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MATA37_PART1-success - PREFACE TO THE THIRD EDITION'i'he...

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Unformatted text preview: PREFACE TO THE THIRD EDITION 'i'he most significant Change in this third edition is the incinsioh ofa new (starred) Chapter 17 on planetary motion, in which calculus is employed for a substantial physics problem. in preparation for this, the old Appendix to Chapter 4 has been repiaced by three Appendices: the first two COver vectors and conic sections, while poiar coor- dinates are now deferred untii the third Appendix, which also discusses the polar coordinate equations of the conic sections. Moreover, the Appendix to Chapter 1?. has been extended to treat vector operations on vector-vaiued curves. Another iarge change is merely a rearrangement of old material: “The Cos- mopolitan integral," previously a second Appendix to Chapter 23, is now an Appendix to the chapter on “Integration in Elementary Terms" (previously Chap- ter i8, now Chapter 19}; moreover, those problems from that chapter which used the matcriai from that Appendix now appear as problems in the newly placed Appendix, A few other changes and renumbcring of Problems result from corrections, and elimination of incorrect problems. I was both startled and somewhat dismayed when i reailzed that after altow- ing l3 years to elapse between the first and second editions of the book, I have ailowed another 14- years to elapse before this third edition. During this time I seem to have accumuiated a not-so-sliort list of corrections, but no longer have the originai communications, and therefore cannot properiy thank the various in- dividuals involved (who by now have probably lost interest anyway). 1 have had time to make only a few changes to the Suggested Reading, which after ail these years probably requires a complete revision; this wiil have to wait until the next edition, which I hope to make in a more timely fashion. MICHAEL SPIVAK PREFACE TO THE FOURTH EDETION Promises, promises! In the preface to the third edition I noted that it was 13 years herween the first and second editions, and then another [4 years before the third, expressing the hope that the next edition would appear sooner. ‘r‘l'eil, here it is another 14 years later before the fourth, and presumably final, edition. I Ait'hough smnil changes have been made to some material, especialiy in Chap- ters 5 and 20, this edition tlilfers mainly in the introduction of additional problems, a complete update ofthe Suggested Reading, and the correction ofnttmcrotts er— rors. These have been brought to my attention over the years by, among; others, Nils von Barth; Philip Loewen; Fernando h-chias; Lance h-Iilter, who provided a long list, particularly for the answer hook; and Michael h-Ialtenfort, who provided an amazingly extensive list of misprints, errors, and criticisms. ' Most of all, however, E am indebted to my friend Ted Shifrin, who has been using the book for the text in his renowned course at the University of Georgia for all these years, and who prodded and helped the to Iinaily make this needed revision I musz also thank the students inhis course this last academic year, who served as guinea pigs for the new edition, resuhing, in particuiar, in the current proofin Prohiem 8-20 for the Rising Sun Lemma, far simpler than Reisz’s original proof, or even the proof in [38} of the Suggested Reading, which itself has now been updated considerably, again with great help from Ted. MICHAEL RPIVAK. CONTENTS PREFACE vi PART 1 Prologue 1 Basic Properties of Numbers 3 2 Numbers of Various Sons 21 PART II Foundations 3 Functions 39 Appendix Ordered Pain‘ 54 4 Graphs 56 Appendix 1" Vectors 75 AppetidixZ, The ConicSecrious 80 Appendix 3. Polar Cam'rfiuarer 84 Limits 90 Continuous Functicns l 15 Three Hard Theorems 322 - Least Upper Bounds E33 Appendix. Unifirm Contbmiyl 144- oo‘JGm PART III Derivatives and Integrals 9 Derivatives 149 10 Difl'crcmialion 168 1] Significance of the Derivative E88 Appendix: Corwaxilwmd Concnviti 219 12 Inverse Functions 231) Appendix. Paranwlric Repmmmlian omeveS 244 13 integrals 253 Appendix. Riemarrrt Sum: 28?. £4 The Fundamental Theorem of Calculus 285 xiii (Ibrahim: 15 The Trigonomen‘ic Functions 303 ‘5‘l6 Jr is Irrational 324 ”:17 i’ianctary Motion 330 $8 The Logaz‘ilhm anti Exponential Functions 339 19 Integration in Eéamcmary Terms 363 Appwul‘i't The Comopoiilmr Ink-gm! 402 _ 1 PARTIV Infinite Sequences and Infinite Series ' G AL C I L l S 20 Approximation by Polynomia} Functions 4-H *2! e is Transcendental 442 22 Infinite Sequences 452 23 Infinizc Series 475 '24 Uniform Cnnvcx‘g‘emc and Power Series 499 25 Complex Numbers 526 26 Complex Functions 541 2'? Compiex Power Series 555 PART V Epilogu e 28 Fields 58? 29 Construction nfthc Real Numbers 588 30 Uniqueness ol‘lhc Rea? Numbers 601 Suggcxled Reading 609 Aimee): {(0 5313.7wa pmMcuuj 619 Glosmfr qfé'wnbols 665 hedex 669 To be smart-ions that you are ignorant i: a great step to knowledge. BENJAMIN DISRAELI CHA?TER BASiC PROPERTIES OF NUMBERS The title of this chapter expresses in a few worth the mathematical knowledge required to read this book. in fact, this short chapter is simply an cxplanation of what is meant by tin: “basic properties of numbers,” ail of whichmaddition and muitiplicat‘ion, subtraction and divasion, solutions ol‘cquations and inequalities, factoring and other algebraic manipulations—are aircaciy Familiar to us: chcrr titclcss, this chapter is not a review Despite the fnn‘tiliarity of the subject, the survey we are about to nntlcrtalte wiil probably seem quite novel; it (locsjnot aim to present an extended review of old material, but to condense this knowledge into a Few simple and obvious propertics'ofnumbers. Some may eve-1% item 100 obvious to mention, but a surprising number ol‘tiivcrsc and important Riots turn out to be. consequences ofthc ones we shail emphasize. Ol‘thc tweivo properties which we shail study in this chapter, the first him: are concerned with the fttndamcntai operations of addition and tnuitiplication. For the moment we consider only addition: this operation is performed on a pair ol‘ nttnihcmwthe sum (l -t~ b exists for any two given numimrs (t and 2‘) {which may possibly be the some number, of course} it might seem reasonabic to regard addition as an operation which can be performed on scvcrai numbers at once, and consider the sum at + —t— (1,. of J] numbers (It. . . . ,a,. as ti basic concept. It is more convenient, however, to consider addition of pairs of numbers only, And to ciefinc other sums in terms oi‘sttms of this type. For the Sum of three numbers a, 12, and c, this may be tlont: in two dili‘orcnt ways. Ont: can hast add it and r, obtaining b + c, and than atltl {t to this llLI111i3ttl‘, obtnining l! + (t’: -l- a); or one can first add (i and b, and than aclcl the sum (1 +1: to t, obtaining {a + b) + 5-. Of comic, the two compound sums obtained are equal, and this fact is the my first property we shall iist: (Pl) if a, I), and 0 art: any l‘ttm'lhm‘s, than n +(b +6) = (a to) +12 3‘1“: statcmcnt of this property cical‘ly renders a soparale concept of (111‘. sum 0! three numbers superfluous; we simply agree that a + b + t denotes the number a +(b-t-c) = {a +b) ~t~ct Addition of Four numbers requires similar, though sliglttiy more involved, considerations, The symbol a -t- b + c + d is defined to moan (i) ((a-l-b)+c}+d. or {9.) (rt + (b + 6)) + d. or {3) a 4- {(b + c) + d), or 4) a + (b+ {c+d)). ) (a ~i~bi + (Mind). AA infogtte This definition is unambiguous since these numbers are all equal, Fortunately, this fact need not be listed separately, since it foilows from the property Pi already listed. For example, we know from Pl that ((,+b)+e=u+(b+cl. and it follows immediately that (i) and (2) are equaL The equality of (2) and (3) is a direct consequenee of Pi, although this may not be apparent at first sight (one must let I; -t- 0 play the role of I: in Pi, and d the role of c). The equalities (3) m (4) = (5) are also simple to prove. it is probably obvious that an appeal to Pl \viil aiso suffice to prove the equality of the i4 possible ways ol‘sunun‘tng live numbers, but it may not be so clear how we can reasonably arrange a proofti'tat this is so without actnaily listing these 14 sums. Such a procedure is feasible, but would soon cease to be iixve considered collections oi'six, seven, or more numbers; it wouid be totally inadequate to prove the equality of ali pos‘sibie sums of an arbitrary finite collection of numbers n}, .,,,(t.,. This fact may be taken for granted, but for those who would like to‘ worry about the proofiand it is worth worrying about once) a reasonable approach is outlined in ?roblem 24. Henceforth, we shall usually make a tacit appeal to the results of this problem and write sums n; + -- - + m, with a hiithe disregard for the arrangement of parentheses. The number 0 has one property so important that we list it next: (F2) If a is any number, then n+0w0-l-rtnrt. An important role is also piayed by 0 in the third property of our list: (P3) For every number u, there is a number we such that a ,4. (Wu) m (me) -i- r.- = 0. Property P2 ought to represent a distinguishing characteristic of the number 0, and it is comforting to note that we are aiready in a position to prove this, Indeed, ifa number x satisfies 0+.r=a for any one number n, then .r = (3 (and consequently this equation also iioids for all numbers a). The proof of this assertion involves nothing more than subtracting a from both sides of the equation, in other words, adding “(I to both sides; as the following detailed proof shows, all three properties ill—P3 must be used to justify this operation. if u «l- ' = n, then (—a) -l- (a +th w {on} -t~ n m 0: hence {(ma) + a) + N m- 0‘. hence hence .7 , Basic Properties Qannberx 5 As we have just hinted, it is convenient to regard subtraction as an operation derived from addition: we consider rt —— b to be an abbreviation for a +-(-—b). it is then possible to find the solution of certain simple equations by a series of steps (eachjustified by Pi, P2, or P3) similar to the ones just presented For the equation a + x w a. For example: ' if x + 3 = 5, then (,r 4- 3) -i- {—3} = 5 + (”3); hence x+(3+ (m3))=5~w3 m2; hence x- + {l m— 2; hence .r m 2. Naturally, such elaborate solutions are oi'interest oniy until you become eonvineed that they can always be supplied. in praotiee, it is usually gust a waste tigi‘timc to solve an equation by indicating so expliciily the reliance on properties Plst P2, and P3 (or any of the further properties we shall list}, ‘7 Only one other property ofaddition remains to be listed. When considering the sums of three numbers a, b, and e, oniy two sums were mentioned: {a + b) + c and rt + {b + e). Actually, several other arrangements are obtained if the order of a, b, and e is changed. That these sums are all equal depends on (P4) If a and I: are any numbers, then a +1) =b+a. The statement (£134 is meant to emphasize that although the operation of ad» dition of pairs of numbers might eoneeivabiy depend on the order oiT the two numbers, in fact it does not, it is helpful to remember that not all operations are so well behaved. For example, subtraction does not have this property: usualiy a —- b aé b we. in passing we might ask just when a —- .5 does equal b ,. 1’s, and is is amusing to discover how powerless we are if we rely only on properties E’l—Pd to justify our manipulations, Algebra of the most elementary variety shows that a — b = b — a only when a = b. Nevertheless, it is impossible to derive this fact from properties Pl—Pél; it is instructive to examine the nienientary algebra care- fully and determine which stems) cannot be justified by leP4, We wili indeed be able to justify all steps in detail when a few more properties are iisted. Oddly enough, however, the crucial property involves multiplication. The basic properties of multiplication are fortunately so similar to those for ad- dition that little comment will be needed; both the meaning and the consequences should be clear. (As in elementary algebra, the product of (i and b will be denoted by a - b, or simply rib.) (P5) If (I, b, and c are any numbers, then o-{b‘c :{adflro (P6) if n is any number, then 'ofogue Moreover, i oi 0. (The assertion that ] 57$ 0 may seem a strange fact to list, but we have to list it, because there is no way it could possibly be proved on the basis of the other properties listcdmthcsc properties wouid all iioid if there were only one number, namely, 0.) (P?) For every number :2 ye 0, there is a tiumbcr a“i such that (P8) If a and b are any numbers, than a-bzb-a. One detail which deserves emphasis is the appearance of the condition a 5x5 0 in P? This condition is quite necessary; since 0-5; = 0 {or ail nttrn‘hcrs b, there is no numberfl" satisfying 9 - 0""1 m 1. 'fhis restriction has an important consequence for division. just as subtraction was defined in terms of addition, so division is defined in terms of multipiication: ihc symbol (1].!) mcans a . 1;". Since 0"' is meaningless, 12/0 is also moaninglcss—division by O is always nndcfined. Property P7 has two important consequences. if a - b n a t c, it does not ticccssai‘iiy follow that b w c; for if a = 0, then both a - b and a t c are (3, no matter what I; and c are. However, if a gé 0, then i: = c; this can he deduced From P7 as follows: If a-b=¢iicanda;&0, thorn a" - (a-b) =a“i -{a-C): hence (0"t -a)-b:{n'"' -a)'c; hence I . b m l 'c; hence b =c. It. is aiso a consequence of PT" that if a -b = 0, then either a m 0 or b = 0. In fact. if ovmeandayEO, then flm2~{a-b)m0; hence (ff-ll >rt) -b = 0; hence l -.‘i = 0; hence it = 0. (It may happen that rt = 0 and b m 0 art: both true; this possihiiity is not oxcludcd when we say "either (.- = {J or b m 0”; in mathematics "or” is always used in the sensc of “one or the other, or both“) This iattcr consequence of P”? is constantly used in the solution of equations. Suppose, for example, that a number x is known to satisfy {x - l)(x —2}=G. Then it follows that either it w l m 0 or x — 2 = (J; hcncc x m i or x m 2. I . Basic Propertm quitmhcrs '7 On the basis of the eight properties listed so for it is still possible to prov: vciy little. Listing the next property, which combines the operations of addition and multiplication, will alter this situation drastically. (P9) If a, b, and c are any numbers, then a-{b-i—c)=a -b~iva-c. (Notice that the equation (in +c) > n =1; -o + c -a is also true, by Pig.) As an example of the usefulness of P9 wonwiil 110w determine just whcn'a m b m ii—o: 'ii it If a ~b=‘f‘b«~a, ; than «innit:tb—ai-ibwwbwa); ‘ hence a=b+bwo: ‘ hence a+a=(b+b-al+n=b+b. Consequently (l'(i+l)mbt(§+1}. and therefore a = I). A second use of F9 is the justification of the assertion n . O = 0 which we have already made, and even used in a proof on page 6 {can you find where?) This {act was not listed as one of the basic properties, even though no proofwzis ocht‘cd when it was first mentioned. With lePS alone a proof was not possiblchsince the number 0 appears only in PE and P3, which concern addition, while the {assertion in question involves multipiication. With P9 tho proof is simpic, though perhaps not obvious: We have a»0+av0=a-(0+0) , mail; l. as we have already noted, this immediatciy implies {by adding ~(a -{l) to both sides) that a - D m 0. A series of further consequences of P9 may help explain the somewhat myste- rious rule that the product of two negative numbers is positive. To begin with, we will establish the more easily acceptable assertion that (—a) . l: = ~«{a >12). To prove this, note that (—a)-b+nvb=-[{—a}+a]-b :04: :0 it follows immediately (by adding ~(rt ~ b) to both sides) that (—(1) - ii = —(o -b). Now note that (ma) - (“bi + [—(a -bll w—w (Ma) . (—1)) + {—0) ' b m (—a) - {(—12} -i~ b] (—a) -0 0. ll 1! ‘rot'ogue Consequently, adding (a - b} to both sides, we obtain (~a) . (Mb) : a - b. The fact that the ptoduct oi“ two negative numbers is positive is thus a consequence of Pl-—P9. in other words, g'fwe want PI to P9 to be true, the rate]??? the [tiniest :11“!th negative tztttttbt'tzt t'sflrced upon us. The various consequences of P9 examined so far, although interesting and im— portant, do not really indicate the significance: of P9; after ail, we could have listed Each of these properties separately. Actually, P9 is the justification for almost all algebraic manipulations. For example, aithough we have shown how to solve the equation (x m DU #2) =0, we can hardly expect to be presented with an equation in this form, We are more likely to he canfronted with the equation x2 m 3t- + 2 = 0. ‘ The “faum’i‘eazion” x3 -« 3x + 2 = (x w 1)(x m 2) is reaily a triple use of ?9: (xWi)t{x—'.Z)=x-(x-2}+{-—i)'{x-'2} wax-x“rtm2)+(—1)-x+t—§)-t~2) mt-2+x[(w2}+{w1)]+2 ms‘2—3x—tw2. A finai illustration ofthc importance of F9 is the fact. that this property is acttiaily used every time one multiplies arabic numerals. For example, the calculation 13 x24 ""53 26 312 is a concise arrangement for the Foliowing equations: l3-24=i3-(2~10+£i} $13.2,10+§3.4. m26'10‘i‘52. (Note that moving 26 to the tail in the above calculation is the same as writing 26 - i0.) The multiplication 13 -4 w 52 uses P9 also: i3-4:(I~iO+3)-4 =i-lO-4+3-4 writ-10+l2 =4tiU-t~l-10+2 ={4+l}-10+2 $5,104,; m52. I. Basic Properties effifiunbert 9 The properties i’lwI’Q have descriptive names which are not essentiai to remetw her, but which are often convenient for reference. We will take. this opportunity to iist properties Pl-~P9 together and indicate the names by which they are commonly designated. (Pl) (Associative iaw for addition) a t- (b + c) e (a + b) +3'c. (P2) (Existence of an additive a + 0 = O + a = a. identity) ‘ ' (P3) (Existence of additive invcises 33 a -t- {~03 = (—tt) ut- a =lt}. {P4} (Commutative law for addition): rt + b m I: + ct. % (P5) {Associative law for multiplica— a - (h c) = {a - b) . c. tiou) (P6) (Existence of a muilipiicative n - i = l -a = u: l ¢ 0. identity) (i’?) (Existence of multiphcalive a .9"! w it” iCt m 1. for (1 ¢ ti. inverses) (1‘8) (Commutative Law for muiti- a J; :1; -a, plication) (P9) (Distributive iaw) a - (b + c') = (i - l) +(t -c. The three basic properties of numbers which remain to he iistctl are concerned with inequalities. Aitltough inequalities occur rarely in elementary mathematics, they play a prominent role in calculus. The two notions of inequality; (2 «r. I) (a is 3359 than 1;) and (t > b {a is greater than b}, are intimately related: (I < b means the same as b > a (thus 1 e: 3 and 3 > 1 art: mereiy two ways of writing the same assertion). The numbers a satisfying a :> 0 are called positive, while those numbers ri' satisfying a < 0 are caiied negative. While positivity can thus he defined in terms of <, it is possible to reverse the procedure: a < b can he defined to mean that b w a is positive. In fact, it is convenient to consider the colicction of ail positive numhcrs, denoted by P, as the basic concept, and state aii properties in terms of P: (P10) (Trichotoniy law) For every number a, one and only one of the foilowing hoids: ti) a m 0, (it) fl is in the coliection P, (iii) -(z is in the collection P. (P1 i) (Closure under addition) 11” a and b are in P, then (1 +1) is in P. (F12) (Closure under muitiplication) If a and b are in P, then a . l: is in P. Hologue These three properties should he Complententecl wiih the following definitions: a>b if a—bisinP; a<b if h>a; 33b if a>boramb; :95}; if a<b0ra=b.* Note, in parricular, that a > 0 if and only if a is in P. All the familiar facts about inequalities, however elementaty they may seem, are consequences of i’lU—PlZ. lFor example, if a and l: are any two numbers, then precisely one of the following holds: {5} a — b = 0, (ii) a — b is in the collection F, (iii) wk: ~— b) = b m a is in the collection P. Using the definitionsjust made, it follows that precisely one oft...
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