The CLeaR Project at the University of Maryland, College Park
CLeaR is an acronym for the developers (Drs. K. Coombes, R. Lipsman
and J. Rosenberg) of a computer supplement for the
sophomore Multivariable Calculus course.
The goals of this project are:
- To enhance students' understanding of the
fundamental concepts of mathematics presented in the traditional
multivariable calculus course at the sophomore level;
- To help students learn about the geometry of
curves and surfaces;
- To reintroduce topics from physics into the curriculum in
an accessible and interesting way;
- To enable students to see the deep connections between
mathematics and science;
- To provide students with a tool, in the form of a
modern mathematical software system, for exploring, understanding, and
using mathematics; and
- To improve students' abilities to communicate their
understanding of mathematics to faculty, peers, and others.
Since 1995, CLeaR has been experimenting with the mathematical software
system Mathematica in Math
241, the sophomore multivariable
calculus course at UMCP. These experiments have resulted in the
writing of a book Multivariable Calculus and Mathematica, with
Applications to Geometry and Physics that was published
in May 1998 by
Springer/TELOS. The book is
used at UMCP in conjunction
with the text Calculus with Analytic Geometry, 5th
Edition, by Ellis & Gulick, although it is completely suitable
for use with most sophomore level multivariable calculus
texts--traditional or reform.
The bulk of the book consists of eight chapters on multi-variable
calculus and its applications. Some chapters cover standard material
from a non-standard point of view; others discuss topics that are hard
to address without using a computer.
Each chapter is accompanied by a Mathematica problem set.
The problem sets
constitute an integral part of the book. Solving the problems exposes
students to geometric, symbolic, and numerical features of
multi-variable calculus. Many of the problems are quite challenging.
Each problem set concludes with a glossary of Mathematica commands,
accompanied by a brief description, which are likely to be useful in
solving the problems in that set. A more complete Glossary, with
examples of how to use the commands, is included at the back of the
book. In addition, the book contains Mathematica Tips, Sample
Notebook Solutions, and an Index. Finally, there is an accompanying
- electronic versions of the Sample Notebook Solutions,
- Mathematica notebooks for each chapter, containing the
Mathematica input lines that recreate all of the output and
figures from that chapter, and
- a Mathematica Notebook containing all the sample input
lines from the Glossary.
Here is a brief description of the books' contents.
- Chapter 1, Introduction and Problem Set A, Review of
One-Variable Calculus, describe the purpose of the book and its
prerequisites. The Problem Set reviews both the elementary
commands and the fundamental concepts of one-variable calculus
needed to use Mathematica to study multi-variable calculus.
- Chapter 2, Vectors and Graphics, and Problem Set B, Vectors
and Graphics, introduce the mathematical idea of vectors in the plane
and in space. They explain how to work with vectors in
Mathematica and how to graph curves and surfaces in space.
- Chapter 3, Geometry of Curves, and Problem Set C,
Curves, examine parametric curves, with an emphasis on geometric
invariants like speed, curvature, and torsion, which can be used to
study and characterize the nature of different curves.
- Chapter 4, Kinematics, and Problem Set D, Kinematics,
apply the theory of curves to the physical problems of moving
particles and planets.
- Chapter 5, Directional Derivatives, and Problem Set E,
Directional Derivatives and the Gradient, introduce the differential
calculus of functions of several variables, including partial
derivatives, directional derivatives, and gradients. They also
explain how to graph functions and their level curves or surfaces with
- Chapter 6, Geometry of Surfaces, and Problem Set F,
Surfaces, study parametric surfaces, with an emphasis on geometric
invariants, including several forms of curvature, which can be used to
characterize the nature of different surfaces.
- Chapter 7, Optimization in Several Variables, and Problem Set G,
Optimization, discuss how calculus can be used to develop
numerical algorithms. They also explain how to use Mathematica
to test and apply these algorithms in concrete problems.
- Chapter 8, Multiple Integrals, and Problem Set H, Multiple
Integrals, develop the integral calculus of functions of several
variables. They show how to use Mathematica to set up multiple
integrals, as well as how to evaluate them.
- Chapter 9, Physical Applications of Vector Calculus, and Problem
Set I, Physical Applications, develop the theories of
gravitation, electromagnetism, and fluid flow, and then use them with
Mathematica to solve concrete problems of practical interest.
- Chapter 10, Mathematica Tips, gathers together the
answers to many Mathematica
questions that have puzzled our students.
- The Glossary includes all the commands from the problem set
glossaries---together with illustrative examples---plus some
- The Sample Notebook Solutions contain sample
solutions to one or more problems from each problem set. These
samples can serve as models when you are working out your own
solutions to other problems.
- Finally, there is a comprehensive Index of
Mathematica commands and mathematical concepts that are found
in the book.
How it works!
In Chapters 2-9, students are
introduced to those aspects of elementary differential geometry,
optimization and physics that, while
vitally important and most relevant to the needs of practising
scientists and engineers, are usually omitted, or only treated
briefly, in a traditional text: namely, numerical,
geometric, symbolic, and qualitative methods. The software systems
render these topics, almost untreatable in an old format, easily
and stimulatingly accessible to undergraduate students.
In solving the problem sets the student brings to bear
newly acquired skills in the computer system to solve non-traditional
problems in multivariable calculus, elementary differential
geometry, optimization and physics. The emphasis is on the geometric,
numeric, and qualitative aspects of the subject. The
problems, each of which is a small project, are designed to
force the student to engage in critical, analytic, and interpretive
thinking beyond rote manipulation of algebra and calculus formulas.
Students do all their work in campus computer Labs. All
platforms are available, and students select those they feel most
comfortable with. Because of
the remarkable interface, faculty barely notice any difference in
the output generated by students working on different platforms.
formal instruction on the platforms or software system is presented
in class. Students learn about them from primer material, and from
on-line help, Graduate Assistant tutors (acting as first-aiders), each
other, and faculty assistance in office hours.
The effects of the project, aside from achieving the goals indicated
above, include: creating a mathematical computational culture among
students (they use the tools they take away from this course in
other courses, in lab reports, and later on in their jobs);
fostering cooperative learning (students are encouraged to work in
teams, and they quickly become acclimated to cooperative
problem-solving in a team setting); enhanced visual and communication
skills (the interface allows the student to integrate textual, symbolic,
and graphical material in an informative and effective way). Most
importantly, the intellectual level of the course has been
raised---without a drop in student performance.