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DOCUMENT RESUME -.. a ED 164.303 E 025. 458 AUTHOR Beck, A.; And Others ,TITLE Calculus, Part 3,'Studentl'Text, Unit No. 70. Revised Edition. , INSTITUTION Stanford Univ.,'CaliY. School Mathematics Study 't.-Group. SPONS AGENCY -National Science Foundation, Washington, D.C. PUB DATE ,65: NOTE 36bp.; For related documents, see SE 025 456-4:59; Contains light and braken'type EDRS PRICE MF-$0.83 HC-$19.41 Plus Postage. DESCRIPTORS *Calculus; *Cuiriculum; *Instructional Materials; *Mathematical Applications; Mathematics .Educatioi; S4Condary Education; *Secondary School Mathematics; *Textbooks IDENTIFIERS *School Mathematics Study Group ABSTRACT. This is part three of a'three-part SMSG calculuS text for high sahool students. One of the goals of the text is to present -calculus as a mathematical discipline -as yell, as presenting its practical uses. The authbrs emphasize the importance of being able to ipterpret the conceptS and theory interms of models to which they .apply. The text demonstrates the origins of the ideas of the calculus in pradtical problems; attempts to express these ideas precisely and_ ` develop them logically; and finally, returns to the problems and applies the theorems resulting from that development. Chapter topics include: (1)vectors and curves; (2) mechanics; (3) numerical analysis; (4) sequences and series; and (5) geometricl optics and waves. (MP) 1 - ********************************45*******************************/ * Reproductions supplied by EDRS are the best that can' be made ,* + ) - * from the original doCumente ../ * *****************************************************#*****,J U S DEPARTMENT lEDUCATIOPtilmi AI id NATIONALINS1 i I EDUCAT . THiSDOCUMENTHA! DUCEDEXACTLYAS TAE PERSONOR ORGAP ATIkr;T POINTSOF V STATEDDO NOT NECE SENTiOFFifiALNATION EDUCATIONPOSITION Calculus Part 3 Student's Text REVISED EDITION ' The following is a list of all thosewho.participiated in the preparation of this volume: A. Beck Olney High School, Philadelphia, Pa. A. A. Blank . NewYork University, New York, N.Y. 1- F. L. Elder lest Hempstead Jr., Sr. High School, N.Y. C. E. Kerr Dickinson College, Carlisle, Pa. M. S. Klamkin Ford. Scientific Laboratory, Dearborn,MiCh.' /. I. 'Kolodner Carnegie Institute of Technology, -Pittsburgh,Ir--- M. D. Kruskal PrinCeton University, Princeton, N.J. C. W. Leeds, III Berkshire School, Sheffield, Mass. Pa,, M. A. Linton, Jr. William Penn Charter School, Philadelphia, H. M..,,Marston Douglass College, New Brunswick; N.J. Purdue University, Lafayette, Ind. L Marx _ R. Pollack New York Uniyeriiry, New York, N.Y. T. L.Reynolds. College of William and Mary, Williamsburg,Va. R. L Starkey Cubberley High Schoolffalo Alto, Calif. V. Tiversky Sylvania Electronics Defease-La.bs , Mr.Vied, Calif H. Weitzner New York University, New York,gy. Stanford, California Distributed for the School Mathematics Study Group - by A. C. Vroman, 3-67 Pasadena Avenue, Pasadena, California e Financial support for School Mathematics .. Study Group has been provided by the- National Science Foundation. Permission to,make verbatimuse of material in this book must be secured fromthe Director of SMSG. Such'permissiori will be grantedexcept in unusual circumstances. Publications incorporating SMSG materials must include bothan acknowledgment of the SMSG copyright (Yale Universityor Stanford University, as the case may be) anda disclaimer of SMSG endorsement. Exclusive license willnot be granted save in exceptional circumstances,and then only by specific action of the AdvisoryBoard of S 0 1965 by The Board of Trustees of the Leland Stanford Junior University. All rights reserved. Printed in the United States of America. 14, 2' TABLE OF CONTENTS Chapter 11. -VECTORS AND CURVES . 669 11-1. Introduction 669' 11-2. Vector Algebra 672 11-3. Vector Geometry 681 11-4. Products of Two Vectors 690 11-5.7 Vector Calculus and Curves 704 11-6. Curves in the Plare 719. Miscellaneob.s Exercises 743 Chapter 12. MECE=CS - 747% %. 12-1, Intros --fo- 747 12-2.. ElemenTlary Mechanical Problems . , 755 12 -3. Constraints:, Use of Energy Conservation . - 772 12-4. Angular Momentum and Central Forces. ., '790 14iscelleineous Exercises 803, Chapter 13: NUMERICAL ANALYSIS 805 13-1. Introduction . 805 13-2. Iteration 8.08 13-3. Theorem with Remainder 819 13-4. Numerical Integration 829 13 -5. Numerical Solution of First Order Differential Equations 842 Ndscellaneous Exercises . 848 Chapter 14. SEQUENCES AND SERIES . 851 . _ 7 14-1. Introduction . .-. - .. 851. 71-2. Convergence of Sequences . 852 14-3. Series , 863 14-4. Conditional and Absolute Convergence ...873 -,.. 14-5. Parentheses and Rearrangements .ea. .87`( 14-6. Sequences of Functions, Uniform Convergence 883 891 14-7. Power Series I 0 NIscellaneou-s Exercises .., 856 Chanter 15. GEOMZTRICAL OPTICS AND WAVES -901 15-1. Introduction 901 15-2. Gea=atrical Optics 903 15-3. Refraction 927 13-4. Kepler-Lambert, Principle_ 939 15 -5. Huyghents Principle 950 15-6. Periodic Waves, . 959 15-7. Method of Stationary Phase 977 15..78. Mathematical Model for Scattering 1001 1 C law O Chdpter 11 VECTORS AND CURVES . Introduction. 2 Given the curvey = x , we can easily show that the point Awith coordinates (0,0) is at absolute minimum. We also show that the area between the curve and thq liney = x + 2 is 2 (Figure 11-1s). 1. igure 11 rla But suppose we take the figure and rotate it about the point A until the liney = x 2 is horizontal (FigUre 11-1b). As a result of the change yt y Figure 11-lb the point AIs no longer a minimum On the other the area of the . shaded region certainly should not change. This simple example suggests that there are two types of properties associated with curves in the plane-or in space. Certain properties depend on the orientation of the curve with respect to a. set of coordinateaxes, and others do not. In this chapter and the next we shall emphasize thoseproper- . ties which do not depend on the coordinates.You should not suppose that' ,either type of property is more important or more fundamental4Zhan tIle other, for they have different uses.In Section 1-1 to find a box with the largest volume, subject given conditions, we plotted the volume of the-box asa function of the length of-a side (Figure 1-1a).If we were to rotate the reference axes, then the new curve.would be of aittle use'in solving the pro- blem. In, contrast, areas and volutes of regions do_ not depend on the.coordi- nates. For-some problems coordinates are not useful until.the solutiorris almostcomplete; such a pro lem is that of describing themotion of twomutually ( attracting objects. a) We shall introduce new tools particularly appr$riate to the study of.pro- perties independent of coordinate axes.We combine partsof euclidean synthetic, -geometry and the analytic-geometry of De§,car-'-=.s. Synthetic geometry does not employ 'coordin/Ite axes but Is%-very awkward :antitative studies.! Cartesian 670 geometry is an ideal tool for the definition and analysis of curves and sur- races,, but the analysis. rests on the choice of coordinates, even when the re- t / .sults do not. To combine the advantages of both systems we introduce the can- cept of, vector as a translation of.the plane or space,-although later the ce,,n- cept;will be seen to have many other interpretations. We shall devqlop an algebra of vectors with operations corresponding to simple geometric construc- tions. In this way, we can describe space independently of coordinates and yet have the power of algebraic operations. All of this could have been done betor the calculus, but we finally ittroduce a vector calculus, and, bring to bear the f I tools we have developed in earlier chapters. Exercises 11 -1 1. Which of the following quantities are independent of the choice of _ . coordinates? (a) The distance between two points, (b) The distance of -a point from the origin. (c) The angle between two lines. (d) The angle of inclination of,a line: (e) The area of e standard region under the graph y = f(x) . (f) The area bounded by two curves. IL A *5- ,--671 11-2.. Vector Algebra. We shall use the geometricalidea of translationas our primary repreAen- tation of the concept of vector.A vector V is then thoughtof as a mapping of space V : Qonto itself in which all points are translatedthe same distance (the length of7 , written !VD in 'thesame direct:Ion. if rf(P1)= Qi and -1(72)= Q2 It^ then the directed linesegments andP2Q2 have the same length and are parallel (directed line segments are said to be parallel if theynot Figure 11-2a only lie on parallel lines,but also -have the same orientation).We can describe V by the walin which it thal6sjust one point sinceany directed segment -15, where Q = V(P) , defines the direction and ,length of V . It does no harm to picture Vas a flQating directed segment (ofa specified length and direction) which may be attached to an initial pointifconvenient. The interpretation of vectoras a translation clearly makesno use of a coordinate system.Nonetheless, it is often convenientto have a represen- tation of a vector in a given coordinate system. IfV(P1) = QIand y 12` f. X-F 2 Figure 11-215 J We shPll use the word "space"to denote either the Euclidean planeE2 or Euclidean three-dimensional space E3, specifying only when necessary. 1 Cj672 V(P2) = Q2then from the parallelism ofPiQi. and P2Q2 the projections of the two directed segments on any given line are equal in-..length and havethe same orientation. Thus if' P an d P have the x-coordLItes x1 and x 1 2 2 respectively, ' -they. Q1 and Q2 have the x-coordinates x, + and x2 + respect-ively (Figure 1-2b)--.; Here. Itl is the common len..zh of the two pro- jections and t is positive if the projections have the orientation as the x.:axis, and negative if they have the opposite orientation.
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  • Communications of the LUNAR and PLANETARY LABORATORY

    Communications of the LUNAR and PLANETARY LABORATORY

    Communications of the LUNAR AND PLANETARY LABORATORY Number 70 Volume 5 Part 1 THE UNIVERSITY OF ARIZONA 1966 Communications of the Lunar and Planetary Laboratory These Communications contain the shorter publications and reports by the staff of the Lunar and Planetary Laboratory. They may be either original contributions, reprints of articles published in professional journals, preliminary reports, or announcements. Tabular material too bulky or specialized for regular journals is included if future use of such material appears to warrant it. The Communications are issued as separate numbers, but they are paged and indexed by volumes. The Communications are mailed to observatories and to laboratories known to be engaged in planetary, interplanetary or geophysical research in exchange for their reports and publica- tions. The University of Arizona Press can supply at cost copies to other libraries and interested persons. The University of Arizona GERARD P. KUIPER, Director Tucson, Arizona Lunar and Planetary Laboratory Published with the support of the National Aeronautics and Space Administration Library of Congress Catalog Number 62-63619 NO. 70 THE SYSTEM OF LUNAR CRATERS, QUADRANT IV by D. W. G. ARTHUR, RUTH H. PELLICORI, AND C. A. WOOD May25,1966 , ABSTRACT The designation, diameter, position, central peak information, and state of completeness are listed for each discernible crater with a diameter exceeding 3.5 km in the fourth lunar quadrant. The catalog contains about 8,000 items and is illustrated by a map in 11 sections. hiS Communication is the fourth and final part of listed in the catalog nor shown in the accompanying e System of Lunar Craters, which is a_calalag maps.