, engineering or a mathematics class that require the use of gradients.
Prerequisite Knowledge or Skills: Multivariable calculus up to partial derivatives. Elementary knowledge of vectors.

Evaluation and Observation

Content Quality

Rating: 4 stars
Strengths: Visual applets that show the gradient for various functions and challenge the reader to sketch steepest incline paths are useful to show the concepts behind the gradient. The text presentation of the gradient is fairly standard, though in an unorthodox order.
Concerns: 1. In the first example on computing a 3d gradient it is claimed that the
gradient is defined except when z=0. In fact, the gradient is defined only
for z>0, since we take a square root of z in the function and in the gradient.

2. The section on the gradient in other coordinate systems is under
development, but even for the sections that look done the ?unit vectors
in the respective directions? should be defined.While the applets provided
are interesting and useful, they are actually a small part of the module,
which consists largely of standard equations and (sometimes) their derivations.

3.  While it is certainly useful to have this material online,
it can be found in any textbook on vector calculus, with better typography. 
(A simple, universally available method for putting complicated mathematical
equations in web pages unfortunately does not yet exist.)

4.  Most interesting properties of the gradient are assumed rather
than derived or motivated.  For example, the first page of the module,
?Rapid Change Vector?, simply lists the geometric properties defining the
gradient of a function in terms of the greatest increase of the function. 
Later pages ask the student to ?recall? that the gradient is normal to
level curves or surfaces.  When the module turns to (Cartesian) co-ordinate
systems, the reader is simply informed of the formula for the gradient
in terms of partial derivatives.  It seems that the authors of the
module have missed an opportunity to use graphics, and particularly the
interactivity possible with Java applets, to motivate some of these steps.

Potential Effectiveness as a Teaching Tool

Rating: 4 stars
Strengths: The first activity on the ?gradient as the normal vector to level curves? drives home the point that the gradient is just that. The second activity shows the use of the gradient in constructing most direct paths in a potential field. An activity in which students are to determine the direction of the gradient vector at various points helps reinforce these ideas.
Concerns: The second activity on the ?gradient as the normal vector to level curves? requires some instruction how to generate the desired paths. Moreover, some more interesting examples of functions could have been chosen. There is only one example in which the paths of steepest ascent are not straight lines.

Ease of Use for Both Students and Faculty

Rating: 4 stars
Strengths: No problems in the use of the module. All applets loaded correctly.
Concerns: 1. On the second page of the skiing example there is no new information on the skiers? paths. The caption to the left seems to indicate there should be.

2. On the ?Geometric Approximation to the Gradient? page the authors refer to a figure/activity that is on the next page without mentioning this. This forward referencing can cause confusion.

3. Tfact that the site is unfinished is cause for some concern.