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Bringing Calculus Up-to-DateAuthor(s): M. E. MunroeSource: The American Mathematical Monthly, Vol. 65, No. 2 (Feb., 1958), pp. 81-90Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2308879.

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BRINGING CALCULUS UP-TO-DATEM. E. MUNROE, Universityf llinois

1. Introduction.There are many conversations,committeemeetings,etc.,today about the modernizationof the undergraduatecalculus course; but alltoo often he attack on the problemfalls hortof being comprehensive. alculushas been in cold storage forover fifty ears now. These have been highlypro-ductive years in mathematics, nd the result s that more changesare in orderthan most people would like to admit.The relevant branches ofmodern mathematicswould seemno be real func-tion theory nd differentialeometry, nd a survey of thesefields uggests atleast the five points listed below. It is significant hat point (i) can be gleanedfrom study of real function heory nd is already coming nto vogue in cal-culus, while points (ii)-(v) come largely fromdifferential eometry nd are toogenerally overlooked. Calculus is too often erroneously lassifiedas a part ofreal function heoryonly.

(i) It is essential to distinguishbetween a function and its values f(a).(ii) The coordinate variables x and y of analytic geometry re themselvesmappings;x maps points ntotheir bscissas; y maps points nto their rdinates.(iii) There is an importantdifferencen point of view toward coordinate

variables in analytic and differential eometry. n analytic geometryx and yare definedover the entireplane, and y =f(x) is a conditionalequation describ-ing a locus. In differential eometry he locus is preassigned;x and y are re-strictedto this locus, and y=f(x) is an identity.A calculus problemstartsasanalytic geometry nd finishes s differential eometry.Thus, somewhere nthe middlethesymbols changemeaning. More details on this pointin Section2.)(iv) Modern differentialeometryhas finallyproduced a definition f thedifferentialhat is quite satisfactory orpurposesof ntegration nd differentia-tiontheory. n this definitionheconceptof differential as beendivorcedfromthat ofapproximate ncrement. Details in Section 3.)(v) In the modern, dvanced studyof integration fdifferentialorms hetheory fthe integral s liftedfrom eal function heory, nd thealgebraofthedifferentialormss shown to fit ntothisframework.When thisabstracttheoryis specialized to the simplified, oncrete cases discussed in calculus, moderndifferentialheorybecomesan effectiveand simpler)tool forcertaindevelop-ments n the theory f the integral. See Sections 5 and 6.)2. Functions and variables. To begin with, definethe word functiontomean a mappingofnumbers nto numbers.A later generalizationwill include

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82 BRINGING CALCULUS UP-TO-DATE [Februarymappingsofn-tuples f numbers nto numbers.This is thetraditionaluse of thewordfunctionn calculus.One also encounters paceswhoseelements re called points.These take theformof lines, planes, curves, surfaces,3-space, space-time,etc. An essentialpart of the machinery s the mappingof these points into numbers.Let thesemappingsbe called variables.Linguistically ncongruous s it mayseem,this istraditionally hemostcommon use ofthe wordvariablein calculus.This defini-tion followsthe principle hat language is establishedby usage, not by logic.It retainsthe word and givesa moreenlightened escription fwhatithas beenusedto mean. It should be noted thatprobability heoryhas alreadyestablishedthisusage with thephraserandomvariable.The coordinatevariablesx and y are variables in the sense just defined. nplane analytic geometry ach of these symbols stands for a mapping of theentireplane onto thereal number ystem. fx has thismeaningandf s a func-tion,thenf(x) is used to denote a compositemapping:

x fpoint * number anothernumber.The form(1) y = f(x)is now a conditionalquation.Here there re twosymbols,yandf(x), each stand-ingfor mapping;but in analyticgeometry1) is not regardedas an assertionthat these are two names for he same mapping.Rather, (1) appears (or shouldappear) onlyas a noun clause in the phrase, "the locus ofy =f(x)." This locusis, ofcourse,the set ofall pointsp in the common domain ofx and y forwhichy(p) =f[x(p) ]. Note that this ast "=" meanswhat it should.Suppose, now, that (1) has been given in propercontextand its locus hasbeen foundand named C. The next step in the logical analysis of a calculusproblem s to restrict he mappingsx and y to C. Note that this could not bedone beforebecause the superstructure escribed above was used to defineC.It would help ifthe restrictedmappingsweregivennewnames; say, start withX and Y and boil them down to x and y. However,this is probablyaskingfortoo muchof a change in set habits. In any case, withx and y restricted o C,(1) becomesan identity-an assertionthat twomappingsare the same. At thisstage one has a special case ofthe typestructure tudied in moderndifferentialgeometrynd so can turn othe iterature fthatdisciplinefor urthernlighten-ment.What follows s an informal ummaryof the specializationof this workto calculus. For a morecompletediscussionof thegeneraltheory ee, for xam-ple, Chevalley,Theory fLie Groups,Princeton,1946.

3. Differentials.A one-dimensionalmanifold s a structureconsistingof apoint set C and a set of coordinate ariablesdefinedthereon. Precise postulatescan be given,but a very oose descriptionwill suffice or hepresentdiscussion.The set C is a smooth curve,and each ofthe coordinatevariablesis continuous


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1958] BRINGING CALCULUS UP-TO-DATE 83and locallyone-to-one. ince each is locallyone-to-one, ach pair u, v is relatedat least locally by an identity ftheform =f(u), and it is convenient o assumethat each such connecting unction has threederivatives.Note that a locus intheplane with therectangular oordinatevariables restricted o it willgenerallyform uch a structure.Consider threeclasses ofvariables on C.

X: all variables on C,Y: thoserelated to a coordinate by a differentiableunction,Z: thoserelated to a coordinate by a thrice-differentiableunction.Let D be a mappingof Y into X, and letDu denotethe map of u byD. Ifp is apoint on thecurve, denote thevalue ofDu at p by (Du),. Such a mappingD iscalled a derivativeperator t p provided

(a) [D(au + bv)], = a(Du)p + b(Dv)p,(b) [D(uv)i] = u(p)(Dv)p + v(p)(Du)p.

Here u and v are variables and a and b real numbers.Regarding a constanta as a variable,observe that(2) (Da)p = 0,because by (a), [D(au)]p=a(Du)p, and by (b), [D(au)J,=a(Du)p+u(p)(Da)p;and ifu is chosen so that u(p) #0, then (2) follows.Observe also that settingu=v in (b) yields(3) [D(u2)]p = 2u(p)(Du)p.

Now, let u be a coordinate, nd let v=f(u). Assumef has threederivatives;thenthere s a differentiableunction such that(4) v= v(p)+f'[u(p)][u - u(p)] + g(u)[u u(p)]2.Recalling (2) and (3), operate on (4) withD at p to obtain(5) (Dv)p = f'[u p) (Du) p.Each of the othertermson the rightvanishes eitherbecause of (2) or becauseit containsthefactor (p) - u(p). Note that inthisstepD must be defined n Ybecause it mustoperateon g(u). However, (5) is provedonlyforvEZ becausethis is required n orderto set up (4).Let D. be the operatorsuch that D.u = 1 at each point on the curve. By(5) thisdeterminesD. on vwherev=f(u), and indeed(6) D.v = f'(u).The variable D.v is called the derivativefv withrespect o u. The notationf'denotes a purelyfunction heoretic oncept; f' is defined rom interms f imitsofdifferenceuotients.On the otherhand,D. is an operatordefinedbypurely


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84 BRINGING CALCULUS UP-TO-DATE [Februaryalgebraic conditions.Thus, (6) is not a definition; t is a theorem elatingthesetwo basically differentotions.Consider now D1 and D2, two derivative operatorsat p. If for ome variableu, (Diu),=a(D2u),, then forv=f(u),

(Div) = f'[u(p) ] Dlu) = af' u(p) D2u)p = a(D2v),.That is, at p any two derivative operators are constant multiples one of theother. So, the set of derivative operators at p has the algebraic structure f astraight ine. Identify t withT., the tangent ine to the curve at p, and definethetangent undle or hegiven curve as the set of all orderedpairs (p, D) wherep is a point of the curve and D is a derivative operatorat p.Finally, the differentials defined s follows.For u a variable on the curve,duis a variable on the tangentbundle. Its value at (p, D) is denoted by du,(D),and du is definedby(7) dup(D) = (Du)p.

Substitute (6) into (5) to obtain(8) (Dv), = (Duv)v(Du)v.By (7), (Dv), and (Du)p are values of appropriate differentials;o (8) may bewritten

dvp(D) = (Duv)vduv(D).This is for n arbitrary on the curve and D on Tp; so(9) dv = Duvduover the tangentbundle. An appropriatenamefor 9) is "Fundamental Theoremon Differentials." t anyrate, n the modern heory tis a theorem, ota revolv-ing definition hat changes meaning with every change of "independent" vari-able.

Multidimensional ases follow he same pattern.Let C be an n-dimensionalmanifold-intuitively, a uniformly -dimensional, mooth set with variablesattached. The basic formrelating variables will be v=f(u1, * , us). Derivativeoperators re defined ythesame postulatesas before, nd bythen-dimensionalgeneralization f (4) it is shown thattheyform family f n-dimensional ectorspaces. A derivative operatoris determined f specified n n variables,and theoperatorsd/duidefinedby0uj f1 for j= iOui tO for j - iform basis in the tangentbundle. Differentials re defined gain by (7), andthe fundamental heoremreads


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1958] BRINGING CALCULUS UP-TO-DATE 85n av(10) dv= E dui.

4. Some immediate advantages. The ultimate advantage of saying thingscorrectly ather than incorrectly eed not be discussed. Let it be understood,then, that the following list-which is quite probably nonrepresentative-merely points out a few of the common difficulties hat can be cleared up bythemodernapproach.(a) Differentialsdentified ith ariables.A variablevon Cgenerates differ-ential dv on the tangentbundle,and this differentialetains ts identitywhetherv=f(u) or v= g(w). Note that df and dg are not defined.Understanding f thiswill help to clear up many notationobscurities.(b) Seconddifferentials.o one seems to know exactly what these ought tobe. Function theoristssometimes want d2v-D.vdu2; at other times (whenchanges ofvariable are in order) they want d2v=D'vdu2+D.vd2u. In working,say, withparametric quations thestudent s apt to use the first f these formsas though t had the invarianceproperties fthe second. It seems advisable toleave these out of calculus completely, and the development outlined abovedoes thisvery nicely.There thedifferentialf dv s not definedbecause thedo-main of dv is the tangentbundle while the differentials defined only for avariablewhose domain is C.(c) Chain rules.Givenf(x, y, u, v) =g(x, y, u, v) =0, it is commonpracticeto find, ay, au/axn terms of otherpartial derivatives by writing ut variousexpressionsmodeled after 10), making substitutions nd equating coefficientsof an appropriatedifferential. his is an effective echniqueforderiving hainrules, and in the moderntheory t is quite acceptable. However, if (10) is adefinition atherthan a theorem, t must be proved invariantundercoordinatechanges.One must oftenknow that the differential orm s invariant under theverychain rulehe is seekingto find(d) Increments nd approximations. he undergraduate s afraid to use thedifferential-insuch manipulations as (c), for example-because of the ap-proximation bugaboo introducedwith the now outmoded definition.In themoderntheorythis fear is never introduced, nd the approximateincrementproblem s properly lassified-under Taylor's theorem.5. Line and surface ntegrals.Let u and vbe variables on a curve C. Parti-tion C by pointspo,pi, * * *, pn. or each i, let qi be the pointon the tangentline Tj whosedistancefrompi is the lengthof thearc Pipi+ion C. Intuitively,"unroll" each increment f arc onto a tangent ine. As noted in Section 3, the

set of derivativeoperators t pi forms "line,"to be identifiedwiththetangentline T, To obtaintheusual geometricnterpretation,pecify hatifs is the arclengthvariable on C, thenD8 is identifiedwiththe unittangentvector.In any


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86 BRINGING CALCULUS UP-TO-DATE [Februarycase, each pointqi constructed bove is identifiedwitha derivative operator atpi; so the notationdv,i(qi) has an obvious meaning. Now, form ums

n-1(11) ~~~~~~E(pj)dvp,(qi) .i=O

Take an appropriate imit of these, and call the result(12) fudv.

The importantfeature to this development of the line integral s that thepoints qi on thetangent ines are determined y an intrinsic otion arc length)on C and have no connectionwith thevariables u and v.Thus, the substitutiontheorem, e udv=fc uDwvdw, s an immediate consequence of the fundamentaltheoremon differentials9). The approximating ums (11) are invariantunderthe substitution;hence so is the integral.The familiar a8'f(x)dxs now a specialcase of (12)-in which C is an interval on the x-axis. In this developmentthedifferentials clearlythe same thing n differentialnd integral alculus.In a double integral-carefully to be distinguished rom n iterated ntegral-exteriormultiplicationf differentials ustbe introduced.This is an operationdenoted by * and characterizedby the rule(13) du *dv = - dv*du.This stems from hefact that du *dv s to be a signed measure on orientedplaneregions.Settingu = v in (13) yields the corollarydu*du= 0. Using theserulesofmultiplication plus a distributive aw) and substituting romthe two-dimen-sional case of (10), one has

du*dv= -dx + - dy * dx+- dya8x ay a

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1958] BRINGING CALCULUS UP-TO-DATE 87a and b can be chosenso that (adu,, bdv,)forms set of rectangular oordinatesin T,. In general, a and b depend on p; for example,on the tangentplanes tothe unitsphere (d+, sin4 dO) generatesrectangular oordinates. n any case, if(adu,, bdv,) s a rectangular et on Tp,and if A is a region n T, positivelyori-ented with respect to this coordinatesystem,thendefineab(du *dv),(A) to bethe area of A. Since theJacobian ofa rotation s unity, 14) shows that this isconsistentfor all such rectangularsets. Values of other exteriorproduct oper-ators are then determined y (14).Now, the substitution ule formultiple ntegralsconsists merelyof makingthe substitution 14) behindthe ntegral ign.A definitionfthe surface ntegralpatterned after that givenabove for the line integralmakes this remarkablyeasy to justify.Take a smooth surface S and partition it in an arbitraryfashion-notnecessarilyfollowing oordinate ines-into subsets Si. At some point pi in Sitake a tangent plane. Here a minorcomplicationappears; one cannot "unroll"the Si as he did the arcs and preservemeasure.However, define urface rea-thereare many tricks foraccomplishingthis-and constructon each tangentplane a set having the same area as Si. Call thisplane set Ai. The shape ofAiis immaterialbecause (see definition bove) (du *dv),i(Ai) depends onlyon thearea and orientation fAi, not on its shape. Now, form ums

n(15) E w(pi)du*dv) ,(Ai).i=lTake the usual limit, nd call it

fwdu * dv.Since theAi dependinno wayon u and v,the sums (15) are invariantunderthesubstitution 14); and thereemergesthe familiar esult

frwdu*dv = w dx*dy.s s (x, y)6. Theorems ofGreen,Gauss, and Stokes. This viewof themultiple ntegralbrings n elegantunification o the theoryrelating n integralover a manifoldto one overtheboundary.There is a masterformula:

(16) ... udv1* .. * vk =.. . f du*dv* dVkwhere M is an oriented (k+1)-dimensional manifoldand B is its boundary.This is provedin theusual way by reducing he integral n theright o an iter-ated integraland performingne integration.Now, ifA is a region n theplane and C its boundary,then (16) yields


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88 BRINGING CALCULUS UP-TO-DATE [Februaryf(udx + vdy)= "A *dx+ dv*dy).

However,au 1u _ audu*dx = -dx++ -ddy *dx = - dx*dy,-ax ay _ 49y

-dv av 1 avdv*dy = [ dx + dy *dy= -dx*dy;so the result s Green's theorem.A similar maneuverwith the form

f(udx + vdy wdz)where C is an appropriate space curve yields the classical theoremof Stokes.The divergence heorem s also a special case of (16) as is easily shownby directcomputation.Minus signsdo not appear in the divergence heorembecause inall this work the permutationsof differentialsre cyclic permutations,nd acyclic permutation fthreethings s an even permutation.When presented n this light,all these theorems ppear as generalizationsof the fundamental heoremof calculus. For, this latter may be writtenu(q)-u(p) = fdu, and thisis a specializationof (16) in which the differencen theleft s regardedas an integraloverthe zero-dimensional oundary consisting ftwo points.Subtractionresultsbecause the boundary s consideredoriented.

7. Notation. Roughly, the structure tudied in calculus seems to consist ofmappings variables) "up" from oint sets into thereal number ystem ogetherwith othermappings (functions)"across the top." It is well to preservethisstratificationn thenotation,and the usual use off and g forfunctions nd thelast lettersof the alphabet forvariables is all to the good. However, there areotherdistinctions hat common notationalpracticefailsto make.The derivativeof a function s anotherfunctionwhose values are appropriatelimitsof difference uotients.Note that a function s not differentiatedwithrespectto" anythingnparticular.Thus, an appropriatenotationfor he deriva-tive off isf', and itwouldbewell ftheprime ymbolwerereservedforfunctionsonly.Derivativesof variables are generatedby the derivativeoperators ntroducedabove, and thebasic form s Dxy.The differentialormdy/dx s not merely nalternative o this. t meansdy . dx,but it follows t once from he fundamentaltheoremon differentialshatdy/dx Dxy.Hybridnotationsuch as y' and df/dx bounds in the literature, ut it is notreallywell-defined. ne step in the modernization f calculus is to break somewell-formedtoo well-formed) abits in thisrespect. Frequently,the confusionstemsfromwriting =f(x) and thenconfusing hesymbolsyandf. As forhabits,


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1958] BRINGING CALCULUS UP-TO-DATE 89however, t is sad to relatethattheauthor ofa recent textbookmade essentiallythe comment ust made hereabout confusing and f and thenproceeded to doexactlythat threepages later.Similardistinctions eed to be made in themultidimensional ases. Iffmapsn-tuples ofnumbers nto numbers, ts arguments re distinguished y positiononly. For example,f(a, b) =a-b and f(b, a) =b-a definethe same function .Thus, fi is appropriate notation for the partial derivative of f with respect toits ith argument.As noted above, a/ax s a derivativeoperator nmanifold he-oryand should be reservedfor pplicationto variables.

8. Existence theorems. Little has been said so far about the role of realfunction heory n a programofmodernizing alculus. In a sense thisis a ques-tionquite independent f the ones raised so farhere,but certainly hefollowingquestion is pertinent o the presentdiscussion.The algebraic theoryof deriva-tive operators and differentialsutlinedabove yieldsformulas nd techniquesin a very elegantfashion, ut it certainly s not selfcontained. s any additionalreal function heory required as a backgroundto tightenup the logic in thisdevelopment?In integralcalculus one must develop the notion of integralof a functionalongsidethatof ntegral fa variable (outlinedabove). The twoare fairly asilyrelated,and the usual existenceproof ppears in the function heory.In differentialalculus the existencequestionsare answereda priori n thepostulates fora manifold.That is, differentiabilityonditionsare imposed byfiat on the connectingfunctions n such a way that the algebraicallydefinedderivativeoperatorsdo represent ifferentiationsn thefunction heoretic ense.However, to keep from operatingin a vacuum, one must show that certainthingsare manifolds.For purposesofcalculus one getsan adequate supplyof manifolds y takingtherectangular oordinates n (n+k)-space and considering he locus ofn equa-tions in them. The implicit function theorem (a standard item in advancedcalculus) gives conditionsunder which k of these variables become local co-ordinateson the locus,and theBrouwertheorem n invariance of domain (nowappearingin the better advanced calculus texts) guaranteesthat the locus hasthe properdimension.*To verify he hypothesesof these theorems, ne mustknow thedifferentiabilityroperties ftheelementary unctions. hese maybeestablished ntheusual way through heorems ndifferentiabilityf sums,prod-ucts, composites, tc.It is probably fair to say, then,that the introduction f themoderntheoryofthedifferentialhanges very ittlethenumberor thenature ofthereal func-tion theoryexistenceproofsrequiredfora logicallysound developmentof cal-culus. This, ofcourse, s subject to the stipulationthat calculus is restricted o

*Added nproof:A recent oteby Yamabe,thisMONTHLY, vol. 64,1957,p. 725,givespre-cisely he result eeded.


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90 COEFFICIENTS OF RECIPROCAL POWER SERIES [Februarythe well-behaved cases. A general attack on the topological problemsof mani-fold theory s muchmore difficult.

9. Two facets to the problem. Most programsfor the improvement f cal-culus amount to the injection nto the course of more and betterreal functiontheory. t is the purpose ofthe presentpaper not to discourage this, but to sug-gest that a more imperative project is the injection of better-if not more-differentialeometry. his latter s more mperativebecause thegarden varietycalculus containsmoredownright rrors n differentialeometry han in realfunction heory.On the otherhand, it is more difficult or t least two reasons.First,to bring hedifferentialeometry p todatewillchange the appearance ofa calculus text; it will modifydefinitions, erminology, otation, procedures.Second, while most calculus teachers are well grounded n real function heory,formanyofthemgraduate study came before ome of the importantdevelop-ments n differentialeometry.Despite thesedifficultiesn efforthould be made. In failing o bring alculusup to date we are transmittingo the nextgenerationnot the nformation vail-able to our contemporaries, ut that available to our grandfathers.

ON THE COEFFICIENTS OF RECIPROCAL POWER SERIESJOHN LAMPERTI, Californianstitute fTechnology1. Introduction.upposethatthe two equences fnumbers n and {fn}satisfy he relations

n-1(l) ~~~~~~Un =Ui,fn-i + bon) n =0,1,I*i=O

whereJ1 f i=jbj 0O f i js

Upon takingthe generating unctions f the sequences,00 00U(Z) = UnZn and F(z) = E fnZ,n=O n=l

relation (1) may be re-expressed s(2) U(z) = U(z)F(z) + 1,wherethepowerseries and theirmultiplication re to be interpretedn a purelyformalway. If U(z) and F(z) satisfy 2), thenwe shall say that F(z) is the re-

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