CHAPTER 2

LITERATURE REVIEW

2.1 INTRODUCTION

Our goal is to create a system to assist students in learning how to compute the

integrals that appear in vector calculus. To have a better understanding of vector

calculus, students must learn to apply appropriate formulas and also to understand

geometric regions well. We begin with a review of six well known computer algebra

systems (CASs): Derive, Maple, Mathcad, Mathematica, MuPAD and REDUCE, and

discuss their capabilities for solving problems in the area of vector calculus. We also

give an overview of three well known vector calculus packages designed to assist in

understanding of vector calculus: Vector Calculus & Mathematica (Davis, Porta &

Uhl, 1999), General Vector Analysis (GVA) (Qin, 1999) and vec_calc (Yasskin &

Belmonte, 2003). Additionally, to build an effective package, it is important to design

it based on computer learning theories, e.g. behaviorism theory, cognitive theory and

humanism theory.

2.2 COMPUTER ALGEBRA SYSTEMS (CASs)

A computer algebra system (CAS), the computer software developed in the field of

computer algebra (symbolic computation), is a math engine that helps to rapidly

perform the computations fundamental to algebra, trigonometry, and calculus:

evaluating, factoring, combining, expanding, and simplifying terms and expressions

containing symbols, integers, fractions, and real and complex numbers. The CASs

such as Derive, Maple, Mathcad, Mathematica, MuPAD and REDUCE, are in

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addition able to perform integration, differentiation, matrix and vector operations,

standard deviations and many other more complex computations involved in calculus,

linear algebra, differential equations and statistics. Additionally, they also allow the

creation of 2D and 3D plots of polynomials, trigonometric functions, exponentials,

etc. (http://www.mackichan.com/products/CAS.htm).

The representation of mathematical objects in a symbolic rather than numeric

computational form has existed since the early days of computer science. Indeed,

many numerical methods are based on symbolic forms. Thus, the notion of “symbolic

computation” is not new to numerical methods. The 1970s and 1980s have seen the

development, however, of environments that place a greater emphasis on computation

with mathematical objects in an implicit or symbolic form. Symbolic computation is

based on defining objects not as numerical quantities, but as entities that have certain

mathematical properties (Fiume, 1995). For instance, among the many properties of

is that it has a non-terminating, non-repeating numeric representation. A famous

mathematician, Abu Ja’far Muhammad bin Musa al-Khwarizmi (c. 830), said

regarding :

And this is an approximation and not a precise determination [of ]. Nobody can determine the exact value of that and know the circumference, except Allah. For this curve [the circle] is not straight and cannot be determined except approximately. That is called an approximation, just as the root of a number is an approximation and not the exact value; nobody knows it except Allah (Joachim von zur Gathen & Gerhard, 1999).

However, there are some fundamental non-numeric mathematical relationships

in which is involved. For example:

sin xx

x

Furthermore, Leibnitz’ formula states:

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and Euler’s formula is:

2

6

n1

1n2

There are a great many such relationships that exist. The algebraic effect of applying

certain functions on is also well known such as sin = 0 and e i + 1 = 0 (Fiume,

1995).

An environment supporting symbolic computation facilitates the manipulation

and combination of these relationships. The output of a symbolic computation is often

another symbolic quantity such as a series or other mathematical object that is

unevaluated, in the sense that a numeric representation has not been explicitly

computed. The ability to defer numeric evaluation and concentrate on symbolic

manipulation distinguishes these computing environments from traditional numerical

approaches. At some point, an expression may be evaluated, yielding a numerical

quantity, but this is neither always necessary nor desirable (Fiume, 1995).

CASs have historically evolved in three stages. An early forerunner was

Williams’ (1961) PMS, which could calculate floating point polynomial gcd’s. The

first generation, beginning in the late 1960s, comprised MACSYMA from Joel

Moses’s MATHLAB group at MIT, SCRATCHPAD from Richard Jenks at IBM,

REDUCE by Tony Hearn, and SAC-I (now SACLIB) by George Collins. MuMATH

by David Stoutemyer ran on a small microprocessor; its successor Derive is available

on the hand-held TI-92. These researchers and their teams developed systems with

algebraic engines capable of doing amazing exact (or formal or symbolic)

computations: differentiation, integration, factorization, etc. (Joachim von zur Gathen

& Gerhard, 1999).

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The second generation started with Maple by Keith Geddes and Gaston

Gonnet from the University of Waterloo in 1985 and Mathematica by Stephen

Wolfram. They began to provide modern interfaces with graphic capabilities, and the

hype surrounding the launching of Mathematica did much to make these systems

widely known. The third generation: AXIOM, a successor of SCRATCHPAD, by

NAG, MAGMA by John Cannon at the University of Sydney, and MuPAD by Benno

Fuchssteiner at the University of Paderborn, incorporated a categorical approach and

operator calculations. MuPAD is designed to work ultimately also in a multiprocessor

environment (Joachim von zur Gathen & Gerhard, 1999).

Today’s research and development of CASs is driven by four goals: wide

functionality (the capability of solving a large range of different problems), ease of

use (user interface, graphics display), speed (how big a problem you can solve with a

routine calculation, say in a day), and robustness (correct answer without the program

crashing). They also have a wide variety of applications in fields that require

computations that are tedious, lengthy and difficult to solve correctly when done by

hand. For instance, CASs are used in high energy physics, for quantum

electrodynamics, quantum chromodynamics, satellite orbit and rocket trajectory

computations and celestial mechanics in general (Joachim von zur Gathen & Gerhard,

1999). Additionally, their power of visualization and of solving nontrivial examples

makes them more and more appealing for use in education. Many topics in

mathematics such as calculus, vector calculus and linear algebra, can be beautifully

illustrated with this technology (Lambe, 1997).

CASs have their own idiosyncrasies, ranging from a partly inconsistent syntax

to actual errors. In order to use them successfully, it is very important to be aware of

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not only of their positive features, but also of their current limitations. In this research,

we refer to the following six well known CASs (Cohen, 2003):

2.2.1 Derive

Derive is a mathematical computer program. It processes algebraic variables,

expressions, equations, functions, vectors, and matrices like a scientific calculator

processes floating point numbers. Additionally, it can also perform numeric and

symbolic computations, algebra, trigonometry, calculus and plot graphs in two and

three dimensions (http://www.derive.com).

The main strengths of Derive are symbolic algebra and powerful graphics. It is

an excellent tool for doing and applying mathematics and documenting mathematical

work. With regards to education, Derive is an ideal tool for teaching and learning of

mathematics. It also increases the student’s ability in the understanding of

mathematics by providing seamless integration of numeric, algebraic and graphic

capabilities. Furthermore, it has proven to be highly supportive for the development of

understanding of advanced mathematical concepts (Kutzler & Kokol-Voljc, 2003).

2.2.2 Maple

Maple is a very large CAS originally developed by the Symbolic Computation Group

at the University of Waterloo, Canada and now distributed by Waterloo Maple Inc. It

is one of the most fully developed CASs currently on the market. Its correct

application saves the users from long manual calculations, which are obviously open

to errors (http://www.maplesoft.com).

Maple is also both a programming language (optimized for mathematical

applications) and a very powerful graphical tool for the visualization of complex

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mathematical relations. The structure of Maple is made up of three components:

kernel, library and interface ( Kofler, 1997; Corless, 1995; Schwartz, 1999; Heal,

Hansen & Rickard, 1996).

2.2.3 Mathcad

Mathcad is produced by Mathsoft Engineering & Education, Inc. It is a technical

calculation tool for professionals, educators and college students worldwide.

Additionally, it is also as versatile and powerful as a programming language

(http://www.mathsoft.com).

In Mathcad, the equations look the way the user sees it on a blackboard or in a

reference book. The syntax is not difficult to learn; simply point and click and the

equations will appear. Furthermore, Mathcad equations are able to solve numerous

mathematics problems, symbolically and numerically. The users can place text

anywhere around the equations to document their work and also visualize the

equations with Mathcad’s two and three dimensional plots (Mathcad 11 User’s Guide,

2002).

2.2.4 Mathematica

Mathematica is a very large CAS developed by Wolfram Research Inc. First released

in 1988, it has had a profound effect on the way computers are used in various fields

such as physical sciences, biological sciences, social sciences, engineering, commerce

and computer science (http://www.wolfram.com). The creator of Mathematica,

Wolfram (1999), said:

Ever since the 1960s individual packages had existed for specific numerical, algebraic, graphical and other tasks. But the visionary concept of Mathematica was to create once and for all a single system that could handle all the various

29

aspects of technical computing in a coherent and unified way. The key intellectual advance that made this possible was the invention of a new kind of symbolic computer language that could for the first time manipulate the very wide range of objects involved in technical computing using only a fairly small number of basic primitives.

2.2.5 MuPAD

MuPAD, developed mainly at the University of Paderborn, Germany, is a modern,

full-featured CAS in an integrated and open environment for symbolic and numeric

computing. It has a comfortable notebook interface that includes a graphic tool for

visualization, an integrated source-level debugger and hypertext help. Its aim is to

address mathematicians, engineers, computer scientists and more generally all those in

need of mathematical computations for education or profession

(http://www.sciface.com).

There are two ways to use MuPAD or a CAS in general. The first, users may

use the mathematical knowledge incorporated in MuPAD by calling the system

functions interactively. For instance, users can compute a symbolic integral by calling

an appropriate function to perform the task. The second, users can easily utilize the

MuPAD’s programming language to add functionality to the system by implementing

their own algorithms as MuPAD procedures. This is useful for special purpose

applications if no appropriate system functions exist (Gerhard, Oevel, Postel &

Wehmeier, 2000).

2.2.6 REDUCE

REDUCE is an interactive program designed for general algebraic computations of

interest to mathematicians, scientists and engineers. Its capabilities include expansion

and ordering of polynomials and rational functions, substitutions and pattern matching

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in a wide variety of forms, automatic and user controlled simplification of

expressions, calculations with symbolic matrices, arbitrary precision integer and real

arithmetic, facilities for defining new functions and extending program syntax,

analytic differentiation and integration, factorization of polynomials and facilities for

the solution of a variety of algebraic equations (Neun, 2000).

For further information on the above CASs, please refer Appendix A (Wester,

1999).

2.3 THE CAPABILITIES OF CASs IN MULTIDIMENSIONAL INTEGRATION AND VECTOR CALCULUS

In this research, we created a review table that consists of the above mentioned CASs

(Wester, 1999). The purpose of this table is to provide a benchmark for the

comparison of their capabilities in multidimensional integration and vector calculus

(refer Table 2.1).

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Table 2.1A review table of the capabilities of CASs in multidimensional

integration and vector calculus

Maths operations De Mp Mc Mm Mu Re

1 Interval integral: f dx

A1, C1 A1, C1 A1, C1 A1, C1 A1, C1 A1, C1

2 Curve integral:C f ds

A1, C2 A1, C2 A1, C2 A1, C2 A1, C2 A1, C2

3 Work:C F dR

A1, C2 A1, C2 A1, C2 A1, C2 A1, C2 A1, C2

4 Area integral: f dA

a Iterated integral

A1, C1 A1, C1 A1, C1 A1, C1 A1, C1 A1, C1

b General 2D area integral as sum of iterated integrals

C3 C3 C3 C3 C3 C3

5 Surface integral:S f dS

A1, C2 A1, C2 A1, C2 A1, C2 A1, C2 A1, C2

6 Volume integral (iterated integral):V f dV

A1, C1 A1, C1 A1, C1 A1, C1 A1, C1 A1, C1

Notations:

(i) The accuracy (A) of the solution:

A1 Correct solution A2 Partial success, but also partially incorrect

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(ii) The capability (C) to perform the task:

C1 Easy (built-in function exists)C2 Not so easy (need to do some programming)

C3 Doesn’t have capability

(iii) CASs:

De Derive 6Mp Maple 9Mc Mathcad 11Mm Mathematica 5.2Mu MuPAD Pro 3.0Re REDUCE 3.8

From Table 2.1, we can see that all the six CASs do not have the capability to

solve a general 2D area integral as a sum of iterated integrals. For example, none of

the six CASs have the capability to find the area of the intersection of the inequalities

y 1+x, y 1-x, y -1+x, y 0.5, y -1-x and y -0.5 automatically. Thus, in this

research, we want to develop the algorithms needed to compute general two

dimensional integrals as a sum of iterated integrals successfully and the program can

also explain the steps to do this (refer Appendix C).

2.4 VECTOR CALCULUS PACKAGES/COURSEWARES

In this section, we discuss briefly three existing well known vector calculus

packages/coursewares that are not embedded in CASs:

2.4.1 Vector Calculus & Mathematica (VC&M)

In their years of teaching mathematics courses such as calculus, vector calculus and

matrix theory, Professor Bill Davis (Ohio State University, USA), Professor Horacio

Porta (University of Illinois, USA) and Professor Jerry Uhl (University of Illinois,

USA), had watched the students and the world change, while the mathematics and its

33

teaching method did not. Their former and current mathematics students agreed that

the mathematics they were studying wasn’t meeting their needs, wasn’t interesting and

exciting enough and was taken only to fulfill prerequisites. Therefore, the three

professors began to write and teach courses based on interactive electronic lessons.

They wanted students to understand and appreciate the underlying mathematical

concepts as well as be able to solve mathematical problems. They wanted to engage

the students in the enjoyment of learning mathematics and excite them about its

possibilities, while grounding them firmly in its truth and intuitions. The product of

their work is the courseware Vector Calculus & Mathematica (Davis, Porta & Uhl,

1999).

Furthermore, based on their teaching experience, these professors found that

for some students, learning mathematics using the traditional ways worked well, but

unfortunately for most they didn’t. Are the students who didn’t do well dumb? Why

didn’t they do well? Who did well? The answer is that lots of these people process

information (physically) differently from the ones perceived as mathematically gifted.

Thus, the purpose of the creation of Vector Calculus & Mathematica was an attempt

toward making the ideas of mathematics such as calculus, vector calculus and matrix

theory clearer by accommodating some different learning styles. The target group of

this courseware is generally categorized as the “visual learners” (Davis, Porta & Uhl,

1999).

The contents of Vector Calculus & Mathematica is divided into two parts. The

first part includes the topics from calculus such as parametric plotting, 2D integrals

and the Gauss-Green formula. The second part includes the topics from vector

calculus such as vectors, perpendicularity, gradient, 2D vector fields and their

trajectories, flow measurements by integrals, sources, sinks, swirls, singularities,

34

transforming 2D integrals, transforming 3D integrals, spherical coordinates, 3D

surface measurements and 3D flow (Davis, Porta & Uhl, 1999).

Vector Calculus & Mathematica is composed of four modules (Davis, Porta &

Uhl, 1999):

i. Basics

This module is for the basic new ideas of the content of Vector Calculus &

Mathematica.

ii. Tutorials

This module contains the sample uses of the basic ideas.

iii. Give It a Try

This module states the problems for the users to do by computer or otherwise.

Doing problems from this module is how the users learn in this courseware.

The Give It a Try problems are the heart of the courses.

iv. Literacy Sheet

The problems in this module should be handled away from the computer after

the users have completed the assigned Give It a Try problems. These are the

problems the users can always do at their leisure.

The Basics and the Tutorials are full of successful Mathematica code which

users can copy, paste and edit and adapt to any situation they want. Copying, pasting

and editing existing Mathematica code minimizes the need to learn lots of

Mathematica and makes Vector Calculus & Mathematica easy to navigate. It is not

necessary at all for the users to type all the Mathematica code from scratch.

Copying-pasting-editing plays a major role in the Vector Calculus &

Mathematica system. Users must know when and what to copy, what to edit and what

the results mean. This intellectually heavy activity separates the mathematics of this

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courseware from the programming of Mathematica. Furthermore, a good use of

copying-pasting-editing reinforces mathematical understanding. In other words, users

need to read the code they copy and paste for understanding of what is being

calculated. When they copy and paste and edit, they need to read the commands for

mathematical content and ask themselves what that calculation did, how it did it, and

how it pertains to the problem at hand.

In using this courseware, users are free to use any method they are comfortable

with. It is recommended that they try different approaches until they have settled in on

a method that works best for them. However, two approaches are recommended by the

authors (Davis, Porta & Uhl, 1999):

i. Random Approach

The users can browse through Give It a Try to get a problem and then explore

the Basics and Tutorials for ideas, techniques and Mathematica code that will

assist them in solving the problem. Almost all of the problems in the courses

have their roots in the Basics and Tutorials.

ii. Sequential Approach

The users can read the Basics and Tutorials before beginning their work on the

Give It a Try problems.

There are also several techniques of solving Give It a Try problems (Davis,

Porta & Uhl, 1999):

i. Learning via editing and experimenting with existing code.

Many potential CAS users are turned off by what appears to be complicated

syntax of programming code. They feel that in order to use a CAS, they must

take a course in its programming language. Vector Calculus & Mathematica

gets around this apparent obstacle through copying, pasting and editing from

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the Basics module and Tutorials module. For instance, below is Mathematica

code in the Basics module and Tutorials module to plot f(x) = sin x for -2 x

2 :

Clearf, x;fx_ SinxPlotfx,x, 2 , 2 ,

PlotStyleThickness0.015, DarkGreen, AspectRatio 13;

Sin[x]

-6 -4 -2 2 4 6

-1

-0.5

0.5

1

Now, the following code results from copying and pasting all the code above

and then editing the red letters, i.e., changing Sin[x] to Cos[x], to present a plot

of f(x) = cos x for -2 x 2 .

Clearf, x;fx_ CosxPlotfx,x, 2 , 2 ,

PlotStyleThickness0.015, DarkGreen, AspectRatio 13;

Cos[x]

-6 -4 -2 2 4 6

-1

-0.5

0.5

1

The users can continue to experiment with plots of other functions of their own

choice. How does the size of a for a cos(x) where a R, influence the character

of the resulting plot? How are the plots resulting from negative a’s different in

character from those resulting from positive a’s? In other words, users can get

lots of extra insight when they copy the Mathematica code found in the Basics

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module and Tutorials module and experiment with it by changing the

appropriate values and rerunning.

ii. Visualization

One of the advantages of learning with Vector Calculus & Mathematica is that

users will have the opportunity to learn through graphics which they can

interact with. The advantage of learning visually with pure thought

uncorrupted by strange words is that words develop into an idea only after the

idea has already settled in the user’s mind. There are several stages in this

learning process: Frequently, visual learners are able to grasp the concepts by

the visualization of the problem. Then they describe the concepts in words.

Finally, to perform calculations, they must be able to translate those pictures

and words into mathematical actions.

iii. Using the printed supplement to support the electronic lessons.

For most users, it is not a good idea to try to acquire new ideas by trying to

read the printed supplement first. Moreover, only few students have ever

learned mathematics by just reading a printed book, any more than they

learned the stuff by just sitting and listening in some class lecture. The chances

are that users will not be able to learn only by reading the printed supplement.

The first exposure to a new idea should be on the live computer screen where

users can interact with the Basics and Tutorials to their own satisfaction. What

users see on the live screen doesn't carry the same power when it appears on

the static printed page. The reason is they can't interact with the printed page.

The lively lessons on the computer give them complete freedom to wrestle

with ideas, to experiment, play, and make mistakes as they do it.

iv. Memorization of rote calculational procedures.

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Memorization has never been the path to learning. On the other hand,

memorization which follows understanding is memorization with conceptual

understanding and it will “hook” into the long term memory. By using this

courseware, users learn by doing and by discovery. They get the chance to

announce mathematical ideas in their own words. All this results in long-

lasting knowledge, not just short-lived memorized procedures for calculations

which they forget easily.

v. Using Mathematica’s word processor

When users work on the Give It a Try problems, they are encouraged to use

Mathematica's word processor to write up the reason they are doing a

particular task. One of the advantages of writing clear explanations is that

users can come back later and understand what they did to solve a problem

whereas a sheet of paper covered with just formulas will mean nothing.

vi. Using the Literacy Sheets

Users can use the Literacy Sheets after they have completed the assigned Give

It a Try problems and have some machine experience with the ideas of a

lesson. It is recommended to use the machine or to look in the book to find

help in answering the questions in the Literacy Sheets, but the computer

shouldn't be a “crutch” for their understanding. Also, after they are done with a

lesson, they should try to be able to answer the Literacy Sheet questions orally

or with pencil and paper only. This is a good way of “polishing” on what they

already know. It is also a good way to get a feeling for paper and pencil skills.

Additionally, the users can get together with others to have Literacy Sheet

sessions to discuss the mathematics of a lesson.

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Many people have the notion that they must be able to do mathematics by hand

before they have the right to do it by machine. This notion is not correct at all. Vector

Calculus & Mathematica puts the users in the position of learning visually through

graphics and automatic calculations which is difficult to do by hand calculations. In

other words, the users cannot learn visually just by using pencil and paper. Once they

are free of the confinement of hand calculation, they are free to understand the

underlying mathematics and also free to find out what the mathematics is good for.

Hand calculations, as reflected in the Literacy Sheet problems, come into the picture

after they have used the machine in their learning. Hand calculations without

understanding are at best useless (Davis, Porta & Uhl, 1999).

However, at this point of our human development, and development of the

technologies we use in learning and calculating, the computer is a powerful new tool.

But that doesn't mean old tools, like pencil and paper, should be thrown away. It just

means that users probably do the deeper work on the computer, and be able to do

sample calculations and concept descriptions on paper. A study done by the College of

Education at the University of Illinois, found that it is impossible to distinguish Vector

Calculus & Mathematica students from traditionally-trained students on the basis of

their ability at hand calculation (Davis, Porta & Uhl, 1999).

2.4.2 GENERAL VECTOR ANALYSIS (GVA)

Many problems in plasma physics involve substantial amounts of analytical vector

calculation. The complexity usually originates from both the vector operations

themselves and the underlying coordinate systems. A computer algebra package called

General Vector Analysis (GVA) (Qin, Tang & Rewoldt, 1999), developed at the

Princeton Plasma Physics Laboratory, Princeton University, USA, is used to perform

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automatic symbolic computation of 3D vector analysis in general coordinate systems.

The general coordinate system could be any possible mathematically well defined

coordinate system where its Jacobian is non-vanishing everywhere. It includes well

known orthogonal coordinate systems like cartesian coordinates, cylindrical

coordinates, and spherical coordinates, and also other non-orthogonal coordinate

systems such as the flux coordinates used extensively in thermal fusion devices.

The modern viewpoint of 3D vector calculus is used to simplify and unify the

algorithm. A general coordinate is defined by the Reimann metric matrix expressed in

its own coordinates. A vector field is isotopic to a one form (covariant components) in

the 3D manifolds and its image under the Hodger star operator (contravariant

components). All vector calculus operators can be expressed in the language of

differential forms in a simple and unified manner. To define a new coordinate system,

users only need to define the corresponding Reimann metric matrix.

Users of this package need not to familiarize themselves to this mathematical

theory, unless they want to set up new coordinate systems of their own. The only thing

they need to realize here is the fact that in this package a vector is a 2x3 list, the first

part of which is the covariant component and the second part is the contravariant

component.

Some of the basic functions in this package:

i. Vector analysis in general coordinates

ii. Define coordinates

iii. Perturbative vector analysis with respect to any small parameter, e.g., inverse

aspect ratio in tokamak to any order

iv. Vector analysis in Shafranov coordinate, flux coordinate, straight tokamak

coordinate, circular concentric tokamak coordinate, etc.

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2.4.3 vec_calc

This Maple package was written by Philip Yasskin and Arthur Belmonte (2003) of

Texas A&M University, Texas, USA. It provides procedures for vector calculus

computations such as vector differential operations, curve integrals, line and surface

integrals and coordinate system conversions. This package was written to accompany

the book: Multivariable CalcLabs with Maple for Stewart’s Multivariable Calculus,

Fifth Edition (Yasskin & Belmonte, 2003).

2.5 THE CAPABILITIES OF VECTOR CALCULUS PACKAGES/COURSEWARES

In order to get an idea of the capabilities of the existing vector calculus

packages/coursewares mentioned above and also to provide a benchmark for future

development of ILMEV, a review table was created (refer Table 2.2). We hope to

continue to develop ILMEV based on the findings from this table (refer to the column

for ILMEV).

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Table 2.2A review table of the capabilities of the existing vector calculus packages/

coursewares and ILMEV

VC&M GVA vec_ calc

ILMEV

1. Plotting a function. To enter a function, we need to know the appropriate code to:

i. define a function ii. define the appropriate domain iii. define other features such as labels for the x and f(x) axes and the thickness and colors of lines

YesYesYes

YesYesYes

YesYesYes

NoNoNo

2. Show some explanations & steps in obtaining the solution automatically

No No No Yes

3. Computation of a wide range of simple geometric integration problems automatically, i.e., minimizing user intervention

No - No Yes

4. Description of the computation of multi-dimensional integrals analytically

No - No Yes

5. Description of the output of the entire problem solving process automatically

No No No Yes

6. Learning mathematical concepts & developing skills with less consideration of a CAS behind it

No No No Yes

7. Integration of multimedia elements to assist in the problem solving process:

i. Textii. Graphicsiii. Soundiv. Animation

YesYesNo No

YesYesNoNo

YesYesNoNo

YesYesYesYes

8. Copying, pasting & editing codes Yes Yes Yes Yes

9. Learning via experimenting with different input values

Yes Yes Yes Yes

10. Perform a particular task by clicking on a button

No Yes No Yes

11. Random approach in doing a lesson Yes Yes Yes Yes

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2.6 COMPUTER BASED LEARNING THEORY

The fundamental theory of learning that involves the use of computer technology such

as behaviorism theory, cognitive theory and humanism theory, is important to ensure

the effectiveness of the learning process. Thus, in this research we incorporated

computer based learning theories in the development of ILMEV to achieve our

research objectives as mentioned earlier.

2.6.1 Behaviorism Theory

Behaviorism theory was introduced by John B. Watson. This theory describes learning

as changes of behavior that can be seen and measured. Additionally, behavior can also

be controlled by imposing regulations, or encouraged through fortification. Skinner, a

behaviorism theorist, added that when a reaction is produced without any stimulation

beforehand and later, and a positive fortification is imposed such as flattering or

motivational remarks, the probability that the reaction will reoccur is higher. On the

other hand, if the reaction is followed by punishment or condemnation, the probability

that the reaction will reoccur is lower. Furthermore, instructors are also advised to

deliver the teaching materials in stages and give an immediate response to every

conduct shown by learners (Amir, 1986; Ee, 1994; Rothstein, 1990; Slavin, 1997).

In this research, ILMEV was developed to assist the users by applying some of

the concepts in the behaviorism theory such as positive stimulations (flattering and

motivational remarks), imposing the mathematical process in stages, and providing a

constant and immediate feedback.

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2.6.2 Cognitive Theory

Discovery learning theory is a cognitive theory introduced by Bruner. This theory

stresses on empowerment for the understanding of a concept of the material taught to

ensure the effectiveness of learning. It is well known that the principle structure of a

particular subject is built based on its concept. Thus, it is recommended that

instructors develop interesting activities when presenting their subject materials in

order to make the concepts of the subject matter easier to understand. This will

increase the understanding of the mathematical concepts and consequently, strengthen

the students ability to memorize and enjoy the learning process (Amir, 1986; Ee,

1994; Rothstein, 1990; Slavin, 1997).

In ILMEV, we introduce interesting activities such as visualization of graphs,

playing (experimenting) with a function by inputting various input values and

checking some steps when obtaining the solution. These activities encourage the

“discovery” of a particular mathematical concept during the process of learning.

2.6.3 Humanism Theory

Humanism learning theory focuses on the learner’s mind, emotion and examination

and interpretation of a particular situation or event. This theory stresses that every

individual is responsible to his own action. In computer based learning, humanism

theory also states the importance of the role of instructors to assist users in using

computer aided materials to enhance their understanding and reduce stress. The use of

a computer enables users to increase their self-expression and freedom, assists them to

be responsible in determining what to learn, increases their creativity and expands

their interest and curiosity. Additionally, users can easily learn whatever they desire

without feeling intimidated or humiliated. On the other hand, it is also important for

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users to know the learning methods and perform self assessment in order to make

learning more meaningful (Amir, 1986; Ee, 1994; Rothstein, 1990; Slavin, 1997).

In this research, we incorporated in ILMEV some of the concepts mentioned in

this learning theory: increase self-expression and freedom in learning, be responsible

in determining what to learn, increase creativity in solving mathematics problems and

enhance interest and curiosity in learning.

Please refer Chapter 4 for examples of ILMEV in action with respect to these

various behaviors.

2.7 CONCLUSION

In this research, we have chosen to create a system, ILMEV, to help students to learn

multidimensional integration. Many students with strong mathematics backgrounds

and a deep interest in some physical science find learning this material difficult. The

central difficulty is in the understanding of geometric regions in two and three

dimensional space where the assistance of computer visualization is substantially

needed. Our review of the literature shows that all common CASs have many tools

and packages to help in computing multidimensional integrals, but these are typically

difficult to use. Additionally, none of these systems can compute many the textbook

examples on integrals without substantial user intervention. The solution of these

problems require the development of an algorithm for computing important integrals

and the building of an interface, based on solid learning theory, to assist student in this

complex computing tasks.

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