Basic Insights in Vector Calculus with a Supplement on Mathematical Understanding PDF
Preview Basic Insights in Vector Calculus with a Supplement on Mathematical Understanding
BASIC 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. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE 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, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. 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 https://www.worldscientific.com/worldscibooks/10.1142/11892#t=suppl 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,
The list of books you might like
