Table Of ContentBASIC INSIGHTS IN VECTOR
CALCULUS
With a Supplement on Mathematical Understanding
Other World Scientific Titles by the Author
Invitation to Generalized Empirical Method:
In Philosophy and Science
ISBN: 978-981-3208-43-8
The (Pre-)Dawning of Functional Specialization in Physics
ISBN: 978-981-3209-09-1
BASIC INSIGHTS IN VECTOR
CALCULUS
With a Supplement on Mathematical
Understanding
Terrance Quinn
Middle Tennessee State University, USA
Zine Boudhraa
Montgomery College, Maryland, USA
Sanjay Rai
Montgomery College, Maryland, USA
Published by
World Scientific Publishing Co. Pte. Ltd.
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Library of Congress Control Number: 2020030528
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
BASIC INSIGHTS IN VECTOR CALCULUS
With a Supplement on Mathematical Understanding
Copyright © 2021 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN 978-981-122-256-6 (hardcover)
ISBN 978-981-122-257-3 (ebook for institutions)
ISBN 978-981-122-258-0 (ebook for individuals)
For any available supplementary material, please visit
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Desk Editor: Liu Yumeng
Printed in Singapore
Terrance Quinn: In memory of my parents, George and Bernice, and two of
my brothers, Patrick and John, all of whom encouraged me in my interest in
mathematics and science, from an early age.
Zineddine Boudhraa: In memory of my mother, Habiba, as well as the
memory of all of those who lost their battles to cancer.
Sanjay Rai: To my parents, Shri Sarva Deo Rai, my late father, and Smt.
Munni Rai, my mother, neither of whom had the opportunity for education,
but through their own experiences in life, understood the need and potential
of education in changing the lives of a generation.
We also dedicate this book to students and teachers, in the hope that it will be
enjoyable and helpful in your growth in mathematics.
Preface
This book was developed from a series of lecture notes for teachers of
undergraduate vector calculus, and students who want to understand the
essentials of the classical theorems. With that said, we believe that graduate
students in mathematics and mathematical sciences generally will also find
the material helpful. In order to reach some level of mastery of, say,
differential forms, one needs to have source insights for the classical vector
calculus theorems. Promoting those source insights is the main purpose of
our book. To help set the stage, we begin by providing some context.
Figure P.1 Positively-oriented closed curve C.
P.1 General context
The vector calculus theorems emerged gradually through a cross-fertilization
of physics and mathematics. In each of the theorems, an integral of a vector
quantity along a “boundary” is found to be equal to a combination of
derivatives in the region or volume “interior” to the “boundary.” For instance,
Green’s theorem is for a vector field
defined on a region R in the (x, y) plane. The boundary of R is a “positively-
oriented” closed curve1 C (see Figure P.1). With appropriate hypotheses in
place,
where
The vector calculus theorems are standard content for undergraduate
multivariable calculus books presently on the market.2 There is, however, a
gap in the textbook literature. Filling that gap will meet a need for students in
both pure and applied mathematics. Note that the gap is not in topics treated,
but in certain learning outcomes.
Currently available texts generally do a good job at providing applications, as
well as examples for developing symbolic and computational techniques.
And in most books, there are proofs — or at least partial proofs — of the
vector calculus theorems.
In modern mathematics, proofs are needed. As you will find by adverting to
your own experience, however, prior to proof, there is basic insight. For
mathematicians and mathematics students, basic insight is a familiar event.3
It is an insight, through a diagram or symbolism, or some combination in
one’s senses. Basic insight is not a guess, but this is not to suggest that
guessing does not occur. Basic insight, however, is something more. It
sometimes comes only after considerable effort, such as a breakthrough to
what might turn out to be a key result in a Ph.D. thesis. Or, one may discover
a possible solution to a problem in a homework set. The emergence is not
routine. In basic insight, we grasp possibility.
If, however, one goes on to ask ‘Is it so?’, then one has shifted to a new kind
or “level” of inquiry. Again, it is a matter of adverting to your own
experience. Asking ‘Is it so?’ is asking about something that, to some extent,
one already understands, at least as a possibility. Is the “it” that one has
grasped correct? One has had an insight. It is not nothing. But does one’s
understanding hold up? Are there counter-examples that might invalidate
one’s conjecture, or cases not accounted for? Is correction or revision
needed? Is what one understands partly mathematical but also partly
descriptive of diagrams, such as when “outward unit normal” is only
descriptively defined in coordinate geometry?4 And so on. All of this is
familiar to the mathematics student and the professional. In modern
mathematics, asking “Is it so?” calls for proof. That is, one seeks “middle
terms”5 by which that which one has discovered as a possibility is understood
to cohere with other results that have already been understood to belong to an
(axiomatic) context.
Basic insight, then, is crucial. It is a beginning. It emerges from a ‘What is
it?’ inquiry-mode, wherein one grasps possibilities. Being asked to read a
proof prior to being invited to basic insight is being asked to read a solution
to a problem that has not yet had. And, without basic insight, the possibility
of competence with symbolic and computational techniques is significantly
undermined.
In calculus textbooks presently available, however, the emphasis tends to be
on symbolic technique and computation.6 This Preface is not the place for a
literature review. To help illustrate the problem, it will be enough to advert to
two common ways that “curl” is introduced.
P.2 Beginning with the coordinate definition of the 3-d vector
curl
Frequently, the discussion of “curl” begins by providing a formula for the 3d
“vector curl” for F = Pi + Qj + Rk. Assuming that all partial derivatives exist,
