Notes on Calculus Analytic Interpretation Helps Understand Physics Applications

8/10/2019 Notes on Calculus Analytic Interpretation Helps Understand Physics Applications

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p 80 djvu p 96 Courant

3. Extensions, Notation, Fundamental Rules.The above definition of the integral as the limit of a sum ledLeibnitz to express the integral by the following symbol:f /(x)&.

The integral sign is a modification of a smmation sign whichhad the shape of a long S. The passage to the limit from a sub-division of the interval ino finite portions Ax, is suggested bythe use othe letter d in place of A. We must, however, guardourselves against thinking of dx as an "infinitely small quan-tity" or "infinitesimal ", or of the integral as the "sum ofan infinite number of inn{tely small quantities ". Such a con-ception would be devoid of any clear meaning; it is only a naivebefogging of what we have previously carried out with precision.

!!!!!!!!!!!!!!!!!!!!!!!!!!!COURANT ON FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS AND CALCULATING DEFINITE INTEGRALS BY USING DEFINITE BY COURANT!!!!!!!!

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4. THE INDEFINITE INTEGRAL, THE PRIMITIVE FUNCTION, AND

THE FUNDAMEntAL THEOREMS OF THE DIFFERENTIAL ANDINTEGRAL CALCULUS.As we have already mentioned above, the conneon betweenthe problem of integration and the problem of dierentiation isthe corner-stone of the dierential and integral calculus. Thisconnexion we will now study.1. he Integral as a Function of the Upper Limit.The value of the definite integral of a functionf (z) depends onthe choice of the two limits of integration a and b. It is a func-tion of the lower limit a as well as of the upper limit 5. In order


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to study this dependence more closely we imagine the lowerlimit a to be a definite fixed number, denote the variable of in-egration no longer by z but by u (cf. p. 82), and denote the upperlimit by z instead of b in order to suggest that we shall let the

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txo FUNDAMENTAL IDEAS [CHar.upper limit vary and that we wish to investigate the value ofthe integral as a function of the upper limit. Accordingly, weff(u) du = (x).

We call this function q)(x) an indefinite integral of the functionf(x). When we speak of an and

not of te indefinite integral, wesuggest that instead of the lowerlimit a any other could be chosen,in which case we should ordi-narily obtain a different valuefor the integral. Geometricallythe indefinite integral for eachvalue of x will be given by thearea (shown by shading in fig. 17)under the curve yf(u) andbounded by the ordinates u---- aand u x, the sign being determined by rules given earlier(p. 81).

If we choose another lower limit a in place of the lower limita, we obtain the indefinite integraltF (x) f (u) du.The difference tF(x) -- ((x) will obviously be given bywhich is a constant, since a and a are each taken as fixexl givennumbers. ThereforeT(x) = $ (x) + const.;Different indefinite integrals of the same function differ onlyby an additive constant.

NOTE: Thus in the mathematical proof, we take the lower boundary as a constanta and make the upper

boundary vary (and thus replace by x)(and them being indefinite integrals, another graphical representation of + C, insteadof just vertical shifts in graph) and then take a different >lower boundary< andtake the integral betweenlower boundaries, all to show that "the difference between indefinite integralsdiffer only by a constant", sinceof course in the integral between both lower boundaries they are both constants"a" and thus are a constant, and do notvary.


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We may likewise regard the integral as a function of the lowerlimit, and introduce the functionin which b is a fixed number. tiere again two such in-

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tegrals with different upper limits b and differ only by anadditive constantff(u) du.

NOTE: This is just to show that the indefinite integrals differ only by a constant still even when upper boundaries are given a fixedconstant instead.

2. The Derivative of the Indefinite Integral.We will now differentiate the indefinite integral (I)(x) with

respect to the variable x. The result is the following theorem:The inclefrite integralq) (x) --- f.f(u) duof a continuous function f(x) always possesses a derivative (I)' (x),that is, differentiation of the indefinite integral of a given eon-tiuous function always gives us back that same function.This is the root idea of the whole of the differential and integralcalculus.

NOTE: Thus he considers the root idea of the whl of differential and integral calculus to be that the differentiation of the indefinite integral of a given continuous funciton always givesus back that same function.

The proof follows extremely simply from the inter-pretation of the integral as an area. We form the dierenquotientq) (x + h) -- q) (x)h 'and observe that the numeratorq) (x + h)- q) (x) =J, +hf (u, du _ f,f(u)du = f+f (u)duts the axea between the ordinate corresponding to x and theordinate corresponding to x + h.

Now let x o be a point in the interval between x and x + h atwhich the function f(x) akes its greatest value, and x a a pointat which it takes is leaat value in that interval (of. tlg. 18).

NOTE: Thus we are differentiation with respect to x, not with respect to udu, soit's x, whichis the upper boundary, that we're differentiating with respect to. The integralin teh numeratorthen becomes from x to x+h which shows that the derivative which we divide by h


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will bethe function itself.

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Then the area in question will lie between the values hf(xo) andhf(xx) , which represent the areas of rectangles with the intervalfrom x to x q- h as base and f(xo) and f(xx) respectively as alli-tudes. Expressed analytically,f(Xo) q) (x q- h) - q) (x) _f(x).-- hThis can also be proved directly from he definition of the in-tegral without appealing to the geometrical interpretation.*To do this we write+hf(u)du = lira f(uv)Auv,where % , ,x, u..... un x q- h are points of division ofthe interval from x io x q- h, and the greatest of the absolutevalues of the differences A% %- %_ends to zero as nincreases. Then A%/h is certainly positive, no maer whetherh is positive or negative. Since we know that f(x0) f(u)f(x)

and since the sum of the quantities A% is equal to h, it followsthatf(xo) _ and hus if we let tend to infinity we obtain the inequalitiesstated above for1 f+(I) (x + h) - (I) ()f(u) du orIh now ends o zero, both f(%) and f(x) must end to thelimit f(x), owing to the ontinnity of the fimcfion. We thereforesee at once that)'(x) = )(x + ) - )() =_y(x),as stated by our theorem.

Owing to the differentiability of (I)(x), we have She followingtheorem, by 3, No. 5, p. 97:The integral of a eotinuous funetioi(x) is itself a continuousfunction of the upper limit.

NOTE: This I haven't seen before that the integral of a continuous function f(x)is itself a continuous funciton of the upper limit.


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For the sake of completoness we would point out that if weregard the definite integral not as a function of its upper limitbut as a function of its lower limit, the derivative is not equalto f(x), but is instead equal to --f(x). In symbols: if we put4(x)then 4' (x) -- -- f (x).The proof follows {mmediaiely from the remark thatf (u)du : --f (u)du.

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group of all primitive functions is answered by the followingtheorem, sometimes referred to as the fundamental theorem ofthe different!a! and integral calculus:The difference of two primitives F(x) and Fz(x) of tksame fuw-

tion f(x) gs always a constant:Thus from any one primitive function F(x) we caobtaothers in the form(x) + by sule. choicof tconstant c. Oonversely, for every valuofte constant o the expression F(x) F(x) - c represents a primi-tive function of f(x).

NOTE: Thus two different sections cover what was covered in Kilmogrov on the fundamental theorem relating definite and indefinitne integrals as well.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

from Mathematics for Nonmathematician Morris Kline

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The major idea characterizing the integral calculus is the inverse to thatunderlying the differential calculus: namely, instead of finding the derivativeof a function from the function, one proceeds to find the function from thederivative. Of course, all really significant ideas prove to have extensions andapplications far beyond what is immediately apparent, and we shall find thisto be true of the integral calculus also.


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NOTE: Thus when we find the integral, we're always assuming that the function we're finding an integral of (the integrand) is a derivative.

!!!!!!!!!!!!!!!!!!!On Integrals from Kolmogrov Aleksandrov etc

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We suppose that a material point is moving along a straight line withvelocity v = f(t), where t is the time. We already know that the distancea covered by our point in the time between t = t l and t = 1 2 is givenby the definite integralf t.a = fit) dt.t,Now let us assume that the law of motion of the point is known to us;

that is, we know the function s = F(t) expressing the dependence on thetime t of the distance s calculated from some initial point A on the straightline. The distance a covered in the interval of time [11' 1 2 ] is obviouslyequal to the differencea = F(1 2 ) - F(t l ).

In this way we are led by physical considerations to the equalityf t.f(t) dt = F(t 2 ) - F(I I ),t,

which expresses the connection between the law of motion of our pointand its velocity.

NOTE: Thus this section is about connection between differnetial and integral calculus, and this section also isabout the general mehtod for finding definite integrals, so these twotopics are intimately related, or even one and the same.

NOTE: Thus by physical considerations, if we have a function of velocity and also know it's indefinite integrals as a funciton of positionor distance, then we know that the definite integral (distance covered in interv

al from t0 to t1) from t0 to t1 interval is F(t1) - F(t0), andthus physically this makes sense.

From a mathematical point of view the function F(t), as we alreadyknow from 5, may be defined as a function whose derivative for allvalues of t in the given interval is equal to f(I), that isF'(I) = f(t).Such a function is called a primitive for f(I).


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We must keep in mind that if the functionf(t) has at least one primitive,then along with this one it will have an infinite number of others; for ifF(t) is a primitive for fit), then F(t) + C, where C is an arbitrary constant,is also a primitive. Moreover, in this way we exhaust the whole set ofprimitives for f(t), since if F I (I) and F 2 (1) are primitives for the samefunction f(I), then their difference 4>(t) = FI(t) - F 2 (1) has a derivative1O. INTEGRAL133q,'(t) that is equal to zero at every point in a given interval so that q,(t)is a constant. *

NOTE: Thus this part says that we can reasonably cancel out the constants C in the integral, whentaking the difference F(t1) - F(t0) because there's an infinite amount of indefinite integrals with constant's C andthere must be one therefore, in which these constants cancel out. (this is donestating the use of hte mean value theorem).

From a physical point of view the various values of the constant Cdetermine laws of motion which differ from one another only in the factthat they correspond to all possible choices for the initial point of themotion.

We are thus led to the result that for an extremely wide class of functionsf(x), including all cases where the function fix) may be considered as thevelocity of a point at the time x, we have the following equalitytr f(x) dx = F(b) - F(a),a(30)where F(x) is an arbitrary primitive for f(x).

This equality is the famous formula of Newton and Leibnitz, whichreduces the problem of calculating the definite integral of a function tofinding a primitive for the function and in this way forms a link betweenthe differential and the integral calculus.

NOTE: Thus these + C constants, physically interpreted means that they only differ from one another in thatthey correspond to all possible choices for the initial point of the motion.

NOTE: Thus as they differ only in their initial points of motion, we get the in

terval by subtractingon the interval, where the initial points won't affect the values.

!!!!!!!!!!On physical interpretation of derivative and why best not to think graphically!!!!!!!!!!!!!


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3. THE DERIVATIVEThe concept of the derivative, like that of the integral, is ofintuitive origin. Its sources are (1) the problem of constructinghe tangent to a given curve at a given point and (2) the problemof finding a precise definition for the velocity in an arbitrarymotion.


2. ae Derira[ire as a Velocity.Just as naive intuition led us to the notion of the directionof the tangent to a curve, so it causes us to assign a velocity to amotion. The definition of velocity leads us once again to exactly

the same limiting process as we have already called differentia-tion.Let us consider, for example, the motion of a point on astraight line, the position of the point being determined by asingle co-ordinate y. This co-ordinate y is the distance, with itsproper sign, of our moving point from a fixed point on the line.The motion is given if we know y as a function of the time t,y--f(t). If this function is a linear functionf (t) ---- cq- b, we callthe motion a uniform motion witlz tlze velocity , and for every pairof values t and t x which are not equal to one another we can writee _f(t) --f(t)The velocity is therefore the difference quotient of the functionct q-b, and this difference quotient is completely independent

of the particular pair of instants which we fix upon. But whatare we to understand by the velocity of motion at an instant tff the motion is no longer nullorton.

In order to arrive at this definition we consider the differencequotient f(q) --f(t), which we shall call the average velocity inthe time interval between t a and t. If now this average velocitytends tea defmibe limit when we let the instant t a come closerand closer to t, we shah natqlmlly define this ]imlt as the velocity


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at the time t. In other words: tlze velocity at tlze time t is tlze deri-vativef, (t) -_- nmf(q) - f(t)From this new meaning of the derivative, which in itself hasnothing to do with the tangent problem, we see that it really is


appropriate to define the limiting process of differentiation as apurely analytical operation independent of geometrical intuitions.

Here again the differentiability of the position-function is anassumption which we shall always tacitly make, and which, infact, is absolutely necessary if the notion of velocity is to haveany meaning.


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7. The Derivative and the Difference Quotient.The fact that in the limiting process which defines the deri-vative the difference Atends to 0 is sometimes expressed bysaying that the quantity Abecomes infitely small. This expres-sion indicates that the passage to the limit is regarded as a pro-cess during which the quantity r is never zero, yet approacheszero as closely as we please. In Leibnitz's notation the passageto the limit in the process of differentiation is symbolically ex-pressed by replacing the symbol A by the symbol d, so that wecan define Leibnitz's symbol for the derivative by the equationdy Aydz oAz'


Def def this 12.5!

If, however, we wish to obtain a clear grasp of the meaning ofthe differeial calculus we must beware of regarding the deri-vative as the quotient of two quantities which are actually"infinitoly small ". The differerwe quot/eA__y absolutely mustbe formed with differences ax which are not equal to O. Afterthe form/ng of this difference quotient we must imagine thepassage to the limit carried out by means of a transformation orsome other device. We have no right to suppose that first Ax

goes through something like a limiting process and reaches avalue which is infinitesimally small but still not 0, so that Azand Ay are replaced by "innltely small quantities" or "in-fmitesima]s" dz and alp, and that the quotient of these quanti-ties is then formed.

NOTE: Thus the main argument for not conceiving of dy/dx as a quotient of infinitisemals is becausewe must necessarily form the difference quotient FIRST with delta x not equal to0, BEFORE taking the limit.


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alp, and that the quotient of these quanti-ties is then formed. Such a conception of the derivative is in-compatible with the clarity of ideas demanded in mathematics;in fact, it is entirely meaningless. For a great many simple-minded people it undoubtedly has a certain charm, the charm ofmystery which is always associated with the word "infinite ";and in the early days of the differential calculus even Leibnitzhimself was capable of comb'ming these vague mystical ideaswith a thoroughly clear understanding of the limiting process.It is true that this fog which hung round the foundations of thenew science did not prevent Leibnitz or his great successorsfrom finding the right path. But this does not release us fromthe duty of avoiding every such hazy idea in our building-upof the differential and integral calculus.

The notation of Leibnitz, however, is not merely agractivein itself, but is actually of great flexibility and the utmostusefulness. The reason is that in many calculations and formaltransformations we can deal with. the symbo/s dy and clxin exactly the same way as if they were ordinary numbers. Theyenable us to give neater expression to many calculations which

can be carried out without their use. In the following pageswe shall see this fact verified over and over again, and shallfind ourselves justified in making free and repeated use of it,provided we do not lose sight of the symbolical character ofthe signs dy and dx.

NOTE: Thus we in many transformaitons we dan deal with dy and dx "as if" they were ordinary numbers.


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Def def this 12!

While tiao idea of the differential as an inCh;rely small quan-tity has no meaning, and it is accordingly futile to define thederivative as the quotient of two such quantities, we may stilltry to assign a sense to the equationf'(x) ---- dy/dx in such a waythat the expression dy/dx need not be thought of as purelysymbolic, but as the actual quotient of two quantities dand d.For this purpose we first define the derivative f'(x) by our limit-ing process, then tbin]of x as fixed and consider the incrementh Az as the independent variable. This quantity h we callthe differential of z, and write h dx. We now define the ex-pression dy = y'dx = ]f'(x) as the differential of te function y;dy is therefore a number which has nothing to do with infinitelysmall quantities. So the derivative y,y' = f'(x) is now really the quotientof tkdifferentials dy and dx; butin this statement there is nothing

remarkable; it is, in fact, merelya tautology, a restatement of theverbal definition. The dierentialdy is accordingly the linear part ofthe increment Ay (see fig. 16).

NOTE: Thus even when thinking of dy as a differential, it apprently isn't meaning infinitey small quantities, butis in fact, a tautology ( a mere restatement of the verbal definition).

We shall not make any hn-mediate use of these differentials.0 Fi. :6.--The differential aNevertheless, it may be pointed out for She sake of completenessthat we may also form second and higher differentials. For ifwe .bln]of as chosen in any manner, but always the samefor every value of x, then dy= hf'(x) is a function of x, of whichwe can again form the differential. The result will be calledthe second differentiaof y, and will be denoted by the symboldd2f(x). The increment of ]f'(x) being h(f'(z - ]) --f'(x)],

the second differential is obtained by replacing the quantity inbrackets by its linear part hf"(x), so that d2y--- -- hf"(x). Wemay naturally proceed further along the same lines, obtainingthird, fourth, ... differentials of y, .&c., which can be definedby the expressions hf'"(x), haftS(x), and so on.

NOTE: See how the differential dy = hf'(x) is a function of x, and is also a function of x of which further differntials


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can be formed.

10. Remarks on Applications to the Natural eienees.In the applications of mathematics to uatural phenomenawe never have o deal with sharply defined quantities. Whethera length is eay a metre is a question which cannot be decided


by any experiment and which consequently has no "physicalmeaning ". Again, there is no immediate physical meaning in say-

ing that the length of a material rod is rational or irrational; wecan always measure it with any desired degree of accuracy inrational numbers, and the real matter of interest is whether ornot we can manage tperform such a measurement using rationalnumbers with relatively small denominators. Just as the ques-tion of rationality or irrationality in the rigorous sense of "exactmathematics" has no physical meaning, so the actual carryingout of limiting processes in applications will usually be nothingmore than a mathematical idealization.

NOTE: Thus Courant's stance is that in physical appications, the actual carryingout fo a limiting process will

usually be nothing mroe than a mathematicla idealization.

The practical significance of such idealzations lies chieflyin the fact that if they are used all analytical expressionsbecome essentially simpler and more manageable. For example,it is vastly simpler and more convenient to work with thenotion of instantaneous velocity, Which is a function of onlyodefinite time-instant, than with the notion of averagevelocity between two different instants. Without such idealiza-tion every rational investigation of nature would be condemnedto hopeless complications and would break down at the veryoutset.

NOTE: Thus he thinks the significance of such idealizations is that they are simpler and more manageable.


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Def def this 12!

We do not intend, however, to enter into a discussion of therelationship of mathematics to reality. We merely wish toemphasize, for the sake of our better undersfnding of the theory,that in applications we have the right to replace a derivativeby a difference quotient and vice versa, provided only that thedifferences are small enough to guarantee a sufficiently closeapproximation. The physicist, the biologist, the engineer, oranyone else who has to deal with these ideas in practice, willtherefore have the right to identify the difference quotientwith the derivative within his limits of accuracy. The smallerthe increment h-----dof the independent variable, the moreaccurately can he represent the increment Ay-----f(x - h)- f(x)by the differential dy hf'(x). So long as he keeps within thelimits of accuracy required by the problem, he is accustomedto speak of the quantities d----- h and dy hf'(z) as "infinitesi-mals ". These "physically infinitesimal" quantities have aprecise meaning. They are finite quantities, not equal to zero,which are chosen small enough for the given investigation,e.g. smaller than a fractional part of a wave-length or smallerthan be distance between two electrons in an atom; in general,smaller than the degree of accuracy required.

NOTE: Thus as long as we keep within the limits of accuracy.

NOTE: Also notice he was against "infinitisemal" interpretation before but now says that"physically infinitesimal" quanitites have a precise meaning, that they are finite quantiites not equal to 0, chosen

small enough for the given investigation, such as fracitnal part of a wavelentght or smaler than the distance between two electrons in anatom ( in general smaller htan degree of accuracy required.)


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Def Def this 12.25!

2. The Question of Applications.The relation of the primitive sum-function to the density ofdistribution perhaps becomes clearer when it is realized thatfrom the point of view of physical facts the limiting processesof integration and differentiation represent an idealization, andthat they do not express anything exact in nature. On the con-trary, in the realm of physical actuality we can form in place ofthe integral only a sum of very many small quantities and inplace of the derivative only a difference quotient of very smallquantities. The quantities Az remain different from 0; thepassage to the limit Az-+ 0 is merely a mathematical simplifi-cation, in which the accuracy of the mathematical representationof the reality is not essentially impaired.

NOTE: Thus in physical acutality in place of the integral we can form only a sum

ofvery small quantities and in place of the derivative we can form only a difference quotient of very small quantites.

As an example we return to the vertical colimn of air. Ac-cording to the atomic theory we find that we cannot thln]r ofthe distribution of mass as a continuous function of z. On thecontrary, we will assume (and this, too, is a simplifying. ideali-zation) that the mass is distributed along the z-axis in the formof a large number of point-molecules lying very close to one

another. Then the sum-function F(z) will not be a continuousfunction, but will have a constant value in the interval betweentwo molecules and will take a sudden jump as the variable zpasses the point occupied by a molecule. The amount of thisjump will be equal o the mass of the molecule, while the averagedistance between molecules, according to results established inatomic theory, is of the order of 10 -s cm. If now we are per-forming upon this air column some measurement in which massesof the order 10 molecules are to be considered negligible, ourfunction cannot be distinguished from a contiwous function.For if we choose two values z and z - Az whose difference Azis less than 10 -cm., then the difference between F(z) andF(z -r) will be the mass of the molecules in the interval;

since the number of these molecules is of the order of 10 , thevalues of F(x) and F(x Az) are, so far as our experiment isconcerned, equal. As density of distribution we consider simplythe difference quotient AF(z)_ F(z -Az)- F(z). it is anAx Ax 'impotent physical assumption that we do not obtain measurablydifferent values for this quotient when Ax is allowed to vary


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ALREADY WENT THROUGH BOTH VOL OF COURANT, PISKUNOV DOESN'T HAVE ANY,

CALCULUS FOR THE PRACTICAL MAN THOMPSON SEEMS TO BE GOOD WITH PHYSICAL INTERPRETATION, EVEN THOUGHMAY NOT BE ANALYTICAL INTERPRETATION SO DEF CONT ON THOMPSON, BUT FIRST SEE HISCONCEPT OF DERIVATIVE ANDTHEN GO ON TO SEE CONCEP TOF INTEGRAL.

MAYBE DO FEYNMAN MAYBE NOT, BECAUSE HIS INTERPRETATION IS TOO SPECIFIC AND FOCUSED ONSIMPLE PROBLEMS OF FINDING VELOCITY FROM DISTANCE AND FINCING DISTANCE FROM VELOCITY, AND THISDOESN'T MAKE IT SEEM TOO TRANSFERABLE THEREFORE TO OTHER PROBLESM YOU MAY WANT TO CONSIDER.

!!!!!!!!!!

FROM Aleksandrov Kolmogrov and more on Derivatives and meaning

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All the three problems discussed, in spite of the fact that they refer todifferent branches of science, namely mechanics, geometry, and thetheory of electricity, have led to one and the same mathematical operationto be performed on a given function, namely to find the limit of the ratioof the increase of the function to the corresponding increase h of theindependent variable as h --+ O. The number of such widely differentproblems could be increased at will, and their solution would lead to thesame operation. To it we are led, for example, by the question of the rateof a chemical reaction, or of the density of a nonhomogeneous mass andso forth. In view of the exceptional role played by this operation onfunctions, it has received a special name, differentiation, and the result

of the operation is called the derivative of the function.

Thus, the derivative of the function y = f(x), or more precisely, thevalue of the derivative at the given point x is the limit* approached by theratio of the increase f(x + h) - f(x) of the function to the increase hof the independent variable, as the latter approaches zero. ...

NOTE: Thus they define it as rate of increase of the function to corresponding


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increase h of the independentvariable as h goes to 0, which is very concise and precise.

On the increment and differential from Kolmogrov

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The first summand on the right side of this equality depends on Llxin a very simple way, namely it is proportional to Llx. It is called thedifferential of the function, at the point x, corresponding to the givenincrement Llx, and is denoted bydy = f'(x) Llx.

NOTE: Thus one important characteristic of the differential is thatit is proportional to delta x (change in x).

NOTE: Another subtle but equally important point is that we call it the differnetial of the fucnction >"at the point x"


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NOTE: Thus in practical problems the differential does have a concrete interpretation. The interpretation of the differentialis as an approximation for the increment of the function. This makes sense. Ifdy = (dy/dx)*delta x = (dy/dx)*dx, that means we'remultiplying the derivative rate of change by a small change in the function, andthus we get an approximation to the increment of hte funcitonchange in y.

For symmetry in the notation it is customary to denote the incrementof the independent variable by dx and to call it also a differential. Withthis notation the differential of the function may be written thus:dy = f'(x) dx.

Then the derivative is the ratio f'(x) = dyJdx of the differential of the

function to the differential of the independent variable.

NOTE: Thus it seems we denote increpemtn of independent variable x by dx and also call it a differential just forpurposes of symmetry.

The differential of a function originated historically in the conceptof an "indivisible." This concept, which from a modern point of viewwas never very clearly defined, was in its time, in the 18th century, afundamental one in mathematical analysis. The ideas concerning it haveundergone essential changes in the course of several centuries. Theindivisible, and later the differential of a function, were represented asactual infinitesimals, as something in the nature of an extremely small

p 119 djvu p 129


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constant magnitude, which however was not zero. The definition givenin this section is the one accepted in present-day analysis. According to thisdefinition the differential is a finite magnitude for each increment Llxand is at the same time proportional to Llx. The other fundamentalproperty of the differential, the character of its difference from Lly, maybe recognized only in motion, so to speak: if we consider an incrementLlx which is approaching zero (which is infinitesimal), then the differencebetween dy and Lly will be arbitrarily small even in comparison withLlx.

NOTE: Thus the differntial is a finite magnitude for each increment change in x(delta x) and is at the same time proportional to delta x.

NOTE: Also as change in x approaches 0 the difference between change in y deltay and dy will be arbitrarily small, evenin comparison with delta x.

This substitution of the differential in place of small increments of thefunction forms the basis of most of the applications of infinitesimalanalysis to the study of nature. The reader will see this in a particularly

clear way in the case of differential equations, dealt in this book in ChaptersV and VI.

NOTE: Thus the subsittuiton of the differntial in place of small increments of hte functions forms the basis fo most applicatoins of analsis to the study of nautre.

!!!!!!!!!!!!!!!!ON DIFFERENTAL EQUATIONS WHAT THE SOLUTION IS!!!!!!!!!!!!!!!!!

djvu p 517 Courant

CHAPTER X1The Differential Equationsfor the Simplest Types of VibrationOn several occasions we have already met with dierentialequations, that is, equations from which an nnknown functionis to be determined and which involve not only this function


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itself but also its derivatives.The simplest problem of this type is that of finding the inde-finite integral of a given functionf (x). This problem requires us tofind a function y = F(x) which satisfies the differential equationy' --f(x) = 0. Furer, we solved a problem of the same type inChap. III, 7 (p. 178), where we showed that an equation of theform y'= ay is satisfied by an exponential function y = ee .As we saw in Chap. V (p. 294), differential equations arise inconnexion with the problems of mechanics, and indeed manybranches of pure mathematics and most of applied mathematicsdepend on differential equations. In this chapter, withoutgoing into the general theory, we shall consider the differentialequations of the simplest types of vibration. These are not onlyof theoretical value, but are also extremely important in appliedmathematics.

DEF DEF THIS 12.5!

It will be convenient to bear the following general ideas anddefinitions in mind. By a solutioof a ditierential equation wemean a function which, when substituted in the differentialequation, satisfies the equation for all values of the inde-

pendent variable that are being considered. Instead of solutionthe term integral is often used: in the first place becausethe problem is more or less a generalization of the ordinaryproblem of integration; and in the second place becauseit frequently happens that the solution is actually found byintegration.

NOTE: Thus by "solution" it is meant a function which when substituted in the differential equation, satisfies theeqution for all values ofth eindependent variable beign considered.

NOTE: The solution is usually found by integration.

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Notes on Calculus Analytic Interpretation Helps Understand Physics Applications