STAT,
"VRT
A COURSE IN
BY
EDOUARD GOURSAT
PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF PARIS TRANSLATED BY
EARLE RAYMOND HEDRICK
PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MISSOURI
VOL. I
DERIVATIVES AND DIFFERENTIALS
DEFINITE INTEGRALS EXPANSION IN SERIES
APPLICATIONS TO GEOMETRY
GINN AND COMPANY
BOSTON • NEW YORK • CHICAGO • LONDON ATLANTA • DALLAS • COLUMBUS • SAN FKANCISCO
STAT.
LIBRARY
ENTERED AT STATIONERS1 HALL
COPYRIGHT, 1904, BY EARLE RAYMOND HEDRICK
ALL RIGHTS RESERVED
PRINTED IN THE UNITED STATES OF AMERICA 426.6
jgregg
GINN AND COMPANY • PRO PRIETORS • BOSTON • U.S.A.
AUTHOR'S PREFACE
This book contains, with slight variations, the material given in my course at the University of Paris. I have modified somewhat the order followed in the lectures for the sake of uniting in a single volume all that has to do with functions of real variables, except the theory of differential equations. The differential notation not being treated in the " Classe de Mathematiques speciales," * I have treated this notation from the beginning, and have presupposed only a knowledge of the formal rules for calculating derivatives.
Since mathematical analysis is essentially the science of the con tinuum, it would seem that every course in analysis should begin, logically, with the study of irrational numbers. I have supposed, however, that the student is already familiar with that subject. The theory of incommensurable numbers is treated in so many excellent well-known works f that I have thought it useless to enter upon such a discussion. As for the other fundamental notions which lie at the basis of analysis, — such as the upper limit, the definite integral, the double integral, etc., — I have endeavored to treat them with all desirable rigor, seeking to retain the elementary character of the work, and to avoid generalizations which would be superfluous in a book intended for purposes of instruction.
Certain paragraphs which are printed in smaller type than the body of the book contain either problems solved in detail or else
*An interesting account of French methods of instruction in mathematics will be found in an article by Pierpont, Bulletin Amer. Math. Society, Vol. VI, 2d series (1900), p. 225.— TRANS.
t Such books are not common in English. The reader is referred to Pierpont, Theory of Functions of Real Variables, Ginn & Company, Boston, 1905; Tannery, Lemons d'arithiiietique, 1900, and other foreign works on arithmetic and on real functions.
iii
7814G2
iv AUTHOR'S PREFACE
supplementary matter which the reader may omit at the first read ing without inconvenience. Each chapter is followed by a list of examples which are directly illustrative of the methods treated in the chapter. Most of these examples have been set in examina tions. Certain others, which are designated by an asterisk, are somewhat more difficult. The latter are taken, for the most part, from original memoirs to which references are made.
Two of my old students at the Ecole Normale, M. Emile Cotton and M. Jean Clairin, have kindly assisted in the correction of proofs ; I take this occasion to tender them my hearty thanks.
E. GOURSAT JANUARY 27, 1902
TRANSLATOR'S PREFACE
The translation of this Course was undertaken at the suggestion of Professor W. F. Osgood, whose review of the original appeared in the July number of the Bulletin of the American Mathematical Society in 1903. The lack of standard texts on mathematical sub jects in the English language is too well known to require insistence. I earnestly hope that this book will help to fill the need so generally felt throughout the American mathematical world. It may be used conveniently in our system of instruction as a text for a second course in calculus, and as a book of reference it will be found valuable to an American student throughout his work.
Few alterations have been made from the French text. Slight changes of notation have been introduced occasionally for conven ience, and several changes and .additions have been made at the sug gestion of Professor Goursat, who has very kindly interested himself in the work of translation. To him is due all the additional matter not to be found in the French text, except the footnotes which are signed, and even these, though not of his initiative, were always edited by him. I take this opportunity to express my gratitude to the author for the permission to translate the work and for the sympathetic attitude which he has consistently assumed. I am also indebted to Professor Osgood for counsel as the work progressed and for aid in doubtful matters pertaining to the translation.
The publishers, Messrs. Ginn & Company, have spared no pains to make the typography excellent. Their spirit has been far from com mercial in the whole enterprise, and it is their hope, as it is mine, that the publication of this book will contribute to the advance of mathematics in America. E R HEDRICK
AUGUST, 1904
CONTENTS
CHAPTER PAGE
I. DERIVATIVES AND DIFFERENTIALS 1
I. Functions of a Single Variable 1
II. Functions of Several Variables 11
III. The Differential Notation 19
II. IMPLICIT FUNCTIONS. FUNCTIONAL DETERMINANTS. CHANGE
OF VARIABLE 35
I. Implicit Functions ........ 35
II. Functional Determinants ...... 52
III. Transformations ... .... 61
III. TAYLOR'S SERIES. ELEMENTARY APPLICATIONS. MAXIMA
AND MINIMA ........ 89
I. Taylor's Series with a Remainder. Taylor's Series . 89 II. Singular Points. Maxima and Minima . . . .110
IV. DEFINITE INTEGRALS ........ 134
I. Special Methods of Quadrature . . . . .134 II. Definite Integrals. Allied Geometrical Concepts . . 140
III. Change of Variable. Integration by Parts . . .166
IV. Generalizations of the Idea of an Integral. Improper
Integrals. Line Integrals ...... 175
V. Functions defined by Definite Integrals .... 192
VI. Approximate Evaluation of Definite Integrals . .196
V. INDEFINITE INTEGRALS 208
I. Integration of Rational Functions ..... 208
II. Elliptic and Hyperelliptic Integrals .... 226
III. Integration of Transcendental Functions . . .236
VI. DOUBLE INTEGRALS ........ 250
I. Double Integrals. Methods of Evaluation. Green's
Theorem 250
II. Change of Variables. Area of a Surface . . . 264
III. Generalizations of Double Integrals. Improper Integrals.
Surface Integrals ....... 277
IV. Analytical and Geometrical Applications . . . 284
viii CONTENTS
CHAPTER PAGE VII. MULTIPLE INTEGRALS. INTEGRATION OF TOTAL DIFFER ENTIALS 296
I. Multiple Integrals. Change of Variables . . . 296 II. Integration of Total Differentials . . . . .313
VIII. INFINITE SERIES . •. 327
I. Series of Real Constant Terms. General Properties.
Tests for Convergence 327
II. Series of Complex Terms. Multiple Series . . . 350 III. Series of Variable Terms. Uniform Convergence . . 360
IX. POWER SERIES. TRIGONOMETRIC SERIES .... 375
I. Power Series of a Single Variable . . . . . 375
II. Power Series in Several Variables ..... S94
III. Implicit Functions. Analytic Curves and Surfaces . 399
IV. Trigonometric Series. Miscellaneous Series . . .411
X. PLANE CURVES 426
I. Envelopes 426
II. Curvature 433
III. Contact of Plane Curves 443
XI. SKEW CURVES 453
I. Osculating Plane ........ 453
II. Envelopes of Surfaces . . • . . . . . 459
III. Curvature and Torsion of Skew Curves .... 468
IV. Contact between Skew Curves. Contact between Curves
and Surfaces ........ 486
XII. SURFACES 497
I. Curvature of Curves drawn on a Surface . . . 497
II. Asymptotic Lines. Conjugate Lines .... 506
III. Lines of Curvature . . . . . . . .514
IV. Families of Straight Lines 526
INDEX . 541
CHAPTER I
DERIVATIVES AND DIFFERENTIALS I. FUNCTIONS OF A SINGLE VARIABLE
1. Limits. When the successive values of a variable x approach nearer and nearer a constant quantity a, in such a way that the absolute value of the difference x — a finally becomes and remains less than any preassigned number, the constant a is called the limit of the variable x. This definition furnishes a criterion for determining whether a is the limit of the variable x. The neces sary and sufficient condition that it should be, is that, given any positive number e, no matter how small, the absolute value of x — a should remain less than e for all values which the variable x can assume, after a certain instant.
Numerous examples of limits are to be found in Geometry and Algebra. For example, the limit of the variable quantity x = (a2 — m2) / (a — m), as m approaches a, is 2 a ; for x — 2 a will be less than e whenever m — a is taken less than e. Likewise, the variable x = a — 1/n, where n is a positive integer, approaches the limit a when n increases indefinitely ; for a — x is less than e when ever n is greater than 1/e. It is apparent from these examples that the successive values of the variable x, as it approaches its limit, may form a continuous or a discontinuous sequence.
It is in general very difficult to determine the limit of a variable quantity. The following proposition, which we will assume as self- evident, enables us, in many cases, to establish the existence of a limit.
Any variable quantity which never decreases, and which ahvays remains less than a constant quantity L, approaches a limit I, which is less than or at most equal to L.
Similarly, any variable quantity which never increases, and which always remains greater than a constant quantity L', approaches a limit l'} which is greater than or else equal to L'.
1
2. DERIVATIVES AND DIFFERENTIALS [I, §2
For example, if each of an infinite series of positive terms is less, respectively, than the corresponding term of another infinite series of positive terms which is known to converge, then the first series converges also ; for the sum 2n of the first n terms evidently increases with n, and this sum is constantly less than the total sum 5 of the second series.
2. Functions. When two variable quantities are so related that the value of one of them depends upon the value of the other, they are said to be functions of each other. If one of them be sup posed to vary arbitrarily, it is called the independent variable. Let this variable be denoted by x, and let us suppose, for example, that it can assume all values between two given numbers a and b (a < b). Let y be another variable, such that to each value of x between a and b, and also for the values a and b themselves, there corresponds one definitely determined value of y. Then y is called a function of x, defined in the interval (a, b) ; and this dependence is indicated by writing the equation y =/(z). For instance, it may happen that y is the result of certain arithmetical operations per formed upon x. Such is the case for the very simplest functions studied in elementary mathematics, e.g. polynomials, rational func tions, radicals, etc.
A function may also be defined graphically. Let two coordinate axes Ox, Oy be taken in a plane ; and let us join any two points A and B of this plane by a curvilinear arc .4 CB, of any shape, which is not cut in more than one point by any parallel to the axis Oy. Then the ordinate of a point of this curve will be a function of the abscissa. The arc A CB may be composed of several distinct por tions which belong to different curves, such as segments of straight lines, arcs of circles, etc.
In short, any absolutely arbitrary law may be assumed for finding the value of y from that of x. The word function, in its most gen eral sense, means nothing more nor less than this : to every value of x corresponds a value of y.
3. Continuity. The definition of functions to which the infini tesimal calculus applies does not admit of such broad generality. Let y =f(x) be a function defined in a certain interval (a, b), and let x0 and x0 -f h be two values of x in that interval. If the differ ence f(x0 -f A) — f(xo) approaches zero as the absolute value of h approaches zero, the function f(x} is said to be continuous for the value x0. From the very definition of a limit we may also say that
I, §3] FUNCTIONS OF A SINGLE VARIABLE 3
a function f(x) is continuous for x — x0 if, corresponding to every positive number e, no matter how small, we can find a positive num ber 77, such that
|/(*o + A)-/(*o)|<«
for every value of h less than rj in absolute value.* We shall say that a function f(x) is continuous in an interval (a, b) if it is continuous for every value of x lying in that interval, and if the differences
each approach zero when h, which is now to be taken only positive, approaches zero.
In elementary text-books it is usually shown that polynomials, rational functions, the exponential and the logarithmic function, the trigonometric functions, and the inverse trigonometric functions are continuous functions, except for certain particular values of the variable. It follows directly from the definition of continuity that the sum or the product of any number of continuous functions is itself a continuous function ; and this holds for the quotient of two continuous functions also, except for the values of the variable for which the denominator vanishes.
It seems superfluous to explain here the reasons which lead us to assume that functions which are defined by physical conditions are, at least in general, continuous.
Among the properties of continuous functions we shall now state only the two following, which one might be tempted to think were self-evident, but which really amount to actual theorems, of which rigorous demonstrations will be given later, f
I. If the function y—f(x) is continuous in the interval (a, b), and if N is a number between f (a) andf(b), then the equation f(x) = N has at least one root between a and b.
II. There exists at least one value of x belonging to the interval (a, b'), inclusive of its end points, for which y takes on a value M which is greater than, or at least equal to, the value of the function at any other point in the interval. Likewise, there exists a value of x for which y takes on a value m, than which the function assumes no smaller value in the interval.
The numbers M and m are called the maximum and the minimum values of f(x), respectively, in the interval (a, b*). It is clear that
* The notation | a \ denotes the absolute value of a. t See Chapter IV.
4 DERIVATIVES AND DIFFERENTIALS [I, §4
the value of x for which /(ce) assumes its maximum value M, or the value of x corresponding to the minimum m, may be at one of the end points, a or b. It follows at once from the two theorems above, that if N is a number between M and m, the equation /(«) = N has at least one root which lies between a and b.
4. Examples of discontinuities. The functions which we shall study will be in general continuous, but they may cease to be so for certain exceptional values of the variable. We proceed to give several examples of the kinds of discontinuity which occur most frequently.
The function y = 1 / (x — a) is continuous for every value x0 of x except a. The operation necessary to determine the value of y from that of x ceases to have a meaning when x is assigned the value a ; but we note that when x is very near to a the absolute value of y is very large, and y is positive or negative with x — a. As the difference x — a diminishes, the absolute value of y increases indefinitely, so as eventually to become and remain greater than any preassigned number. This phenomenon is described by saying that y becomes infinite when x = a. Discontinuity of this kind is of great importance in Analysis.
Let us consider next the function y = sin 1/z. As x approaches zero, I/a; increases indefinitely, and y does not approach any limit whatever, although it remains between + 1 and — 1. The equation sin l/a; = ,4, where A \ < 1, has an infinite number of solutions which lie between 0 and e, no matter how small e be taken. What ever value be assigned to y when x — 0, the function under con sideration cannot be made continuous for x = 0.
An example of a still different kind of discontinuity is given by the convergent infinite series
When x approaches zero, S (x~) approaches the limit 1, although 5 (0) = 0. For, when x = 0, every term of the series is zero, and hence 5 (0) = 0. But if x be given a value different from zero, a geometric progression is obtained, of which the ratio is 1/(1 + a;2). Hence
~
I, §5] FUNCTIONS OF A SINGLE VARIABLE 5
and the limit of S(x) is seen to be 1. Thus, in this example, the function approaches a definite limit as x approaches zero, but that limit is different from the value of the function for x = 0.
5. Derivatives. Let/(x) be a continuous function. Then the two terms of the quotient
k
approach zero simultaneously, as the absolute value of h approaches zero, while x remains fixed. If this quotient approaches a limit, this limit is called the derivative of the function /(#), and is denoted by y', or by /' (x), in the notation due to Lagrange.
An important geometrical concept is associated with this analytic notion of derivative. Let us consider, in a plane XOY, the curve A MB, which represents the function y =/(#), which we shall assume to be continuous in the interval (a, b). Let M and M' be two points on this curve, in the interval (a, b), and let their abscissas be x and x + A, respectively. The slope of the straight line MM' is then precisely the quotient above. Now as h approaches zero the point M ' approaches the point M] and, if the function has a derivative, the slope of the line MM' approaches the limit y'. The straight line MM', therefore, approaches a limiting position, which is called the tangent to the curve. It follows that the equation of the tangent is
Y-y = y'(X-x),
where X and Y are the running coordinates.
To generalize, let us consider any curve in space, and let
be the coordinates of a point on the curve, expressed as functions of a variable parameter t. Let M and M' be two points of the curve corresponding to two values, t and t + h, of the parameter. The equations of the chord MM1 are then
x-f(t) Y
f(t + h) -
If we divide each denominator by h and then let h approach zero, the chord MM' evidently approaches a limiting position, which is given by the equations
X -f(f) Y-
f(t) 4,' ft)
6 DERIVATIVES AND DIFFERENTIALS [i,§5
provided, of course, that each of the three functions f(t), <f> (t), \J/ (t) possesses a derivative. The determination of the tangent to a curve thus reduces, analytically, to the calculation of derivatives.
Every function which possesses a derivative is necessarily con tinuous, but the converse is not true. It is easy to give examples of continuous functions which do not possess derivatives for par ticular values of the variable. The function y = xsinl/x, for example, is a perfectly continuous function of x, for x = 0,* and y approaches zero as x approaches zero. But the ratio y /x = sinl/cc does not approach any limit whatever, as we have already seen.
Let us next consider the function y = x*. Here y is continuous for every value of a;; and y = 0 when x = 0. But the ratio y /x = x~* increases indefinitely as x approaches zero. For abbreviation the derivative is said to be infinite for x = 0 ; the curve which repre sents the function is tangent to the axis of y at the origin.
Finally, the function
y =
is continuous at x = 0,* but the ratio y /x approaches two different limits according as x is always positive or always negative while it is approaching zero. When x is positive and small, el/x is posi tive and very large, and the ratio y /x approaches 1. But if x is negative and very small in absolute value, el/x is very small, and the ratio y / x approaches zero. There exist then two values of the derivative according to the manner in which x approaches zero : the curve which represents this function has a corner at the origin.
It is clear from these examples that there exist continuous func tions which do not possess derivatives for particular values of the variable. But the discoverers of the infinitesimal calculus confi dently believed that a continuous function had a derivative in gen eral. Attempts at proof were even made, but these were, of course, fallacious. Finally, Weierstrass succeeded in settling the question conclusively by giving examples of continuous functions which do not possess derivatives for any values of the variable whatever.! But as these functions have not as yet been employed in any applications,
* After the value zero has been assigned to y for x = 0. — TRANSLATOR.
t Note read at the Academy of Sciences of Berlin, July 18, 1872. Other examples are to be found in the memoir by Darboux on discontinuous functions (Annales de I'Ecole Normale Superieure, Vol. IV, 2d series). One of Weierstrass's examples is given later (Chapter IX).
I, §6] FUNCTIONS OF A SINGLE VARIABLE 7
we shall not consider them here. In the future, when we say that a function f(x) has a derivative in the interval (a, b), we shall mean that it has an unique finite derivative for every value of x between a and b and also f or x = a (h being positive) and f or x = b (h being negative), unless an explicit statement is made to the contrary.
6. Successive derivatives. The derivative of a function f(x) is in general another function of x,f'(x). If f'(x) in turn has a deriva tive, the new function is called the second derivative of /(x), and is represented by y" or by f"(x). In the same way the third deriva tive y'", or /'"(#), is defined to be the derivative of the second, and so on. In general, the rath derivative 7/n), or fw(x), is the deriva tive of the derivative of order (n — 1). If, in thus forming the successive derivatives, we never obtain a function which has no derivative, we may imagine the process carried on indefinitely. In this way we obtain an unlimited sequence of derivatives of the func tion /(cc) with which we started. Such is the case for all functions which have found any considerable application up to the present time.
The above notation is due to Lagrange. The notation Dny, or Dnf(x), due to Cauchy, is also used occasionally to represent the wth derivative. Leibniz' notation will be given presently.
7. Rolle's theorem. The use of derivatives in the study of equa tions depends upon the following proposition, which is known as Roue's Theorem :
Let a and b be two roots of the equation f (x) = 0. If the function f(x) is continuous and possesses a derivative in the interval (a, b~), the equation /'(#) = 0 has at least one root which lies between a and b.
For the function f(x) vanishes, by hypothesis, for x = a and x = b. If it vanishes at every point of the interval (a, b), its derivative also vanishes at every point of the interval, and the theorem is evidently fulfilled. If the function f(x) does not vanish throughout the inter val, it will assume either positive or negative values at some points. Suppose, for instance, that it has positive values. Then it will have a maximum value M for some value of x, say xlf which lies between a and b (§ 3, Theorem II). The ratio
8 DERIVATIVES AND DIFFERENTIALS [I, §8
where h is taken positive, is necessarily negative or else zero. Hence the limit of this ratio, i.e. f'(x^), cannot be positive ; i.e. f'(xi) = 0- But if we consider f'(x\) as the limit of the ratio
> — h
where h is positive, it follows in the same manner that f\x\) ^ 0, From these two results it is evident that/'^) = 0.
8. Law of the mean. It is now easy to deduce from the above theorem the important law of the mean : *
Let f(x) be a continuous function which has a derivative in the interval (a, b). Then
(1) m-f(a) = (b-a)f(c-),
where c is a number between a and b.
In order to prove this formula, let <£ (x) be another function which has the same properties as/(x), i.e. it is continuous and possesses a derivative in the interval (a, b). Let us determine three constants, A, B, C, such that the auxiliary function
vanishes for x = a and for x = b. The necessary and sufficient conditions for this are
A /(a) +B <£(«)+ C' = 0, Af(b) + B<l>(b)+ C = 0; and these are satisfied if we set A = <l>(a)-4> (b), B =/(&) -/(a), C =/(«) 0 (*)-/(&) * (a).
The new function \J/(x) thus defined is continuous and has a derivative in the interval (a, b). The derivative if/'(x) = A f'(x) + B <£'(z) there fore vanishes for some value c which lies between a and b, whence replacing A and B by their values, we find a relation of the form
It is merely necessary to take <£ (a;) = x in order to obtain the equality which was to be proved. It is to be noticed that this demonstration does not presuppose the continuity of the derivative/'^).
•"Formule des accroissements finis." The French also use " Formule de la moyenne" as a synonym. Other English synonyms are "Average value theorem " and " Mean value theorem." —TRANS.
I, §8] FUNCTIONS OF A SINGLE VARIABLE 9
From the theorem just proven it follows that if the derivative f'(x) is zero at each point of the interval (a, b), the function f(x) has the same value at every point of the interval ; for the applica tion of the formula to two values Xi, xz, belonging to the interval (a, b), gives f(xi)=f(x.2). Hence, if two functions have the same derivative, their difference is a constant ; and the converse is evi dently true also. If a function F(x) be given whose derivative is f(oc), all other functions which have the same derivative are found by adding to F(x) an arbitrary constant*
The geometrical interpretation of the equation (1) is very simple, Let us draw the curve A MB which represents the function y = f(x) in the interval (a, b). Then the ratio [/(&)— /(«)]/ (b — a) is the slope of the chord AB, while /'(«) is the slope of the tangent at a point C of the curve whose abscissa is c. Hence the equation (1) expresses the fact that there exists a point C on the curve A MB, between A and B, where the tangent is parallel to the chord AB.
If the derivative /'(a;) is continuous, and if we let a and b approach the same limit x0 according to any law whatever, the number c, which lies between a and b, also approaches x0} and the equation (1) shows that the limit of the ratio
b — a
is f'(xo). The geometrical interpretation is as follows. Let us consider upon the curve y=f(x) a point M whose abscissa is x0, and two points A and B whose abscissa are a and b, respectively. The ratio [/(&) —/(«)] / (b — «) is equal to the slope of the chord AB, while /'(x0) is the slope of the tangent at M. Hence, when the two points A and B approach the point M according to any law whatever, the secant AB approaches, as its limiting position, the tangent at the point M.
* This theorem is sometimes applied without due regard to the conditions imposed in its statement. Let/(x) and 0(^), f°r example, be two continuous functions which have derivatives /'(a;), </)'(x) in an interval (a, 6). If the relation /'(z) <t>(x)—f(x) 4>'(x) = 0 is satisfied by these four functions, it is sometimes accepted as proved that the deriva tive of the function// <f>, or [/'(a;) 0 (cc) - f(x) 0'(z)] / <£2, is zero, and that accordingly f/<t> is constant in the interval (a, b). But this conclusion is not absolutely rigorous unless the function $ (a;) does not vanish in the interval (a, b). Suppose, for instance, that 0 (a;) and <j>'(x) both vanish for a value c between a and 6. A function/(x) equal to Ci<f>(x) between a and c, and to C%<f)(x) between c and b, where Cj and C2 are dif ferent constants, is continuous and has a derivative in the interval (a, b), and we have f'(x)<t>(x) — f(x)<p'(x) = 0 for every value of x in the interval. The geometrical interpretation is apparent.
10
DERIVATIVES AND DIFFERENTIALS
[I, §9
This does not hold in general, however, if the derivative is not continuous. For instance, if two points be taken on the curve y = x*, on opposite sides of the y axis, it is evident from a figure that the direction of the secant joining them can be made to approach any arbitrarily assigned limiting value by causing the two points to approach the origin according to a suitably chosen law.
The equation (!') is sometimes called the generalized law of the mean. From it de 1' Hospital's theorem on indeterminate forms fol lows at once. For, suppose f(a) = 0 and <f> (a) = 0. Replacing b
by x in (!'). we find * \ /• \
where a^ lies between a and x. This equation shows that if the ratio f'(x)/(j>'(x) approaches a limit as x approaches a, the ratic /"(#) / (j> (a;) approaches the same limit, if f(a) = 0 and <f> (a) = 0.
9. Generalizations of the law of the mean. Various generalizations of the law of the mean have been suggested. The following one is due to Stieltjes (Bulletin de la Socie'te Mathtmatique, Vol. XVI, p. 100). For the sake of defmiteness con sider three functions, /(x), g(x), h(x), each of which has derivatives of the first and second orders. Let a, 6, c be three particular values of the variable (a < b < c). Let A be a number defined by the equation
and let
be an auxiliary function. Since this function vanishes when x = b and when x = c, its derivative must vanish for some value f between 6 and c. Hence
|
/(«) |
9 (a) |
h(a) |
1 |
a |
a« |
|
|
/(&) |
9(b) |
h(b) |
-A |
1 |
b |
62 |
|
/(c) |
9(c] |
h(c) |
1 |
c |
c2 |
|
|
/(a) |
9(0.} |
h(a) |
1 |
a |
a2 |
|
|
/(&) |
9(b) |
h(b) |
-A |
1 |
b |
b2 |
|
/(*) |
9(x) |
h(x) |
1 |
x |
x* |
/(a) g(a) h(a) /(&) g(b) h(b) /'(f)
-A
1 a a2 1 b b2 0 1 2f
If b be replaced by x in the left-hand side of this equation, we obtain a function of x which vanishes when x = a and when x = b. Its derivative therefore van ishes for some value of x between a and 6, which we shall call £. The new equation thus obtained is
/ (a) g (a) h (a)
/'(f)
1 2f
= 0.
Finally, replacing f by x in the left-hand side of this equation, we obtain a func tion of x which vanishes when x =• £ and when x •= f . Its derivative vanishes
I, §10] FUNCTIONS OF SEVERAL VARIABLES 11
for some value ij, which lies between £ and f and therefore between a and c. Hence A must have the value
J_
1.2
/ (a) g (a) h (a)
where £ lies between a and 6, and 17 lies between a and c.
This proof does not presuppose the continuity of the second derivatives f"(x), g"(x), h"(x). If these derivatives are continuous, and if the values a, 6, c approach the same limit XQ, we have, in the limit,
1
/ (x0) g (x0) h (x0) f (x0) g' (xo) h' (x0) f"(x0) 0"(xo) h"(xQ)
Analogous expressions exist for n functions and the proof follows the same lines. If only two functions /(x) and g (x) are taken, the formula? reduce to the law of the mean if we set g (x) = 1.
An analogous generalization has been given by Schwarz (Annali di Mathe- matica, 2d series, Vol. X).
II. FUNCTIONS OF SEVERAL VARIABLES
10. Introduction. A variable quantity w whose value depends on the values of several other variables, x, y, z, ••-, t, which are in dependent of each other, is called a function of the independ ent variables x, y, z, •••, t; and this relation is denoted by writing w =f(x, y,z,---, t). For definiteness, let us suppose that w = f(x, y) is a function of the two independent variables x and y. If we think of x and y as the Cartesian coordinates of a point in the plane, each pair of values (x, y) determines a point of the plane, and con versely. If to each point of a certain region A in the xy plane, bounded by one or more contours of any form whatever, there corresponds a value of w, the function f(x, y) is said to be defined in the region A.
Let (x0, y0) be the coordinates of a point M0 lying in this region. The function f(x, y) is said to be continuous for the pair of values (xoi yo) if, corresponding to any preassigned positive number c, another positive number 77 exists such that
|/C*o + h, y0 + k)-f(x0, 2/0) | < e
whenever \h < rj and \k\<rj.
This definition of continuity may be interpreted as follows. Let us suppose constructed in the xy plane a square of side 2^ about M0 as center, with its sides parallel to the axes. The point M',
12 DERIVATIVES AND DIFFERENTIALS [I, §11
whose coordinates are x0 + h, y0 + k, will lie inside this square, if | h | < rj and | k \ < rj. To say that the function is continuous for the pair of values (x0, T/O) amounts to saying that by taking this square sufficiently small we can make the difference between the value of the function at M0 and its value at any other point of the square less than e in absolute value.
It is evident that we may replace the square by a circle about (x0, y0) as center. For, if the above condition is satisfied for all points inside a square, it will evidently be satisfied for all points inside the inscribed circle. And, conversely, if the condition is satisfied for all points inside a circle, it will also be satisfied for all points inside the square inscribed in that circle. We might then define continuity by saying that an rj exists for every c, such that whenever V/i2 + k'2 < 17 we also have
I /(«<> + h> y0 + k) -f(x0,
The definition of continuity for a function of 3, 4, • • • , n inde pendent variables is similar to the above.
It is clear that any continuous function of the two independent variables x and y is a continuous function of each of the variables taken separately. However, the converse does not always hold.*
11. Partial derivatives. If any constant value whatever be substi tuted for y, for example, in a continuous function f(x, y), there results a continuous function of the single variable x. The deriva tive of this function of x, if it exists, is denoted by fx(x, y) or by <ax. Likewise the symbol uv, or fy (x, y), is used to denote the derivative of the function f(x, y} when x is regarded as constant and y as the independent variable. The functions fx(x, y) and fy (x, y) are called the partial derivatives of the function f(x, y). They are themselves, in general, functions of the two variables x and y. If we form their partial derivatives in turn, we get the partial derivatives of the sec ond order of the given function f(x, y). Thus there are four partial derivatives of the second order, fa (x, y),fx¥(x, y),fyx(x, y),f+(x, y\ The partial derivatives of the third, fourth, and higher orders are
* Consider, for instance, the f unction /(x, y), which is equal to 2 xy / (x2 + y2) when the two variables x and y are not both zero, and which is zero when x = y = 0. It is evident that this is a continuous function of x when y is constant, and vice versa. Nevertheless it is not a continuous function of the two independent variables x and y for the pair of values x = 0, y = 0. For, if the point (a-, y) approaches the origin upon the line x = y. the f unction/ (x, y) approaches the limit 1, and not zero. Such functions have been studied by Baire in his thesis.
I, §11] FUNCTIONS OF SEVERAL VARIABLES 13
defined similarly. In general, given a function w = /(x, y, z, ••-, f) of any number of independent variables, a partial derivative of the nth order is the result of n successive differentiations of the function /, in a certain order, with respect to any of the variables which occur in /. We will now show that the result does not depend upon the order in which the differentiations are carried out. Let us first prove the following lemma :
Let w = f (x, y) be a function of the two variables x and y. Then fxij = ftjx, provided that these two derivatives are continuous.
To prove this let us first write the expression
U =f(x + Ax, y + Ay) -f(x, y + Ay) -f(x + Ax, y) + /(x, y}
in two different forms, where we suppose that x, y, Ax, A?/ have definite values. Let us introduce the auxiliary function
<£ 00 =f(x + Ax, u) -/(x, v), where v is an auxiliary variable. Then we may write
Applying the law of the mean to the function <£(w), we
U = Ay <£„ (y + 0Ay), where 0 < 0 < 1 ;
or, replacing <j>u by its value,
U = Ay [/„(* + Ax, y + 0Ay) -fy(x, y + 0Ay)].
If we now apply the law of the mean to the function fy (u, y + 0Ay), regarding u as the independent variable, we find
U = Ax Ay/^ (x + 0' Ax, y + 0Ay), 0 < 0' < 1.
From the symmetry of the expression U in x, y, Ax, Ay, we see that we would also have, interchanging x and y,
U = Ay Aaj/q, (x + 0| Ax, y + ^ Ay),
where 0, and 0[ are again positive constants less than unity. Equat ing these two values of U and dividing by Ax Ay, we have
fxy(x + 0[ Ax, y + ^Ay) =f,,x(x + 0'Ax, y + 0Ay).
Since the derivatives /,.„ (x, y) and fvx(x, y) are supposed continuous, the two members of the above equation approach fxy (x, y) and fyx(x, y), respectively, as Ax and Ay approach zero, and we obtain the theorem which we wished to prove.
14 DERIVATIVES AND DIFFERENTIALS [I, § n
It is to be noticed in the above demonstration that no hypothesis whatever is made concerning the other derivatives of the second order, f^ and fyt. The proof applies also to the case where the function f(x, y) depends upon any number of other independent variables besides x and y, since these other variables would merely have to be regarded as constants in the preceding developments.
Let us now consider a function of any number of independent variables,
«=/(»> y> *>•••> *)j
and let n be a partial derivative of order n of this function. Any permutation in the order of the differentiations which leads to fi can be effected by a series of interchanges between two successive differentiations ; and, since these interchanges do not alter the result, as we have just seen, the same will be true of the permuta tion considered. It follows that in order to have a notation which is not ambiguous for the partial derivatives of the nth order, it is sufficient to indicate the number of differentiations performed with respect to each of the independent variables. For instance, any nth derivative of a function of three variables, to =/(x, y, z), will be represented by one or the other of the notations
where p -f- q + r = n* Either of these notations represents the result of differentiating / successively p times with respect to x, q times with respect to ?/, and r times with respect to 2, these oper ations being carried out in any order whatever. There are three distinct derivatives of the first order, fx, f , fz\ six of the second order, fa, fa fa /3.v, fa fxz ; and so on.
In general, a function of p independent variables has just as many distinct derivatives of order n as there are distinct terms in a homo geneous polynomial of order n in p independent variables ; that is,
as is shown in the theory of combinations.
Practical rules. A certain number of practical rules for the cal culation of derivatives are usually derived in elementary books on
* The notation /a Pyq..r (x, y, z) is used instead of the notation fxfynzr (x, y, z) for simplicity. Thus the notation fxy (x, y), used in place of f'x'y(x, y), is simpler and equally clear. — TRANS.
I, §11] FUNCTIONS OF SEVERAL VARIABLES 16
the Calculus. A table of such rules is appended, the function and its derivative being placed on the same line :
y' = ax0-'1; y' = ax log a,
where the symbol log denotes the natural logarithm ;
y = log x, y' = ->
X
y = sin x, y = cos x,
y •= arc sin x, y = arc tan x,
JL -\- X
y = uv, y' = u'v 4- uv1 ;
_ u f ^ u'v — uv' .
y =/(«), 2/*=/>K;
The last two rules enable us to find the derivative of a function of a function and that of a composite function if fu,fv,fw are con tinuous. Hence we can find the successive derivatives of the func tions studied in elementary mathematics, — polynomials, rational and irrational functions, exponential and logarithmic functions, trigonometric functions and their inverses, and the functions deriv able from all of these by combination.
For functions of several variables there exist certain formulae analogous to the law of the mean. Let us consider, for definite- ness, a function f(x, y) of the two independent variables x and y. The difference f(x + h, y 4- K) — f(x, y) may be written in the form
f(x + h,y + k) -f(x, y) = [/(* + h, y + k) -f(x, y + &)]
to each part of which we may apply the law of the mean. We thus find
f(x + h,y + k}-f(x, y) = hfx(x + 6h, y + k}+ kfv(x, y + O'K),
where 6 and 0' each lie between zero and unity.
This formula holds whether the derivatives fx and /„ are continu ous or not. If these derivatives are continuous, another formula,
16 DERIVATIVES AND DIFFERENTIALS [1,512
similar to the above, but involving only one undetermined number 6, may be employed.* In order to derive this second formula, con sider the auxiliary function <f>(£) = f(x + ht, y + kfy, where x, y, h, and k have determinate values and t denotes an auxiliary variable. Applying the law of the mean to this function, we find
Now <£(>") is a composite function of t, and its derivative 4>'(t) is equal to hfx (x -f- ht, y + kf) + kfy (x + ht, y -f- kt) ; hence the pre ceding formula may be written in the form
12. Tangent plane to a surface. We have seen that the derivative of a function of a single variable gives the tangent to a plane curve. Similarly, the partial derivatives of a function of two variables occur in the determination of the tangent plane to a surface. Let
(2) z . F(x, y)
be the equation of a surface S, and suppose that the function F(x, ?/), together with its first partial derivatives, is continuous at a point (^o? yo) of the xy plane. Let z0 be the corresponding value of z, and AT0 (cr0, 7/0 > «0) the corresponding point on the surface S. If the equations
(3) *=/(*), z/ = <KO> * = ^(9
represent a curve C on the surface S through the point M0, the three functions f(f), <j>(t), "A(0> which we shall suppose continuous and differentiable, must reduce to x0, y0, z0, respectively, for some value t0 of the parameter t. The tangent to this curve at the point M0 is given by the equations (§ 5)
x — x° Y — z — *
Since the curve C lies on the surface S, the equation \j/(t)=F[f(t~), . must hold for all values of t; that is, this relation must be an identity
* Another formula may be obtained which involves only one undetermined number 0, and which holds even when the derivatives/^, and/, are discontinuous. For the applica tion of the law of the mean to the auxiliary function <j>(t) =f(x + ht,y + k) +f(x, y + kt) gives
<£(!) -0(0) = 0'(0), 0<0<1.
or
f(x + h,y + k) -f(x, y) = hfx(x + 6h, y + k) + kfy(x, y + 6k), 0<0<1.
The operations performed, and hence the final formula, all hold provided the deriva tives fx and fy merely exist at the points (x + ht, y + k), (x,y + kt),0^t^\. — TRANS.
I, § 13] FUNCTIONS OF SEVERAL VARIABLES 17
in t. Taking the derivative of the second member by the rule for the derivative of a composite function, and setting t = t0, we have
(5) <j,'(t0)=fi(t0)FXo + <t>'(t0)FVa.
We can now eliminate f'(t0~), <£'(£0)> i//(£0) between the equations (4) and (5), and the result of this elimination is
(6) Z-z0 = (X- ar0) FXg + (Y - y0) F^.
This is the equation of a plane which is the locus of the tangents to all curves on the surface through the point M0. It is called the tan gent plane to the surface.
13. Passage from increments to derivatives. We have defined the successive derivatives in terms of each other, the derivatives of order n being derived from those of order (n — 1), and so forth. It is natural to inquire whether we may not define a derivative of any order as the limit of a certain ratio directly, with out the intervention of derivatives of lower order. We have already done some thing of this kind for fxy (§ 11); for the demonstration given above shows that/rj, is the limit of the ratio
f(x + Ax, y + Ay) -/(x + Ax, y)-f(x, y + Ay) + /(x, y) Ax Ay
as Ax and Ay both approach zero. It can be shown in like manner that the second derivative /" of a function f(x) of a single variable is the limit of the ratio
/(x + hi + h*) -f(x + hi) -f(x
^1^2
as hi and h2 both approach zero. For, let us set
/i(x)=/(z + Ai)
and then write the above ratio in the form
h\
f>(x +
+
hi
The limit of this ratio is therefore the second derivative /", provided that derivative is continuous.
Passing now to the general case, let us consider, for definiteness, a function of three independent variables, w =f(x, y, 2). Let us set
A£W =/(x + h, y, z) -/(x, y, 2), A£W =/(x, y + k, 2) -/(x, y, 2), A^w =/(x, y, 2 -f 1) -/(x, y, z),
where A* w, A* w, Alz u are the^irsi increments of w. If we consider ^, k, I as given constants, then these three first increments are themselves functions of x, y, 2, and we may form the relative increments of these functions corresponding to
18 DERIVATIVES AND DIFFERENTIALS [I, § 13
increments hi, ki, ^ of the variables. This gives us the second increments, A*1 A*w> A*1 Avw' ' ' ' • Tnis process can be continued indefinitely ; an increment of order n would be defined as a first increment of an increment of order (n — 1). Since we may invert the order of any two of these operations, it will be suffi cient to indicate the successive increments given to each of the variables. An increment of order n would be indicated by some such notation as the following :
A<->« = AX' ' ' ' A*p A*' • • • AX1 • • • A^/(z, y, z),
where p + q + r = n, and where the increments h, k, I may be either equal or unequal. This increment may be expressed in terms of a partial derivative of order n, being equal to the product
hihy • • • hpki • • • kgl\ • • • lr
x fxp*z'(x + *i Ai + • • • + d,,hp, y + eiki + • • • + O'qkq, z + ffi'li + • • • + Kir), where every 6 lies between 0 and 1. This formula has already been proved for first and for second increments. In order to prove it in general, let us assume that it holds for an increment of order (n — 1), and let
0 (X, y, 2) = A** • • • Ah/ Ajt • • • AX1 ' ' ' £ f. Then, by hypothesis,
$(x,y,z) = hz---hpki-- -kqli-- •Irfxp-iif,ir(x + 0sh2+ ---- \-6Php, y-\ ---- ,«H ---- ). But the nth increment considered is equal to 0(x + hi, y, z) — <f>(x, y, z); and if we apply the law of the mean to this increment, we finally obtain the formula sought. Conversely, the partial derivative fxT^zr is the limit of the ratio
AX'- .-AX-.. -AX'--- A/-/
hi h? • • • hp ki k2 • • • kg li • • • lr
as all the increments h, k, I approach zero.
It is interesting to notice that this definition is sometimes more general than the usual definition. Suppose, for example, that w =/(x, y) — <f>(x) + ^(y) is a function of x and y, where neither <f> nor ^ has a derivative. Then u also has no first derivative, and consequently second derivatives are out of the question, in the ordinary sense. Nevertheless, if we adopt the new definition, the deriva tive fxy is the limit of the fraction
/(x + h, y + k) -/(x + h, y) -/(x, y + k) +/(x, y)
hk which is equal to
h) + t(V + k) - <t>(x + h) -
hk
But the numerator of this ratio is identically zero. Hence the ratio approaches zero as a limit, and we find/xy = 0.*
* A similar remark may be made regarding functions of a single variable. For example, the f unction /(K) = xs cosl/x has the derivative
f'(x) = 3 x2 cos - + xsin-i
and f'(x) has no derivative for x — 0. But the ratio
/(2ar)-2/(tt)+/(0) o"
or 8 a cos (I/ 2 a) — 2 a cos (I/ or), has the limit zero when a approaches zero.
l)§14] THE DIFFERENTIAL NOTATION 19
III. THE DIFFERENTIAL NOTATION
The differential notation, which has been in use longer than any other,* is due to Leibniz. Although it is by no means indispensable, it possesses certain advantages of symmetry and of generality which are convenient, especially in the study of functions of several varia bles. This notation is founded upon the use of infinitesimals.
14. Differentials. Any variable quantity which approaches zero as a limit is called an infinitely small quantity, or simply an infinitesi mal. The condition that the quantity be variable is essential, for a constant, however small, is not an infinitesimal unless it is zero.
Ordinarily several quantities are considered which approach zero simultaneously. One of them is chosen as the standard of compari son, and is called the principal infinitesimal. Let « be the principal infinitesimal, and ft another infinitesimal. Then 0 is said to be an infinitesimal of higher order with respect to a, if the ratio ft/a approaches zero with a. On the other hand, ft is called an infini tesimal of the first order with respect to a, if the ratio ft/a approaches a limit K different from zero as a approaches zero. In this case
^ = K + e, «
where c is another infinitesimal with respect to a. Hence ft=a(K + c)= Ka + at,
and Ka is called the principal part of ft. The complementary term at is an infinitesimal of higher order with respect to a. In general, if we can find a positive power of a, say a", such that ft /a" approaches a finite limit K different from zero as a approaches zero, ft is called an infinitesimal of order n with respect to a. Then we have
4 ; = K + e,
a
or
ft = an (K -f e) = Ka* + «"«.
The term Ka" is again called the principal part of ft.
Having given these definitions, let us consider a continuous func tion y=f(x), which possesses a derivative f'(x). Let Aa; be an
* With the possible exception of Newton's notation. — TRANS.
20 DERIVATIVES AND DIFFERENTIALS [I, § 14
increment of x, and let A?/ denote the corresponding increment of y. From the very definition of a derivative, we have
where c approaches zero with Ace. If Ax be taken as the principal infinitesimal, AT/ is itself an infinitesimal whose principal part is f'(x) Ax.* This principal part is called the differential of y and is denoted by dy.
dy=f(x)&x.
When /(x) reduces to x itself, the above formula becomes dx = Ax ; and hence we shall write, for symmetry,
where the increment dx of the independent variable x is to be given the same fixed value, which is otherwise arbitrary and of course variable, for all of the several dependent functions of x which may be under consid eration at the same time.
Let us take a curve C whose equation is y = f(x), and consider two points on it, M and M', whose abscissae are x and x -f dx, respectively. In the triangle MTN we have
NT = MN tan Z TMN = dxf'(x).
Hence NT represents the differential dy,
while Ay is equal to NM'. It is evident from the figure that M'T is an infinitesimal of higher order, in general, with respect to NT, as M' approaches M, unless MT is parallel to the x axis.
Successive differentials may be defined, as were successive deriv atives, each in terms of the preceding. Thus we call the differ ential of the differential of the first order the differential of the second order, where dx is given the same value in both cases, as above. It is denoted by d2y:
d*y = d (dy) = [/"(x) dx] dx = f"(x) (dx}*. Similarly, the third differential is
d*y = d(d*y) = [_f»(x)dx*]dx =f"(x)(dx)*,
* Strictly speaking, we should here exclude the case where f'(x) = 0. It is, how ever, convenient to retain the same definition of dy =f'(x)&x in this case also. even though it is not the principal part of Ay. — TRANS.
I, §14] THE DIFFERENTIAL NOTATION 21
and so on. In general, the differential of the differential of order (n — 1) is
The derivatives /'(or), /"(a), ••-, f(n\x), ... can be expressed, on the other hand, in terms of differentials, and we have a new nota tion for the derivatives :
t — dy ,, _ <Py M dny
y ~ dx' ~ dx2' ~dtft>
To each of the rules for the calculation of a derivative corresponds a rule for the calculation of a differential. For example, we have
d xm = mxm-ldx, dax = ax log a dx ;
, , dx
d log x = — j d sin x = cos x dx ; • • • ;
SC
, . dx dx aarcsmcc = •> darctanx = - -•
± Vl - a;2 1 + x2
Let us consider for a moment the case of a function of a function. y =/(«), where u is a function of the independent variable x.
whence, multiplying both sides by dx, we get
yxdx =/(M) X uxdx; that is,
dy =f(u)du.
The formula for dy is therefore the same as if u were the inde pendent variable. This is one of the advantages of the differential notation. In the derivative notation there are two distinct formulae,
&=/(*)> yx=f(u)uxy
to represent the derivative of y with respect to cc, according as y is given directly as a function of x or is given as a function of x by means of an auxiliary function u. In the differential notation the same formula applies in each case.*
If y = f(u, v, w) is a composite function, we have
Vx = Uxfu + Vxfv + Wxfn,
at least if fu,fv,fw are continuous, or, multiplying by dx, yxdx = uxdxfu + vxdxfv + wxdxfw ;
* This particular advantage is slight, however ; for the last formula ahove is equally well a general one and covers both the cases mentioned. — TRANS.
22 DERIVATIVES AND DIFFERENTIALS [I, § 15
that is,
dll = fudu + fvdv +fwdw.
Thus we have, for example,
V du,
V
The same rules enable us to calculate the successive differentials. Let us seek to calculate the successive differentials of a function y = /(u), for instance. We have already
dy=f'(u}du.
In order to calculate d?y, it must be noted that du cannot be regarded as fixed, since u is not the independent variable. We must then calculate the differential of the composite function f'(u) du, where u and du are the auxiliary functions. We thus find
To calculate d*y, we must consider d*y as a composite function, with u, du, d2u as auxiliary functions, which leads to the expression
d*y =f'"(u)du8 + 3f"(u)dud*u +f'(u)d*u ;
and so on. It should be noticed that these formulae for d*y, d*y, etc., are not the same as if u were the independent variable, on account of the terms d*u, dzu, etc.*
A similar notation is used for the partial derivatives of a function of several variables. Thus the partial derivative of order n of f(x, y, s), which is represented by fxf>flzr in our previous notation, is represented by
in the differential notation.f This notation is purely symbolic, and in no sense represents a quotient, as it does in the case of functions of a single variable. _ _ . __
15. Total differentials. Let w =f(x, y, z) be a function of the three independent variables x, y, z. The expression
o /• o /» Q /«
du = ^- dx + ^ dy + -^ dz
ex dy - dz
* This disadvantage would seem completely to offset the advantage mentioned above. Strictly speaking, we should distinguish between d^y and d?uy, etc. — TRANS.
t This use of the letter d to denote the partial derivatives of a function of several variables is due to Jacob! . Before his time the same letter d was used as is used for the derivatives of a function of a single variable.
I, §15] THE DIFFERENTIAL NOTATION 23
is called the total differential of o>, where dx, dy, dz are three fixed increments, which are otherwise arbitrary, assigned to the three independent variables x, y, z. The three products
8f 7 df ' j df j
TT- dx. £- dy, ~ dz
ex dy ' cz
are called partial differentials.
The total differential of the second order d*<a is the total differ ential of the" total differential of the first order, the increments dx, dy, dz remaining the same as we pass from one differential to the next higher. Hence
_ , 7 . ddia ddia cdw
d2u = d(dta) = — — dx -f -^ — dy + -=— dz ; Ox oy cz
or, expanding,
ex* dx oy ex cz
!
Oy Oz
+ 2 - dxdy + 2 dxdz + 2 •=•• dy dz.
Ox oy ox cz Oy Oz
If cPf be replaced by df2, the right-hand side of this equation becomes the square of
We may then write, symbolically,
0x cy oz
it being agreed that df* is to be replaced by 82f after expansion.
In general, if we call the total differential of the total differential of order (n —1) the total differential of order n, and denote it by dn(a, we may write, in the same symbolism,
*.-(£**»* +£*)",
\0x Oy Oz /
where dfn is to be replaced by dnf after expansion ; that is, in our ordinary notation,
DERIVATIVES AND DIFFERENTIALS [I, §15
where
A n'
pqr p\q\r\
is the coefficient of the term ap&'cr in the development of (a. + b + c)n. For, suppose this formula holds for dn w. We will show that it then holds for dn+l<o; and this will prove it in general, since we have already proved it for n = 2. From the definition, we find
dn+l w=d(dn(o)
r zn+if dn+lf
- +
whence, replacing en + 1/by cfn + l, the right-hand side becomes
f ( 7T- dx -f 7f- dy + — rf« I ,
1 C7« ^V <7«
or
cy cz I \ox cy
Hence, using the same symbolism, we may write
- -
cy ' cz
Note. Let us suppose that the expression for dw, obtained in any way whatever, is
(7) dw = P dx -f- Q dy + R dz,
where P, Q, R are any functions x, y, z. Since by definition
d<a 8<a d<a
rfw = ^- <£c + — ay + ^- dz,
dx cy cz
we must have
where dx, dy, dz are any constants. Hence
/«\ S<a - P go) - n 8<a - P
(o) "5~ — - .r, ^~ — y, "5~~ — -ft.
^X ^ KB
The single equation (7) is therefore equivalent to the three separate equations (8) ; and it determines all three partial derivatives at once.
I, §16] THE DIFFERENTIAL NOTATION 25
In general, if the nth total differential be obtained in any way
whatever,
d" w = 2 Cpqr dx" dy" dzr ;
then the coefficients Cyqr are respectively equal to the corresponding nth derivatives multiplied by certain numerical factors. Thus all these derivatives are determined at once. We shall have occasion to use these facts presently.
16. Successive differentials of composite functions. Let w = F(u, v, w~) be a composite function, u, v, w being themselves functions of the independent variables x, y, z, t. The partial derivatives may then be written down as follows :
dia_dFdii dFdv dFdw dx du dx dv dx dw dx
d«> _d_F_d_u d_F_d_v_ ___ dy du dy dv dy dw dy
d_F_d_v
dz du dz do dz dw dz
dw _ dF du dF dv dF dw
dt du dt dv dt dw dt
If these four equations be multiplied by dx, dy, dz, dt, respectively, and added, the left-hand side becomes
d(» , <?W 7 , ^<° 7 ,^WJ.
•3- dx + -r- d y + -^- dz + -^ dt, dx dy dz d£
that is, do* ; and the coefficients of
d_F d]F 0F
du do dw
on the right-hand side are du, dv, dw, respectively. Hence
dF dF dF
(9) do) = ^— du + -r— dv + ^— dw,
cu dv cw
and we see that the expression of the total differential of the first order of a composite function is the same as if the auxiliary functions were the independent variables. This is one of the main advantages of the differential notation. The equation (9) does not depend, in form, either upon the number or upon the choice of the independent variables ; and it is equivalent to as many separate equations as there are independent variables.
To calculate d2w, let us apply the rule just found for dta, noting that the second member of (9) involves the six auxiliary functions u, v, w, du, dv, dw. We thus find
26 DERIVATIVES AND DIFFERENTIALS [I, § lu
d2F dF
= -i^—- du2 4- -z — — du dv + - — — du dw + -^- dzu Ctr cucu cucw en
4- - dudv 4- ^ dv2 + ff dvdw + ^
du dv cv2 cu cw cv
d2F d2F d'2F dF
+ „ „ du dw 4- o Q dv dw -f TT-^ t?w + ^—
^«gw ^y^M> Cw1 cw
or, simplifying and using the same symbolism as above,
7 , ^ , ^ ^ , „
d2w = [7^- du+ ^- dv + — dw\ + TT- »*« + — c?2w 4- ^— «•«•. Vc/w ^y CM; / cu Co cw
This formula is somewhat complicated on account of the terms in d2u, dzv, dzw, which drop out when u, v, w are the independent variables. This limitation of the differential notation should be borne in mind, and the distinction between d2w in the two cases carefully noted. To determine ds w, we would apply the same rule to <22o>, noting that d2w depends upon the nine auxiliary functions u, v, w, du, dv, dw, d2 u, d2 v,d2w; and so forth. The general expres sions for these differentials become more and more complicated ; dnw is an integral function of du, dv, dw, d'2u, • • •, dnu, dnv, dnw, and the terms containing dnu, dnv, dnw are
dF 7 dF , dF 7 — dnu 4- — dn v 4- — dnw. cu cv cw
If, in the expression for d" w, u, v, w, du, dv, dw, • • • be replaced by their values in terms of the independent variables, dnt» becomes an integral polynomial in dx, dy, dz, • • • whose coefficients are equal (cf. Note, § 15) to the partial derivatives of w of order n, multiplied by certain numerical factors. We thus obtain all these derivatives at once.
Suppose, for example, that we wished to calculate the first and second derivatives of a composite function <a=f(ii), where w is a function of two independent variables u = <f> (x, y). If we calculate these derivatives separately, we find for the two partial derivatives of the first order
1ft 8 w _ 8 w d u 8u) _ du> du
dx du dx dy du dy
Again, taking the derivatives of these two equations with respect to x, and then with respect to y, we find only the three following distinct equations, which give the second derivatives :
THE DIFFERENTIAL NOTATION
27
(11)
dx*
dx dy
du\*
ex
du
d*
0 <i> 0 U C U C <a
du* Cx dy du dx dy
d- 22 da dy
&ut
,2*
The second of these equations is obtained by differentiating the first of equations (10) with respect to y, or the second of them with respect to x. In the differential notation these five relations (10) and (11) may be written in the form
(12)
en
cu
If du and d*u in these formulae be replaced by 'du
TT- dy and -^—a
dx dy
respectively, the coefficients of dx and dy in the first give the first partial derivatives of «o, while the coefficients of dxz, 2 dx dy, and dy2 in the second give the second partial derivatives of w.
17. Differentials of a product. The formula for the total differential of order n of a composite function becomes considerably simpler in certain special cases which often arise in practical applications. Thus, let us seek the differential of order n of the product of two functions o> = uv. For the first values of n we have
dw = v dti + u dv, d* a) = v d* u + 2 du dv -f ud* v, • • • ; and, in general, it is evident from the law of formation that d" w = v d" u 4- r, dr dn~^u + C»d*v dn~2n -f • • • +
where Clt C2, • • • are positive integers. It might be shown by alge braic induction that these coefficients are equal to those of the expansion of (a + &)" ; but the same end may be reached by the following method, which is much more elegant, and which applies to many similar problems. Observing that Cl, C2, • • • do not depend upon the particular functions n and v employed, let us take the
28 DERIVATIVES AND DIFFERENTIALS [I, § 17
special functions u = e*, v = &, where x and y are the two inde pendent variables, and determine the coefficients for this case. We thus find
w = ex+y, dw = ex+y(dx + dy), • • -, dn<* = ex + y(dx + dy)n, du = e*dx, dzu = exdx*, •• •, dv — evdy, d*v = eydy2, • • •;
and the general formula, after division by ex+'J, becomes
(dx + di/}n = dx* + C^dydx*-1 + Ctdy2dxn-2 -\ [-dp.
Since dx and dy are arbitrary, it follows that
r _n n(n-l) n(n -1) -• • (n - p + 1)
Cl~l' 1.2 "' p~ 1.2-..p
and consequently the general formula may be written
(13) dn(uv) = vdnu+^dudn-lu + 7^ ^ d2vdn~2u -\ \-ud*v.
1 1 . 4
This formula applies for any number of independent variables. In particular, if u and v are functions of a single variable x, we have, after division by dxn, the expression for the nth derivative of the product of two functions of a single variable.
It is easy to prove in a similar manner formulae analogous to (13) for a product of any number of functions.
Another special case in which the general formula reduces to a simpler form is that in which u, v, w are integral linear functions of the independent variables x, y, z.
u= ax -f by + cz+f, v = a'x + b'y + c'z +/' , w = a"x + b"y + c"z +/",
where the coefficients a, a', a", b, b', • • • are constants. For then we
have
du = a dx + b dy + c dz,
dv = a' dx -f- b'dy + c'dz, dw = a"dx + b"dy + c"dz,
and all the differentials of higher order dnu, dnv, dniv, where n>l, vanish. Hence the formula for dn<j> is the same as if u, v, w were the independent variables ; that is,
I, §18] THE DIFFERENTIAL NOTATION 29
(dF . 8F . 8F , V"> dnw = -5- du + -T- dv 4- 5— dw I .
We proceed to apply this remark.
18. Homogeneous functions. A function <f>(x, y, z) is said to be homogeneous of degree m, if the equation
<£(w, v, w)= tm$(x, y, z) is identically satisfied when we set
u = tx, v = ty, w = tz.
Let xis equate the differentials of order n of the two sides of this equation with respect to t, noting that u, v, w are linear in t, and that
du = x dt, dv = y dt, dw = z dt. The remark just made shovvs that
ihi + y dfo + *^)(">== m(m ~1} ' " (m " n '+1)<m""*(a;' y»*)-
If we now set # = 1, w, v, w reduce to #, ?/, 2, and any term of the development of the first member,
becomes
d"<>
whence we may write, symbolically,
which reduces, for n = 1, to the well-known formula
Various notations. We have then, altogether, three systems of nota tion for the partial derivatives of a function of several variables, — that of Leibniz, that of Lagrange, and that of Cauchy. Each of these is somewhat inconveniently long, especially in a complicated calculation. For this reason various shorter notations have been devised. Among these one first used by Monge for the first and
30
DERIVATIVES AND DIFFERENTIALS
[I, §19
second derivatives of a function of two variables is now in common use. If z be the function of the two variables x and y, we set
P —
t =
dy ex2 ex 8y o if
and the total differentials dz and d'2z are given by the formulae
dz = p dx + q dy, d2z = r dx'2 -f- 2 s dx dy + t dy~.
Another notation which is now coming into general use is the following. Let z be a function of any number of independent vari ables x1} xz, x3) • ••, xn ; then the notation
ex l ex.2 • • • ox is used, where some of the indices alt a.2) • • •, an may be zeros.
19. Applications. Let y — f(x) be the equation of a plane curve C with respect to a set of rectangular axes. The equation of the tangent at a point M(x, y) is
Y-y = y'(X-x).
The slope of the normal, which is perpendicular to the tangent at the point of tangency, is — l/y'; and the equation of the normal is, therefore,
Let P be the foot of the ordinate of the point Jlf, and let T and N be the points of intersection of the x axis with the tangent and the normal, respectively.
The distance PN is called the subnormal ; FT, the subtangent; MN, the normal; and M T, the tangent.
From the equation of the normal the ab scissa of the point N is x + yy', whence the subnormal is ± yy'. If we agree to call the length PN the subnormal, and to attach the sign + or the sign — according as the direc tion PN is positive or negative, the subnormal will always be yy' for any position of the curve C. Likewise the subtangent is — y /y'. The lengths MN and M T are given by the triangles MPN and MPT:
Various problems may be given regarding these lines. Let us find, for instance, all the curves for which the subnormal is constant and equal to a given number a. This amounts to finding all the functions y=f(x) which satisfy the equation yy' = a. The left-hand side is the derivative of 2/2/2, while the
I, EXS.] EXERCISES 31
right-hand side is the derivative of ax. These functions can therefore differ
only by a constant ; whence
y'2 = 2ax + C,
which is the equation of a parabola along the x axis. Again, if we seek the curves for which the subtangent is constant, we are led to write down the equa tion y'/y = l/«; whence
log2/ = - + logC, or y = Ce?,
a
which is the equation of a transcendental curve to which the x axis is an asymp tote. To find the curves for which the normal is constant, we have the equation
or
/a2 - y* The first member is the derivative of - Vo^- y2 ; hence
(x + C)2 + y2 = a2,
which is the equation of a circle of radius a, whose center lies on the x axis.
The curves for which the tangent is constant are transcendental curves, which we shall study later.
Let y = f(x) and Y — F(x) be the equations of two curves C and C", and let M, M' be the two points which correspond to the same value of x. In order that the two subnormals should have equal lengths it is necessary and sufficient that
YY'=±yy';
that is, that Y2 — ± y2 + C, where the double sign admits of the normals' being directed in like or in opposite senses. This relation is satisfied by the cirfves
and also by the curves
which gives an easy construction for the normal to the ellipse and to the hyperbola.
EXERCISES
1. Let p = f(6) be the equation of a plane curve in polar coordinates. Through the pole O draw a line perpendicular to the radius
vector OM, and let T and N be the points where this line cuts the tangent and the normal. Find expres sions for the distances OT, ON, MN, and MT in terms of /(0) and /'(<?).
Find the curves for which each of these distances, in turn, is constant.
2. Let y = f(x), z—<t>(x) be the equations of a
skew curve T, i.e. of a general space curve. Let N FIG. 3
32 DERIVATIVES AND DIFFERENTIALS [I, Exs.
be the point where the normal plane at a point Af, that is, the plane perpendicu lar to the tangent at .M", meets the z axis ; and let P be the foot of the perpen dicular from M to the z axis. Find the curves for which each of the distances PN and JOT, in turn, is constant.
[Note. These curves lie on paraboloids of revolution or on spheres.]
3. Determine an integral polynomial /(z) of the seventh degree in x, given that f(x) + 1 is divisible by (x - I)4 and f(x) - 1 by (x+1)*. Generalize the problem.
4. Show that if the two integral polynomials P and Q satisfy the relation
Vl -p-t = Q Vl - x2, then
dP ndx
Vl - p* Vl - x2 where n is a positive integer. [Note. From the relation
(a) l-P2 = Q2(l-x») it follows that
(b) - 2 PP' = Q [2 Q'(l - x*) - 2 Qx].
The equation (a) shows that Q is prime to P ; and (b) shows that P' is divisible
by Q-]
5*. Let E (x) be a polynomial of the fourth degree whose roots are all dif ferent, and let x = U / V be a rational function of t, such that
where R\ (t) is a polynomial of the fourth degree and P / Q is a rational function. Show that the function U/ V satisfies a relation of the form
dx kdt _ •
VR(X) Vfl!(«)'
where A; is a constant. [JACOBI.]
[Note. Each root of the equation R(U/ V) = 0, since it cannot cause R'(x) to vanish, must cause UV — VU', and hence also dx/dt, to vanish.]
6*. Show that the nth derivative of a function y = $ (u), where u is a func tion of the independent variable x, may be written in the form
where
1.2
~ (*=1, 2,
ctx
[First notice that the nth derivative may be written in the form (a), where the coefficients AI, A*, • • -, An are independent of the form of the function <j>(u).
I, EXS.] EXERCISES 33
To find their values, set 0 (M) equal to w, «2, • • • , un successively, and solve the resulting equations for Ait A^, • • • , An. The result is the form (b).]
7*. Show that the nth derivative of <f> (x2) is "2 (»>(x2) + n(n -
dxn
+ n(n-V--(n-*P+V (2 X)n- 2P 0("-P)(x2) + • • • ,
1 . ju • • • p
where p varies from zero to the last positive integer not greater than n/2, and where 0(0 (x2) denotes the ith derivative with respect to x. Apply this result to the functions er3?, arc sin x, arc tan x.
8*. If x = cos u, show that
d»-i(l -x2)">-* , 1.3.5- • -(2m- 1) .
S ' = (— l)m~i — sin mu.
dxm~l m
[OLINDE RODRIGUES.]
9. Show that Legendre's polynomial,
2 . 4 . 6 • • • 2 n dx" satisfies the differential equation
ax-1 ax
Hence deduce the coefficients of the polynomial.
10. Show that the four functions
yt = sin (n arc sin x), 2/3 = sin (n arc cos x),
y2 = cos (n arc sin x), 2/4 = cos (n arc cos x),
satisfy the differential equation
(1 - x2) y" - xy' + ri*y = Q.
Hence deduce the developments of these functions when they reduce to poly nomials.
11*. Prove the formula
i
dn G*
_(x«-iei) = (-!)" -•
dx»V x»+!
[HALPHEN.]
12. Every function of the form z = x$(y/x) + $ (y/x) satisfies the equation
rx2 + 2 sxy + ty* = 0, whatever be the functions <f> and ^.
13. The function z = x0(x + y) + y^(x + y) satisfies the equation
r - 2 s + t = 0, whatever be the functions 0 and \f/.
34 DERIVATIVES AND DIFFERENTIALS [I, Exs
14. The function z =f[x + </>(y)] satisfies the equation ps = qr, whatever be the functions / and 0.
15. The function z = x»<j>(y/x) + y~n^(y/x) satisfies the equation
rx2 + 2 sxy + ty2 + px + qy = n2z, whatever be the functions <j> and \f/.
16. Show that the function
y - x - ai | 0! (x) + x - az \ fa (x) + • • • + | x - an \ 0n (x),
where fa (x), fa (x), • • • , 0n (x), together with their derivatives, 0i (x), 0£ (x), • • • , 0n(x), are continuous functions of x, has a derivative which is discontinuous for x = a\ , Oz , • • • , an •
17. Find a relation between the first and second derivatives of the function « =/(&!» M), where M = 0(x2, x3); xt, x2, x3 being three independent variables, and /and 0 two arbitrary functions.
18. Let/"(x) be the derivative of an arbitrary f unction /(x). Show that
1 d*u, _ 1 #2» u dx2 v dx2' where u = [/'(x)]-i and » =/(x) [/'(x)]-*.
19*. The nth derivative of a function of a function u-<p(y), where y = ^ (x), may be written in the form
^1.2,
where the sign of summation extends over all the positive integral solutions of
the equation i + 2 j + 3 h -\ + Ik = n, and where p = i + j + • • . + k.
[FA A DE BRUNO, Quarterly Journal of Mathematics, Vol. I, p. 359.]
CHAPTER II
IMPLICIT FUNCTIONS FUNCTIONAL DETERMINANTS CHANGE OF VARIABLE
I. IMPLICIT FUNCTIONS
20. A particular case. We frequently have to study functions for which no explicit expressions are known, but which are given by means of unsolved equations. Let us consider, for instance, an equation between the three variables x, y, z,
(1) F(x, y, z) = 0.
This equation defines, under certain conditions which we are about to investigate, a function of the two independent variables x and y. We shall prove the following theorem :
Let x = x0, y =. y0, z = z0 b& a set of values which satisfy the equa tion (1), and let us suppose that the function F, together with its first derivatives, is continuous in the neighborhood of this set of values* If the derivative Fz does not vanish for x = x0, y = y0, z = z0, there exists one and only one continuous function of the independent variables x and y which satisfies the equation (1), and which assumes the value z0 when x and y assume the values x0 and y0, respectively.
The derivative Fz not being zero for x = x0, y = y0, z = z0, let us suppose, for defmiteness, that it is positive. Since F, Fx, Fv, Fz are supposed continuous in the neighborhood, let us choose a positive number I so small that these four functions are continuous for all sets of values x, y, z which satisfy the relations
(2) \x-x0\<l, \y-y0\<l, \*-z0<l, and that, for these sets of values of x, y, z,
Fz(x,y,z}>P,
*Iu a recent article (Bulletin de la Societe Mathematique de France, Vol. XXXI, 190.'?, pp. 184-192) Goursat has shown, by a method of successive approximations, that it is not necessary to make any assumption whatever regarding Fx and Ft/, even as to their existence. His proof makes no use of the existence of Fx and Fy. His general theorem and a sketch of his proof are given in a footnote to § 25. — TRANS.
35
36 FUNCTIONAL RELATIONS [II, §20
where P is some positive number. Let Q be another positive num ber greater than the absolute values of the other two derivatives Fx, Fy in the same region.
Giving x, y, z values which satisfy the relations (2), we may then write down the following identity :
F(*> V, *) - F(*o, 7/0, «0) = F(x, y, z} - F(x0, y, z) + F(x0, y, z)
-F(x0, T/o, z) +F(x0, 7/0j z) — F(x0, 7/0, «0) 5
or, applying the law of the mean to each of these differences, and observing that F(x0, y0, «0) = 0,
F(x>y>*) = (* — *o)-FTar[»o + 8(x — »o), y, *] + 0 - yo) Fv [*., 2/o + ff(y - y0), «]
+ (z - «0) F2 [>o, ?/o, *o + 0"0 - «0)]. Hence -F(cc, T/, 2) is of the form
(3) S F^' y' ^ = A (x> y> ^ (x ~ xd
I +B(x, y, z) (y - T/O) + C (x, y, z) (z - *0),
where the absolute values of the functions A(x, y, z), B(x, y, z), C(x, y, z) satisfy the inequalities
M|<Q, \B\<Q, \C\>P
for all sets of values of x, y, z which satisfy (2). Now let c be a positive number less than Z, and rj the smaller of the two numbers I and Pe/2Q. Suppose that x and y in the equation (1) are given definite values which satisfy the conditions
and that we seek the number of roots of that equation, z being regarded as the unknown, which lie between z0 — e and z0 + c. In the expression (3), for F(x, y, z} the sum of the first two terms is always less than 2Qrj in absolute value, while the absolute value of the third term is greater than Pe when z is replaced by z0 ± e. From the manner in which 77 was chosen it is evident that this last term determines the sign of F. It follows, therefore, that F(x, y, z0 — e) < 0 and F(x, y, z0 + e) > 0 ; hence the equation (1) has at least one root which lies between z0 — e and z0 + e. Moreover this root is unique, since the derivative Fz is positive for all values of z between z0 — e and z0 + e. It is therefore clear that the equation (1) has one and only one root, and that this root approaches ZQ as x and y approach XQ and ?/0, respectively.
II, § 20]
IMPLICIT FUNCTIONS
37
Let us investigate for just what values of the variables x and y the root whose existence we have just proved is denned. Let h be the smaller of the two numbers I and PI/2Q; the foregoing reason ing shows that if the values of the variables x and y satisfy the
inequalities \x — x^\
< h, the equation (1) will have one
and only one root which lies between z0 — I and z0 -f I- Let R be a square of side 2 h, about the point M0(x0, y0), with its sides parallel to the axes. As long as the point (x, y) lies inside this square, the equation (1) uniquely determines a function of x and y, which remains between z0 — I and z0 + I. This function is continuous, by the above, at the point M0, and this is likewise true for any other point Ml of R; for, by the hypotheses made regarding the func tion F and its derivatives, the derivative Ft(xlf yl} «i) will be posi tive at the point Mlt since \xl — x0<l, \y\ — ya\<l, \zi~ zo\<l- The condition of things at Ml is then exactly the same as at M0, and hence the root under consideration will be continuous for
Since the root considered is defined only in the interior of the region R, we have thus far only an element of an implicit function. In order to define this function out side of R, we proceed by successive steps, as follows. Let L be a con tinuous path starting at the point (x0, y0~) and ending at a point (X, F) outside of R. Let us suppose that the variables x and y vary simul taneously in such a way that the ~ point (x, y) describes the path L. If we start at (x0, y0) with the value
z0 of z, we have a definite value of this root as long as we remain inside the region R. Let M1(xl, y^ be a point of the path inside R, and zt the corresponding value of z. The conditions of the theorem being satisfied for x = xlt y = yl} z = zv, there exists another region Rl} about the point MI, inside which the root which reduces to zl for x = Xi, y = yi is uniquely determined. This new region #! will have, in general, points outside of R. Taking then such a point Mt on the path L, inside R± but outside R, we may repeat the same con struction and determine a new region R2, inside of which the solu tion of the equation (1) is defined; and this process could be repeated indefinitely, as long as we did not find a set of values of x, y, z for which Fz = 0. We shall content ourselves for the present
Fm 4
38 FUNCTIONAL RELATIONS [II, § 21
with these statements; we shall find occasion in later chapters to treat certain analogous problems in detail.
21. Derivatives of implicit functions. Let us return to the region R, and to the solution z = <f>(x, y) of the equation (1), which is a continuous function of the two variables x and y in this region. This function possesses derivatives of the first order. For, keeping y fixed, let us give x an increment Ax. Then z will have an incre ment Az, and we find, by the formula derived in § 20,
F(x + As, y,z + A«) - F(x, ij, z) = Az Fx (x + 0Az, y,e + Az) -f Aa Ft (x, y, z + 0'Az) = 0.
Hence
and when A# approaches zero, As does also, since z is a continuous function of a;. The right-hand side therefore approaches a limit, and z has a derivative with respect to x :
In a similar manner we find
If the equation F = 0 is of degree m in z, it defines m functions of the variables x and y, and the partial derivatives cz/cx, 3z/dy also have m values for each set of values of the variables x and y. The preceding formulas give these derivatives without ambiguity, if the variable z in the second member be replaced by the value of that function whose derivative is sought. For example, the equation
defines the two continuous functions
+ Vl — x'2 — y* and — Vl — x'2 — y2
for values of x and y which satisfy the inequality x- + y2 < 1. The first partial derivatives of the first are
- y
II, §±2] IMPLICIT FUNCTIONS 39
and the partial derivatives of the second are found by merely chang ing the signs. The same results would be obtained by using the
formulae
dz _ x Cz _ y
dx z cy z
replacing z by its two values, successively.
22. Applications to surfaces. If we interpret x, y, z as the Cartesian coordinates of a point in space, any equation of the form
(4) F(x,y, z)=0
represents a surface S. Let (cc0, y0, z^) be the coordinates of a point A of this surface. If the function F, together with its first deriva tives, is continuous in the neighborhood of the set of values x0, yw z0, and if all three of these derivatives do not vanish simultaneously at the point A, the surface S has a tangent plane at A. Suppose, for instance, that Fz is not zero for x = x0, y = y0, z = «0. Accord ing to the general theorem we may think of the equation solved for z near the point A, and we may write the equation of the surface
in the form
z = 4(x, y},
where <f> (x, y) is a continuous function ; and the equation of the tangent plane at A is
Replacing dz /dx and dz /dy by the values found above, the equation of the tangent plane becomes
If Fz = 0, but Fx =£ 0, at A , we would consider y and z as inde pendent variables and a; as a function of them. We would then find the same equation (5) for the tangent plane, which is also evi dent a priori from the symmetry of the left-hand side. Likewise the tangent to a plane curve F(x, y) = 0, at a point (x0, y0~), is
If the three first derivatives vanish simultaneously at the point A.
dF
40 FUNCTIONAL RELATIONS [II, §23
the preceding reasoning is no longer applicable. We shall see later (Chapter III) that the tangents to the various curves which lie on the surface and which pass through A form, in general, a cone and not a plane.
In the demonstration of the general theorem on implicit functions we assumed that the derivative F^ did not vanish. Our geometrical intuition explains the necessity of this condition in general. For, if F^ = 0 but F^ 3= 0, the tangent plane is parallel to the % axis, and a line parallel to the z axis and near the line x = xw y = y0 meets the surface, in general, in two points near the point of tangency. Hence, in general, the equation (4) would have two roots which both approach z0 when x and y approach x0 and y0, respectively.
If the sphere a;2 + y2 -+- «2 — 1 = 0, for instance, be cut by the line y = 0, x = 1 + c, we find two values of z, which both approach zero with e ; they are real if c is negative, and imaginary if c is positive.
23. Successive derivatives. In the formulae for the first derivatives,
3z= _Fx dz_ = _FJL
dx~ F,' cy~ F,'
we may consider the second members as composite functions, z being an auxiliary function. We might then calculate the successive deriv atives, one after another, by the rules for composite functions. The existence of these partial derivatives depends, of course, upon the existence of the successive partial derivatives of F(x, y, K).
The following proposition leads to a simpler method of determin ing these derivatives.
If several functions of an independent variable satisfy a relation F = 0, their derivatives satisfy the equation obtained by equating to zero the derivative of the left-hand side formed by the rule for differ entiating composite functions. For it is clear that if F vanishes identically when the variables which occur are replaced by func tions of the independent variable, then the derivative will also van ish identically. The same theorem holds even when the functions which satisfy the relation F = 0 depend upon several independent variables.
Now suppose that we wished to calculate the successive derivatives of an implicit function y of a single independent variable x defined by the relation
II, §23] IMPLICIT FUNCTIONS 41
We find successively
dF
T~ + ox cy
dF SF ,
~ ~
d'2F dF
—— + 2 — — v' + — — v'2+ —v" =0 * ^ dxdyy -dy» ^ dyy
2
3
dx* ox2 dy y ox dy* * dxdy dys
32 7,1
from which we could calculate successively y', y", y'
Example. Given a function y =/(x), we may, inversely, consider y as the independent variable and x as an implicit function of y defined by the equation y=f(x). If the derivative /'(x) does not vanish for the value XQ, where 2/o =/(zo)i there exists, by the general theorem proved above, one and only one function of y which satisfies the relation y = f(x) and which takes on the value XQ for y = 2/0- This function is called the inverse of the f unction /(x). To cal culate the successive derivatives xy, xyt, ay, • • • of this function, we need merely differentiate, regarding y as the independent variable, and we get
1 = /'(x) xy,
0 = /"(x) (X,)2 + /'(x) ay,
0 =/"'(x) (xy)* + 3f"(x)xyx? +/'(x)x2/3,
whence
1 f"(z) _8[/"(x)]«-
~7^)' ~[7w' [/'(
It should be noticed that these formulae are not altered if we exchange xv and /'(x), Xy2 and /"(x), Xj,s and /"'(x), • • • , for it is evident that the relation between the two functions y = /(x) and x = 0 (y) is a reciprocal one.
As an application of these formulae, let us determine all those functions y=f(x) which satisfy the equation
y'y'" - 3y"* = 0.
Taking y as the independent variable and x as the function, this equation becomes
Xj/> = 0.
But the only functions whose third derivatives are zero are polynomials of at most the second deree. Hence x must be of the form
where Ci, C2, C3 are three arbitrary constants. Solving this equation for y, we see that the only functions y = /(x) which satisfy the given equation are of the form _
y = a ± V bx + c,
42 FUNCTIONAL RELATIONS [II, §24
where a, 6, c are three arbitrary constants. This equation represents a parabola whose axis is parallel to the x axis.
24. Partial derivatives. Let us now consider an implicit function of two variables, denned by the equation
(6) F(x,y,z) = 0.
The partial derivatives of the first order are given, as we have seen, by the equations
9F.9F9* ?l dFdz_
(7) o — p 7T- — — u, -^ h fl ;p — u. 0x 9* 0z <?y </* 0#
To determine the partial derivatives of the second order we need only differentiate the two equations (7) again with respect to x and with respect to y. This gives, however, only three new equations, for the derivative of the first of the equations (7) with respect to y is identical with the derivative of the second with respect to x. The new equations are the following:
.O±JL.?£ + ££/!?)%-—— =o
dx2 dxdzdx dz2 \dx] dz dx2 d2F , d*F dz ^ d2F d_z_ <P_F dz dz d_F d*z _
"" r\ n r\ o I O O O ^J
dx ftr dx dy c~
d2F d2F dz d2 F (dz\2 dF d2z ~ + " fe dy + 'dz2 (dy) +^ dy2 :
The third and higher derivatives may be found in a similar manner. By the use of total differentials we can find all the partial deriva tives of a given order at the same time. This depends upon the following theorem :
If several functions u, v, w, • • • of any number of independent vari ables x, y, z, ••• satisfy a relation F = 0, the total differentials satisfy the relation dF= 0, which is obtained by forming the total differential of F as if all the variables which occur in F ivere independent variables.
In order to prove this let F(u, v, w) = 0 be the given relation between the three functions u, v, iv of the independent variables x, y, z, t. The first partial derivatives of M, v, w satisfy the four equations
dFdu
__ __
du dx dv dx dw dx
d_Fd_u ,
du dy dv dy dw dy
II,§24J IMPLICIT FUNCTIONS 43
dFdu dFdv d_F_d-w_
o ~ « ~o ~ I o ~a — ^) CM tfS 0» tf£ OW CZ
dFdu d_Fd_» ^^! = 0
aw & a? a< a^ st =
Multiplying these equations by dx, dy, dz, dt, respectively, and adding, we find
°-^du + ^-dv + d-/-dw = dF=0.
du dv OW
This shows again the advantage of the differential notation, for the preceding equation is independent of the choice and of the number of independent variables. To find a relation between the second total differentials, we need merely apply the general theorem to the equation dF = 0, considered as an equation between u, v, w, du, dv, dw, and so forth. The differentials of higher order than the first of those variables which are chosen for independent variables must, of course, be replaced by zeros.
Let us apply this theorem to calculate the successive total differ entials of the implicit function defined by the equation (6), where x and y are regarded as the independent variables. We find
*F i ^F j ^3F j
— dx + 7— a;/ + 7— dz = 0,
ox cy cz
dF dF 8F V2> , dF n
•T- dx + -z- dy + -r- dz ) + — <P* = 0,
dx dy dz / cz
and the first two of these equations may be used instead of the five equations (7) and (8) ; from the expression for dz we may find the two first derivatives, from that for d^z the three of the second order, etc. Consider for example, the equation
Ax2 + A'y* + A"z* = l, which gives, after two differentiations,
Ax dx + A 'ydy + A "z dz = 0, A dx2 + A 'dy2 + A "dz2 + A " zd*z = 0,
whence
Axdx + A'ydy,
dg~ -- TTi - ' A"z
and, introducing this value of dz in the second equation, we find A (A x* + A "z2} dx* + 2AA 'xy dx dy + A '(A'y* + A "z2} dy*
44 FUNCTIONAL RELATIONS [II, §24
Using Monge's notation, we have then
Ax A'y
p= ~IV q~ ~IV
A(Ax* + A"z*) _ AA'xy
~ "**'
This method is evidently general, whatever be the number of the independent variables or the order of the partial derivatives which it is desired to calculate.
Example. Let z = /(x, y) be a function of x and y. Let us try to calculate the differentials of the first and second orders dx and d'2x, regarding y and z as the independent variables, and x as an implicit function of them. First of all, we have
dz = — dx + — dy. dx dy
Since y and z are now the independent variables, we must set
d*y = d2z = 0, and consequently a second differentiation gives
0 = ^dx» + 2 ^- dxdy + ?^-dy* + ^d?x. £x2 dxdy dy* dx
In Monge's notation, using p, q, r, s, t for the derivatives of /(x, y), these equations may be written in the form
dz — p dx + q dy, 0 = r dx2 + 2 s dx dy + tdy* + p d2x.
From the first we find
, dz — q dy dx= - — -, P
and, substituting this value of dx in the second equation, rdz* + 2(ps-qr)dydz + (q*r -2pqs
<Px= -
The first and second partial derivatives of x, regarded as a function of y and z, therefore, have the following values :
dx _ 1 8x _ q
dz p dy p
d*x _ r dzx _qr — ps d2x _ 2pqs — pH — q*r
dz2 p8 dy dz p3 dy2 ps
As an application of these formulae, let us find all those functions /(x, y) which satisfy the equation
= 2pqs.
If, in the equation z =/(x, y), x be considered as a function of the two inde pendent variables y and z, the given equation reduces to Xyt = 0. This means
II, § 25]
IMPLICIT FUNCTIONS
45
that xv is independent of y ; and hence xv = 0(z), where <f>(z) is an arbitrary function of z. This, in turn, may be written in the form
which shows that x - y <f>(z) is independent of y. Hence we may write
where ^ (z) is another arbitrary function of z. It is clear, therefore, that all the functions z =/(x, y) which satisfy the given equation, except those for which fx vanishes, are found by solving this last equation for z. This equation represents a surface generated by a straight line which is always parallel to the xy plane.
25. The general theorem. Let us consider a system of n equations
(E)
' ' '} xpi ul> **ll " 'i
* (x x • • • x ' u u •••w) = 0
between the n-\-p variables ui} u3, •••, un; xl} xy, •••, xp. Suppose that these equations are satisfied for the values xv — x\, • • • , xp = xp, u± = wj, • • • j un = u°n ; that the functions Fi are continuous and possess first partial derivatives which are continuous, in the neighborhood of this system of values; and, finally, that the determinant
du
does not vanish for
x,- —
uk =
Under these conditions there exists one and only one system of con tinuous functions u^ = <f>i(xi, x2, •• •, x^), •• •, un — <}>n(x1, x2, • ••, xp~) which satisfy the equations (E) and which reduce to u\, u\, •••, u°n, for x, = x\, •••, xp = x*p*
*In his paper quoted above (ftn., p. 35) Goursat proves that the same conclusion may be reached without making any hypotheses whatever regarding the derivatives cFi/dXj of the functions F{ with regard to the x's. Otherwise the hypotheses remain exactly as stated above. It is to be noticed that the later theorems regarding the existence of the derivatives of the functions 4> would not follow, however, without some assumptions regarding dFf/dXj. The proof given is based on the following
46 FUNCTIONAL RELATIONS [II, §26
The determinant A is called the Jacobian,* or the Functional Deter minant, of the n functions Fu F2, • • -, Fn with respect to the n vari ables ul} it?, • • •, un. It is represented by the notation
D(Flf F2, ...,F,,)
We will begin by proving the theorem in the special case of a system of two equations in three independent variables x, y, z and two unknowns u and v.
(9) Fi(x, y, z, u, v) = 0,
(10) Fi(x,y,z,u,v) = Q.
These equations are satisfied, by hypothesis, for x = x0,y = y0,z = z0, u = u0, v = v0 ; and the determinant
dF\ dFj _ dF\ dFt
du cv dv cu
does not vanish for this set of values. It follows that at least one of the derivatives dF^/dv, dF2/dv does not vanish for these same values. Suppose, for definiteness, that oFl/8u does not vanish. According to the theorem proved above for a single equation, the relation (9) defines a function v of the variables x, y, z, u,
v =f(x, y, *> «)>
which reduces to v0 for x = x0, y = y0, z = zw u = u0. Replacing v in the equation (10) by this function, we obtain an equation between x, y, z, and u,
$(«, y, z, u} = Ft[x, y, z, u, f(x, y, z, «)] = 0,
lemma: Let f\(x\,3kt,---,vp; MI, u2, ••• ,un), •••,/„(«!, x?, •••,xp; MI, u2, •••, un) be n functions of the n + p variables X{ and u^, which, together with the n2 partial deriva tives cfi/GUfr, are continuous near Xi — 0, xz = 0, ••• , xp = 0, HI = 0, •••, un = 0. If the n functions f{ and the n2 derivatives dfi/^Uf. all vanish for this system of values, then the n equations
«i=/i. «2=/2, •"' «•»=/» admit one and only one system of solutions of the form
where 01( 02> •••> 0n a™ continuous functions of the p variables Xi, x2, •• •, xp which all approach zero as the variables all approach zero. The lemma is proved by means of a suite of functions u^ =fi(x1,xz , •••,xp\ u[m~l\ w^"'-1), •••, u^'^) (i = l, 2, •••, n), where M^O) = 0. It is shown that the suite of functions u\m) thus denned approaches a limiting function U{, which 1) satisfies the given equations, and 2) constitutes the only solution. The passage from the lemma to the theorem consists in an easy transforma tion of the equations (E) into a form similar to that of the lemma. — TRANS. *JACOBI, Crelle's Journal, Vol. XXII.
II, §25] IMPLICIT FUNCTIONS 47
which is satisfied for x = x0, y — y0, z = zw u = u0. Now
^ t , .
du 8u dv du
and from equation (9),
du ov ou whence, replacing df/du by this value in the expression for
WP nhfaun
we obtain
d® D(u, v)
~du ~ dF^
dv
It is evident that this derivative does not vanish for the values x0, y<» zo> uo- Hence the equation <I> = 0 is satisfied when u is replaced by a certain continuous function u = <£ (x, y, «), which is equal to MO when x = x0, y = y0, z = z0 ; and, replacing u by <£ (x, y, z) in f(x, y, z, ?/), we obtain for v also a certain continuous function. The proposition is then proved for a system of two equations.
We can show, as in § 21, that these functions possess partial derivatives of the first order. Keeping y and z constant, let us give x an increment Ax, and let AM and Ay be the corresponding increments of the functions u and v. The equations (9) and (10) then give us the equations
'-£ + . + A. + .' + A,, + ." = 0
• + + A. + ,' + A, + ,-' = 0,
' ' '
where e, e', e", rj, rj', rj" approach zero with Aa-, A«, Av. It follows that
, ^ + c ^ + 77" - ^ + «» V-2 + 77 A;/. \ da; / \ go 7 / V g» / \ Ox
8Fi , A/^« , ,\ /^i , £"VaF24-V
\- f. II — p t]
y / \ ou
When Ax approaches zero, AM and Av also approach zero ; and hence e, e', e", 77, 77', 77" do so at the same time. The ratio Aw /Ax therefore approaches a limit ; that is, u possesses a derivative with respect to x :
48 FUNCTIONAL RELATIONS [II, §26
dFl cF2 dFl dF2
du dx dv dv dx
dx dFl dF2 dFi dFz
du dv dv du
It follows in like manner that the ratio Av/Aa; approaches a finite limit dv /dx, which is given by an analogous formula. Practically, these derivatives may be calculated by means of the two equations
8 Ft dFj du dFl dv _
dx du dx dv dx
dF2 dF^du dF^dv
o "T" <~\ ~r\~ T ~ ~^~~ == " ! OX CU OX CV OX
and the partial derivatives with respect to y and z may be found in a similar manner.
In order to prove the general theorem it will be sufficient to show that if the proposition holds for a system of (n — 1) equations, it will hold also for a system of n equations. Since, by hypothesis, the functional determinant A does not vanish for the initial values of the variables, at least one of the first minors corresponding to the elements of the last row is different from zero for these same values. Suppose, for definiteness, that it is the minor which corresponds to dFn/dun which is not zero. This minor is precisely
D(Fl} F2, -••,Fn_1). D(UI, «2, ..., «„_,) '
and, since the theorem is assumed to hold for a system of (n — 1) equations, it is clear that we may obtain solutions of the first (n — 1) of the equations (E) in the form
M1 = ^1(a?1, a;,, •••,*„; Mn), ••-, un_l = fa^fa, x2, • •-, xp; un),
where the functions <£,. are continuous. Then, replacing u^ •••, wn_, by the functions ^1? •••,<£„_! in the last of equations (E), we obtain a new equation for the determination of un,
ai '••>*,; un) = Fn (xlt «,,-.., xp ; ^t, <^2, • • ., 0B_,, MII) = 0.
It only remains for us to show that the derivative d<b/dun does not vanish for the given set of values x\, x°2, • • -, x°p, <; for, if so, we can solve this last equation in the form
«» = ^0*i, *a, •••, *p),
where $ is continuous. Then, substituting this value of un ir <^i5 •••> <f>n-i> we would obtain certain continuous functions foi
II, §25] IMPLICIT FUNCTIONS 49
HI, u2, ••-, un_l also. In order to show that the derivative in ques tion does not vanish, let us consider the equation
- . . .
dun dui dun dun-i ^un dun
The derivatives 8<j>}/8un, d<jj2/dun, •••, d*n_i/d«n are given by the (n — 1) equations
(12)
7J ~7, r * " T o ~o i o — V}
n-1 tj *Pl i i ^-^n-1 ^yn-1 • g-^n-l r\ .
and we may consider the equations (11) and (12) as n linear equa tions for d<f>!/dun, • ••, d<f>n-i/dun, d®/dun) from which we find
cun D (MU t<2, •••,«„- i) D (M!, ?<2, • • • , wn)
It follows that the derivative d®/dun does not vanish for the initial values, and hence the general theorem is proved.
The successive derivatives of implicit functions defined by several equations may be calculated in a manner analogous to that used in the case of a single equation. When there are several independent variables it is advantageous to form the total differentials, from which the partial derivatives of the same order may be found. Consider the case of two functions u and v of the three variables x, y, z defined by the two equations
F(x, y, z, u, v)=0,
The total differentials of the first order du and dv are given by the two equations
3F . , 0F _ . SF . . 8F , .dF
-%- dx + ^- dy + ^- dz + -5- du + -r- dv = 0, dx dy dz du cv
d® d$ .. d& , . d® , , d® -, A
-z- dx + -^- du + -^- dz + -r— du + -5— c?v = 0. Sec cy ^s du cv
Likewise, the second total differentials dzu and d*v are given by the equations
50 FUNCTIONAL RELATIONS [II, §26
dF V2) dF dF
- dx + -.. + — dv) + G/-d*u+™ d*v = 0, cv du co
2> d& d&
+ ?-*d*u + ~d*v = 0,
CU CV
and so forth. In the equations which give dnu and dnv the deter minant of the coefficients of those differentials is equal for all vahies of n to the Jacobian D(F, <£)/Z>(w, v), which, by hypothesis, does not vanish.
26. Inversion. Let MI, «2, • • • , un be n functions of the n independent vari ables xi, x2, • • •, £n, such that the Jacobian D(UI, «2, • • • , u,,)/D(xi, x2, • • •, xn) does not vanish identically. The n equations
/i g\
( Un = 0n(X1( X2, • • -, Xn)
define, inversely, Xi, x2, • • • , xn as functions of u\, M2, •••,«„. For, taking any system of values x?, x£, • • • , x£, for which the Jacobian does not vanish, and denoting the corresponding values of MI, w2, • • •, Un by uj, w!j, • • •, M°, there exists, according to the general theorem, a system of functions
which satisfy (13), and which take on the values x", x", • • -, x°, respectively, when MI = wj, • • • , un = unn. These functions are called the inverses of the func tions 0i, 02, ••-, 0n, and the process of actually determining them is called an inversion.
In order to compute the derivatives of these inverse functions we need merely apply the general rule. Thus, in the case of two functions
u=f(x, y), v = <f>(x,y)t
if we consider u and v as the independent variables and x and y as inverse functions, we have the two equations
whence
, 8f, , Bf , , d<f> , 30 ,
du = — dx + — dy, dv = — - dx + — - dy.
dx cy dx cy
^0 j %f -i c0 . df
— ait av du H — - dv
•, dy dy , dx dx
dx = , dy = •
a/^0_?/a0 a/ a0 _ ^/ ^0
dx dy dy dx dx dy dy dx
We have then, finally, the formulae
50 _ d_f
dx dy dx dy
du 8/a0_a/c0 dv d_fdj>_d_fd_$
dx dy dy dx dx dy dy dx
II, §27] IMPLICIT FUNCTIONS 51
£0 8f
dx dy ex
eu ~" df £0 _ cf d<t> eB^
ex e^ ey ex ex ey ey ex
27. Tangents to skew curves. Let us consider a curve C repre sented by the two equations
l*i(*,y,«)-0,
(14) <
JF,(a5,y,«) = 05
and let x0, T/O, 20 be the coordinates of a point M0 of this curve, such that at least one of the three Jacobians
dF\dF^ _d_F\d_F\ dF1dFt_dF\ 8F* 8Fl gF2 dF_i dF±
dy dz dz dy ' vz dx dx dz ex dy dij dx
does not vanish when x, y, z are replaced by x0, %, zot respectively. Suppose, for defmiteness, that D(Fl} Fj/D(y, z) is one which does not vanish at the point Mn. Then the equations (14) may be solved in the form
y = ^(x)) z = t(x)>
where $ and \j/ are continuous functions of x which reduce to y0 and z0, respectively, when x = x0. The tangent to the curve C at the point 3/o is therefore represented by the two equations
X-x0 = F-7/Q = Z -g0 1 ' ^'(x0) " f (x0)
where the derivatives <#>'(cc) and i//(ce) may be found from the two equations
In these two equations let us set x = x0)y = y0,z = 2^, and replace *'(*„) and ^o) by ( F - T/O) / (X - *„) and (Z - «0)/(X - x0), respectively. The equations of the tangent then become
62 FUNCTIONAL RELATIONS [II, §28
or
X xa Y y0 Z — z0
^(y, *) o ^>(«, «) o fl(«,y
The geometrical interpretation of this result is very easy. The two equations (14) represent, respectively, two surfaces Sj and S2, of which C is the line of intersection. The equations (15) represent the two tangent planes to these two surfaces at the point 1/0; and the tangent to C is the intersection of these two planes.
The formulae become illusory when the three Jacobians above all vanish at the point M0. In this case the two equations (15) reduce to a single equation, and the surfaces Si and S2 are tangent at the point A/0. The intersection of the two surfaces will then consist, in general, as we shall see, of several distinct branches through the point M0.
II. FUNCTIONAL DETERMINANTS
28. Fundamental property. We have just seen what an important role functional determinants play in the theory of implicit functions. All the above demonstrations expressly presuppose that a certain Jacobian does not vanish for the assumed set of initial values. Omitting the case in which the Jacobian vanishes only for certain particular values of the variables, we shall proceed to examine the very important case in which the Jacobian vanishes identically. The following theorem is fundamental.
Let HI, u2, •• • , un be n functions of the n independent variables x\i xii '") xn" In order that there exist between these n functions a relation II (MI} w2> • • •> un) == 0, which does not involve explicitly any of the variables xly xz, • • • , xn, it is necessary and sufficient that the functional determinant
D(UI, M2, •••, ?y)
should vanish identically.
In th.3 first place this condition is necessary. For, if such a rela tion TL(UI, w2, • • •, wn) = 0 exists between the n functions HI, u%, • • •, un, the following n equations, deduced by differentiating with respect to each of the z's in order, must hold :
II, £28.1 FUNCTIONAL DETERMINANTS 53
endUl 8udu2 an 8un __
7; ^ — ~T~ "^ "o ~r ' ' ' T Q /-, — " ,
jfi , jf2 , » = Q.
dui 8xn du2 dxn dun dxn
and, since we cannot have, at the same time,
^5 = — = = ^5 = 0
£«! <7U2 CUH
since the relation considered would in that case reduce to a trivial identity, it is clear that the determinant of the coefficients, which is precisely the Jacobian of the theorem, must vanish.*
The condition is also sufficient. To prove this, we shall make use of certain facts which follow immediately from the general theorems.
1) Let u, v, w be three functions of the three independent variables x, y, z, such that the functional determinant D(u, v, w)/D(x, y, z) is not zero. Then no relation of the form
A du + /u, dv + v dw = 0
can exist between the total differentials du, dv, dw, except for X = p, = v = 0. For, equating the coefficients of dx, dy, dz in the foregoing equation to zero, there result three equations for X, p., v which have no other solutions than X = /u, = v = 0.
2) Let w, u, v, w be four functions of the three independent variables x, y, z, such that the determinant D (u, v, w} / D (x, y, s) is not zero. We can then express x, y, z inversely as functions of u, v, wt and substituting these values for x, y, z in o>, we obtain
a function
a, = $ (u, v, w)
of the three variables u, v, ^v. If by any process ivhatever we can obtain a relation of the form
(16) du = P du + Q dv + R dw
*As Professor Osgood has pointed out, the reasoning here supposes that the partial derivatives an / Si/i , dU /£«„, • • • , ^H / dUn do not all vanish simultaneously for any system of values which cause U (ulf u2, ••-,«„) to vanish. This supposition is certainly justified when the relation II = 0 is solved for one of the variables ut.
54 FUNCTIONAL RELATIONS [II, §28
between the total differentials dw, du, dv, dw, taken with respect to the independent variables x, y, z, then the coefficients P, Q, R are equal, respectively, to the three first partial derivatives of <£ (u, v, w) :
d& d$ 8<b
P = o ' Q = ~o ' •** = o '
Cu cv ow
For, by the rule for the total differential of a composite function (§ 16), we have
d& d<b d&
«<D = - — du + -^— dv -|- — dw : du cv cw
and there cannot exist any other relation of the form (16) between d<a, du, dv, dw, for that would lead to a relation of the form
A. du + p. do + v dw = 0,
where X, /t, v do not all vanish. We have just seen that this is impossible.
It is clear that these remarks apply to the general case of any number of independent variables.
Let us then consider, for definiteness, a system of four functions of four independent variables
(17)
X = Fl(x,y,z, *), Y=Fi (x, y, z, t), Z = F3(x, y, z, t), T=Ft(x,y,z, t),
where the Jacobian D(Fl} F2, F3, Fi)/D(x, y, z, t) is identically zero by hypothesis ; and let us suppose, first, that one of the first minors, say D(F^ F2, Fs)/D(x, y, z), is not zero. We may then think of the first three of equations (17) as solved for x, y, z as functions of X, Y, Z, t ; and, substituting these values for x, y, z in the last of equations (17), we obtain T as a function of A', Y, Z, t:
(18) T=*(X,Y,Z,t).
We proceed to show that this function $ does not contain the vari able t, that is, that 8$ /dt vanishes identically. For this purpose let us consider the determinant
II,
FUNCTIONAL DETERMINANTS
55
A =
|
QFi |
dj\ |
df\ |
dX |
|
dx |
dt/ |
dz |
|
|
dx |
dF, |
dFz dz |
dY |
|
~dx~ |
dFs ty |
dFs dz |
dZ |
|
dx |
dF\ |
dz |
dT |
If, in this determinant, dX, dY, dZ, dT be replaced by their values
ox
tiy
Ct
and if the determinant be developed in terms of dx, dy, dz, dt, it turns out that the coefficients of these four differentials are each zero ; the first three being determinants with two identical columns, while the last is precisely the functional determinant. Hence A = 0. But if we develop this determinant with respect to the elements of the last column, the coefficient of dTis not zero, and we obtain a relation of the form
dT = P dX + Q dY + R dZ.
By the remark made above, the coefficient of dt in the right-hand side is equal to d<i?/dt. But this right-hand side does not contain dt, hence d&/dt = 0. It follows that the relation (18) is of the form
which proves the theorem stated.
It can be shown that there exists no other relation, distinct from that just found, between the four functions X, Y, Z, T, independent of x, y, z, t. For, if one existed, and if we replaced T by $>(X, Y, Z) in it, we would obtain a relation between X, Y, Z of the form U(X, Y, Z)=0, which is a contradiction of the hypothesis that D(X, Y, Z)/D(x, y, z) does not vanish.
Let us now pass to the case in which all the first minors of the Jacobian vanish identically, but where at least one of the second minors, say D(Flt F^)/D(x, y}, is not zero. Then the first two of equations (17) may be solved for x and y as functions of X. Y, z, t, and the last two become
Z = *! (X, Y, z, t), T = ®.2 (A', Y, z, t).
56 FUNCTIONAL RELATIONS [n,
On the other hand we can show, as before, that the determinant
dX dY dZ
ex
ex
fy
¥
ex
dy
vanishes identically ; and, developing it with respect to the elements of the last column, we find a relation of the form
dZ = FdX + QdY,
whence it follows that
In like manner it can be shown that
!r=0' '
dt
= 0;
and there exist in this case two distinct relations between the four functions X, Y, Z, T, of the form
There exists, however, no third relation distinct from these two; for, if there were, we could find a relation between X and Y, which would be in contradiction with the hypothesis that D(X, Y} / D(x, y) is not zero.
Finally, if all the second minors of the Jacobian are zeros, but not all four functions X, Z, Y, T are constants, three of them are functions of the fourth. The above reasoning is evidently general. If the Jacobian of the n functions F1} F2, •• •, FH of the n independ ent variables xly x2, •••, xn, together with all its (n — r + 1) -rowed minors, vanishes identically, but at least one of the (n — r)- rowed minors is not zero, there exist precisely r distinct relations between the n functions ; and certain r of them can be expressed in terms of the remaining (n — r), between which there exists no relation.
The proof of the following proposition, which is similar to the above demonstration, will be left to the reader. The necessary and sufficient condition that n functions of n + p independent variables be connected by a relation which does not involve these variables is that every one of the Jacobians of these n functions, with respect to any n
II, §28] FUNCTIONAL DETERMINANTS 57
of the independent variables, should vanish identically. In par ticular, the necessary and sufficient condition that two functions F i(#i , xz, •••, CCB) and F2(xl, xz,---, #„) should be functions of each other is that the corresponding partial derivatives dF1/dxi and dF2/dXf should be proportional.
Note. The functions F19 F2, •• •, Fn in the foregoing theorems may involve certain other variables y1} y2) ••-, ym, besides xl, x2, •••, xn. If the Jacobian D(Fl} Fz, •••, Fn)/D(xl, x2, ••-, ojn) is zero, the functions JF\, F2, •••, Fn are connected by one or more relations which do not involve explicitly the variables x1} x2, •••, xn, but which may involve the other variables y1} y2, •••, ym.
Applications. The preceding theorem is of great importance. The funda mental property of the logarithm, for instance, can be demonstrated by means of it, without using the arithmetic definition of the logarithm. For it is proved at the beginning of the Integral Calculus that there exists a function which is defined for all positive values of the variable, which is zero when x — 1, and whose derivative is l/x. Let/(x) be this function, and let
u=f(x)+f(y), v = xy. Then
D (u, v) _
D (x, y)
x y =0.
y x
Hence there exists a relation of the form f(x)+f(V) =
and to determine 0 we need only set y = 1, which gives f(x) = <j> (x). Hence, since x is arbitrary,
f(z)+f(y)=f(xy).
It is clear that the preceding definition might have led to the discovery of the fundamental properties of the logarithm had they not been known before the Integral Calculus.
As another application let us consider a system of n equations in n unknowns
(MI, «2,
(19)
. Fn(Ul, W2,
where J/i, JT2) •••) Hn are constants or functions of certain other variables *i» *2» •••» Xmi which may also occur in the functions .F,-. If the Jacobian Z)(Fi, F2, • • •, Fn)/D(u\, «2i • • •> un) vanishes identically, there exist between the n functions F, a certain number, say n — fc, of distinct relations of the form
• • • , Ft)t • • • , Flt = Un-k (F!, • • • , Fk).
58
FUNCTIONAL RELATIONS
[II, § 29
In order that the equations (19) be compatible, it is evidently necessary that Ht + l = Hi (Hn .••,Hk),..-,Hn = UH-t (Hi, • • • , Hk),
and, if this be true, the n equations (19) reduce to k distinct equations. We have then the same cases as in the discussion of a system of linear equations.
29. Another property of the Jacobian. The Jacobian of a system of n functions of n variables possesses properties analogous to those of the derivative of a function of a single variable. Thus the preceding theorem may be regarded as a generalization of the theorem of § 8.
The formula for the derivative of a function of a function may be extended to Jacobians. Let Flf F2, • • •, Fn be a system of n func tions of the variables MI} u2, •••, un, and let us suppose that u^ w2> • •-, ua themselves are functions of the n independent variables xlf x • • • x. Then the formula
D(F
l,
-, Fn) D(UI,
D(xlt
D(x1}
follows at once from the rule for the multiplication of determinants and the formula for the derivative of a composite function. For, let us write down the two functional determinants
cj\
ou
dF
du
dxn dxu
cx
where the rows and the columns in the second have been inter changed. The first element of the product is equal to
,
dFl
i
du,,
that is, to
?!, and similarly for the other elements.
30. Hessians. Let/(x, ?/, z) be a function of the three variables x, y, z. Then the functional determinant of the three first partial derivatives cf/dx, Sf/cy, df/dz,
a2/ a2/ a2/
ax2 ex 5y dx az
a2/ a2/ a2/ ax cy a?/2 a2/ a2/
ex cz cy oz
dydz
a2/
cz-
II, § 30]
FUNCTIONAL DETERMINA NTS
59
is called the Hessian of f(x, y, z). The Hessian of a function of n variables is defined in like manner, and plays a role analogous to that of the second deriva tive of a function of a single variable. We proceed to prove a remarkable invariant property of this determinant. Let us suppose the independent vari ables transformed by the linear substitution
(X= aX+ [3Y+ yZ, y= a'X + p'Y+ y'Z,
(19')
where X, F, Z are the transformed variables, and or, 0, 7, • • • , 7" are constants such that the determinant of the substitution,
a J8 7 A = a' /3' 7' a" /3" 7"
is not zero. This substitution carries the function /(x, y, z) over into a new function F(X, Y, Z) of the three variables X, Y, Z. Let II (X, F, Z) be the Hessian of this new function. We shall show that we have identically
II (X, F, Z) = A2/t(x, ?/, z),
where x, ?/, z are supposed replaced in /i(x, y, z) by their expressions from (19'). For we have
fZF dF dF^
H =
7 £^ ?I\
BY' cZ ) ~\dX' aT' aZ/ D(x, y,
D(X, Y, Z) D(x, y, z) D(X, Y, Z)
and if we consider cf/cx, cf/dy, df/dz, for a moment, as auxiliary variables, we may write
By cz D(x, y,
^, ^, Kl
cx dy dz /
D(x,y,z) U(X, Y, Z)
But from the relation F(X, Y, Z) =f(x, y, z), we find
dF cf ,cf ,,Bf -— = a — + a' — + a" — , dX ex cy dz
dY
dy
whence
3F a/ , a/ , „ c/
-^ = 7 — + 7 — + 7 — » c:Z cx cy cz
d_F dF
er' ez
and hence, finally,
dx dy dz
a a' a"
7 7
= A;
H=
D(x, y, z)
D(X, Y, Z) It is clear that this theorem is general.
'- = A2/i-
60 FUNCTIONAL RELATIONS [II, §30
Let us now consider an application of this property of the Hessian. Let /(x, y) = ox3 + 3 bx*y + 3 cxy2 + dy*
be a given binary cubic form whose coefficients a, b, c, d are any constants. Then, neglecting a numerical factor,
h =
ax + by bx 4 cy bx + cy ex + dy
= (ac - &2)x2 + (ad - bc)xy + (bd - c2)y2,
and the Hessian is seen to be a binary quadratic form. First, discarding the case in which the Hessian is a perfect square, we may write it as the product of
two linear factors :
h = (mx + ny) (px + qy).
If, now, we perform the linear substitution
mx + ny = X, px + qy = Y, the form/(x, y) goes over into a new form,
F(X, Y) = AX* + 3 BX2 Y+3 CXY2 + DY8, whose Hessian is
H(X, Y) = (AC - B2) X2 + (AD - EC] XY + (BD - C2) F2,
and this must reduce, by the invariant property proved above, to a product of the form KXY. Hence the coefficients A, B, C, D must satisfy the relations
If one of the two coefficients B, C be different from zero, the other must be so, and we shall have
-?• -f-
F(X, Y) = — (B*X* + 3 B2 CX* Y + 3 BC* XY2 + C* Y3) = (B^+^Y)\
whence F(X, Y), and hence /(x, y), will be a perfect cube. Discarding this particular case, it is evident that we shall have B = C = 0 ; and the polynomial F(X, Y) will be of the canonical form
AX* + DY3.
Hence the reduction of the form /(x, y) to its canonical form only involves the solution of an equation of the second degree, obtained by equating the Hessian of the given form to zero. The canonical variables X, Y are precisely the two factors of the Hessian.
It is easy to see, in like manner, that the form/(x, y) is reducible to the form AX3 + BX2 Y when the Hessian is a perfect square. When the Hessian van ishes identically /(x, y) is a perfect cube :
/(x, y) = (ax
II, §31] TRANSFORMATIONS 61
III. TRANSFORMATIONS
It often happens, in many problems which arise in Mathematical Analysis, that we are led to change the independent variables. It therefore becomes necessary to be able to express the derivatives with respect to the old variables in terms of the derivatives with respect to the new variables. We have already considered a problem of this kind in the case of inversion. Let us now consider the question from a general point of view, and treat those problems which occur most frequently.
31. Problem I. Let y be a function of the independent variable x, and let t be a new independent variable connected luith x by the relation x = <£(£). It is required to express the successive derivatives of y with respect to x in terms of t and the successive derivatives of y with respect to t.
Let y=f(x) be the given function, and F(t) =/[<£(£)] the func tion obtained by replacing x by <j>(t) in the given function. By the rule for the derivative of a function of a function, we find
dy dy ,. . •37 = ~r~ X 9 m, at ax
whence
dy
dt yt
This result may be stated as follows : To find the derivative of y with respect to x, take the derivative of that function with respect to t and divide it by the derivative of x with respect to t.
The second derivative d2y/dx* may be found by applying this rule to the expression just found for the first derivative. We find :
-I
dLl = ±_ -y^'(0-y^"(0. dx* w) ' [>'(0]'
and another application of the same rule gives the third derivative
62 FUNCTIONAL RELATIONS [H, §32
or, performing the operations indicated,
_
<*»• [>'(OJ6
The remaining derivatives may be calculated in succession by repeated applications of the same rule. In general, the nth deriva tive of y with respect to x may be expressed in terms of <}>'(£), <j>"(t), • ••, <£(n)(£), and the first n successive derivatives of y with respect to t. These formulae may be arranged in more symmetrical form. Denoting the successive differentials of x and y with respect to t by dx, dy, dzx, d*y, ••-, d"x, dny, and the successive derivatives of y with respect to x by y', y", •••, y(n\ we may write the preceding formulae in the form
(20)
y — 7
9 . dx
f _ dx d2y — dy 6?
dx3 x2 - 3 d?y dx dzx + 3dy (d*x)* - dy d*x dx
y ~~ 5
The independent variable t, with respect to which the differentials on the right-hand sides of these formulae are formed, is entirely arbitrary ; and we pass from one derivative to the next by the recurrent formula
, ,
«<»>=
the second member being regarded as the quotient of two differen tials.
32. Applications. These formulas are used in the study of plane curves, when the coordinates of a point of the curve are expressed in terms of an auxiliary variable t.
« =/(*)» y = * co
in order to study this curve in the neighborhood of one of its points it is necessary to calculate the successive derivatives y', ?/", • • • of y with respect to x at the given point. But the preceding formulas give us precisely these derivatives, expressed in terms of the succes sive derivatives of the functions f(t) and <j> (#), without the necessity
II, §32] TRANSFORMATIONS 63
of having recourse to the explicit expression of y as a function of x, which it might be very difficult, practically, to obtain. Thus the first formula
y> = dx = f'(t)
gives the slope of the tangent. The value of y" occurs in an impor tant geometrical concept, the radius of curvature, which is given by the formula
which we shall derive later. In order to find the value of R, when the coordinates x and y are given as functions of a parameter t, we need only replace y' and y" by the preceding expressions, and we find
(dx2 4- dy^Y R = . , — ^r~ "
where the second member contains only the first and second deriva tives of x and y with respect to t.
The following interesting remark is taken from M. Bertrand's Traitt de Calcul differentiel et integral (Vol. I, p. 170). Suppose that, in calculating some geometrical concept allied to a given plane curve whose coordinates x and y are supposed given in terms of a parameter £, we had obtained the expression
F(x, y, dx, dy, d2x, d2y, • • -, dnx, d»y),
where all the differentials are taken with respect to t. Since, by hypothesis, this concept has a geometrical significance, its value cannot depend upon the choice of the independent variable t. But, if we take x = t, we shall have dx — dt, dzx = d3x = • • • = dax = 0, and the preceding expression becomes
f(x, y, y', y", ••••> 2/('°) ;
which is the same as the expression we would have obtained by supposing at the start that the equation of the given curve was solved with respect to y in the form y = *(«). To return from this particular case to the case where the inde pendent variable is arbitrary, we need only replace y', y", • • • by their values from the formulae (20). Performing this substitution in
we should get back to the expression F(x, y, dx, dy, d'2x, d2y, • • •) with which we started. If we do not, we can assert that the result obtained is incorrect. For example, the expression
dxd'2y + dyd2x
64 FUNCTIONAL RELATIONS [II, §33
cannot have any geometrical significance for a plane curve which is independent of the choice of the independent variable. For, if we set x = t, this expression reduces to y" /(I + y'2)$ ; and, replacing y' and y" by their values from (20), we do not get back to the preceding expression.
33. The formulae (20) are also used frequently in the study of differential equations. Suppose, for example, that we wished to determine all the functions y of the independent variable x, which satisfy the equation
(21) (1_^£*_e eg + „.„ = „,
where n is a constant. Let us introduce a new independent variable t, where x = cos t. Then we have
dy dy dt
dx — sin t
d*y dy
smt-jfi.— cost — d?y at* dt <
dx2 sin8 1
and the equation (21) becomes, after the substitution,
(22)
It is easy to find all the functions of t which satisfy this equation, for it may be written, after multiplication by 2 dy /dt,
whence
where a is an arbitrary constant. Consequently
or
— 71 = 0.
II, § 34] TRANSFORMATIONS 65
The left-hand side is the derivative of arc sin (y/a) — nt. It follows that this difference must be another arbitrary constant b, whence
y = a sm(nt + &), which may also be written in the form
y = A sin nt + B cos nt.
Returning to the original variable x, we see that all the functions of x which satisfy the given equation (21) are given by the formula
y = A sin (n arc cos a) + B cos (n arc cos a), where A and B are two arbitrary constants.
34. Problem II. To every relation between x and y there corresponds, by means of the transformation x = f(t, u), y = <f>(t, u*), a relation between t and u. It is required to express the derivatives of y with respect to x in terms of t, u. and the derivatives of u with respect to t.
This problem is seen to depend upon the preceding when it is noticed that the formulae of transformation,
give us the expressions for the original variables x and y as func tions of the* variable t , if we imagine that u has been replaced in these formulas by its value as a function of t. We need merely apply the general method, therefore, always regarding x and y as composite functions of t, and u as an auxiliary function of t. We find then, first,
8<jt 8<ft du
dy _dy dx dt du dt dx dt ' dt df df du dt du dt and then
d?y __ d (dy\ dx dx"2 dt \dxj ' dt
or, performing the operations indicated,
, _ _ , ,
SuBtdt du*\dt) gu dfr* \dt + du dt/\dt*
£t du dt
66 FUNCTIONAL RELATIONS [II, §33
In general, the nth derivative y(n) is expressible in terms of t, u, and the derivatives du/dt, d?vi/dt2, •••, dnu/dtn.
Suppose, for instance, that the equation of a curve be given in polar coordinates p = /(o>). The formulae for the rectangular coor dinates of a point are then the following :
x = p cos <D, y — p sin w.
Let p', p", • • • be the successive derivatives of p with respect to w, considered as the independent variable. From the preceding formulae
we find
dx = cos (a dp — p sin w e?w,
dy = sin o> dp + p cos w d<a, d2x = cosu) dzp — 2 sin w dai dp — p cos w da?, d2y — sinw d2p + 2 cosw rfw ffy — p sin w «7(o2, whence
<&e2 + dif1 = dp2 + p2 rfw2, — dij d^x = 2 du dp2 — p d<a d2p + p2 c?w3.
The expression found above for the radius of curvature becomes
p' + *pm-pp
35. Transformations of plane curves. Let us suppose that to every point m of a plane we make another point M of the same plane cor respond by some known construction. If we denote the coordinates of the point m by (x, y) and those of M by (X, F), there will exist, in general, two relations between these coordinates of the form
(23) X=f(x,y), Y=4>(x, y}.
These formulae define a point transformation of which numerous examples arise in Geometry, such as projective transformations, the transformation of reciprocal radii, etc. When the point m describes a curve c, the corresponding point M describes another curve C, whose properties may be deduced from those of the curve c and from the nature of the transformation employed. Let y', ?/", • • • be the suc cessive derivatives of y with respect to x, and F', F", • • • the succes sive derivatives of F with respect to X. To study the curve C it is necessary to be able to express F', F", ••• in terms of x, y, y', y", ••-. This is precisely the problem which we have just discussed ; and we find
II, § 36]
TRANSFORMATIONS
67
dY
|
Y' |
dx |
dx dy ' |
|
Y" |
dX dx dY' dx |
dx dy |
|
dX dx |
/df df ,V 1 o i — ~ y } \dx dy I |
and so forth. It is seen that Y' depends only on x, y, y'. Hence, if the transformation (23) be applied to two curves c, c', which are tangent at the point (x, ?/), the transformed curves C, C' will also be tangent at the corresponding point (A', F). This remark enables us to replace the curve c by any other curve which is tangent to it in questions which involve only the tangent to the transformed curve C.
Let us consider, for example, the transformation defined by the formulae
Y =
<-— A
x2 + y'2
which is the transformation of reciprocal radii, or inversion, with the origin as pole. Let m be a point of a curve c and M the cor responding point of the curve C. In order to find the tangent to this curve C we need only apply the result of ordinary Geometry, that an inversion carries a straight line into a circle through the pole.
Let us replace the curve c by its tangent mt. The inverse of mt is a circle through the two points Mand O, whose center lies on the perpendicular
Ot let fall from the origin upon mt. The tangent MT to this circle is perpendicular to AM, and the angles Mmt and mMT are equal, since each is the complement of the angle mOt. The tangents mt and MT are therefore antiparallel with respect to the radius vector.
36. Contact transformations. The preceding transformations are not the most general transformations which carry two tangent curves into two other tangent curves. Let us suppose that a point M is determined from each point m of a curve c by a construction
FIG. 5
68 FUNCTIONAL RELATIONS [II, §36
which depends not only upon the point m, but also upon the tangent to the curve c at this point. The formulae which define the trans formation are then of the form
(24) X = /(*, y, y-), Y=<j>(x, y, y1) ;
and the slope Y' of the tangent to the transformed curve is given by the formula
dx dy dy> y
In general, F' depends on the four variables x, y, y', y" ; and if we apply the transformation (24) to two carves c, c' which are tangent at a point (x, y~), the transformed curves C, C' will have a point (X, Y) in common, but they will not be tangent, in general, unless y" happens to have the same value for each of the curves c and c'. In order that the two curves C and C' should always be tangent, it is necessary and sufficient that Y' should not depend on y"; that is, that the two functions f(x, y, y') and <£ (x, y, y') should satisfy the condition
In case this condition is satisfied, the transformation is called a contact transformation. It is clear that a point transformation is a particular case of a contact transformation.*
Let us consider, for example, Legendre's transformation, in which the point M, which corresponds to a point (x, y) of a curve c, is given
by the equations
X = y', Y=xy'-y;
from which we find
Y, _dY _xjf _ ~ dX - y"
which shows that the transformation is a contact transformation. In like manner we find
dY' dx 1
V11' =r
dX y"dx y"
y'
dX
*Legendre and Ampere gave many examples of contact transformations. Sophus Lie developed the general theory in various works ; see in particular his Geometric der Beruhrungstransformationen. See also JACOBI, Vorlesungen iiber Dynamik.
II, §37] TRANSFORMATIONS 69
and so forth. From the preceding formulae it follows that x = Y', y = XY'-Y, y' = X,
which shows that the transformation is involutory.* All these prop erties are explained by the remark that the point whose coordinates are X = y', Y = xy' — y is the pole of the tangent to the curve c at the point (x, y) with respect to the parabola x2 — 2 y = 0. But, in general, if M denote the pole of the tangent at m to a curve c with respect to a directing conic 2, then the locus of the point M is a, curve C whose tangent at M is precisely the polar of the point m with respect to 2. The relation between the two curves c and C is therefore a reciprocal one ; and, further, if we replace the curve c by another curve c', tangent to c at the point m, the reciprocal curve C' will be tangent to the curve C at the point M.
Pedal curves. If, from a fixed point O in the plane of a curve c, a perpen dicular OM be let fall upon the tangent to the curve at the point m, the locus of the foot M of this perpendicular is a curve (7, which is called the pedal of the given curve. It would be easy to obtain, by a direct calculation, the coordinates of the point Jlf, and to show that the trans formation thus defined is a contact transfor mation, but it is simpler to proceed as follows. Let us consider a circle 7 of radius E, de scribed about the point 0 as center; and let ?MI be a point on OM such that Om\ x OM= E2. The point mi is the pole of the tangent mt with respect to the circle ; and hence the transformation which carries c into C is the result of a transformation of reciprocal po- lars, followed by an inversion. When the point m describes the curve c, the point mi, the pole of mt, describes a curve Ci tangent to the polar of the point m with respect to
the circle 7, that is, tangent to the straight line miti, a perpendicular let fall from mi upon Om. The tangent M Tto the curve C and the tangent m\ti to the curve Ci make equal angles with the radius vector OmiM. Hence, if we draw the normal MA, the angles AMO and AOM are equal, since they are the comple ments of equal angles, and the point A is the middle point of the line Om. It follows that the normal to the pedal is found by joining the point Mto the center of the line Om.
37. Projective transformations. Every function y which satisfies the equation y" = 0 is a linear function of x, and conversely. But, if we subject x and y to the projective transformation
* That is, two successive applications of the transformation lead us back to the original coordinates. — TRANS.
70 FUNCTIONAL RELATIONS [II, §38
_ aX + bY+c _ a'X+ b' Y + c'
a" X + b" Y + c" a" X -f b" Y + c"
a straight line goes over into a straight line. Hence the equation y" = 0 should become d'^Y/dX"2 — 0. In order to verify this we will first remark that the general projective transformation may be resolved into a sequence of particular transformations of simple form. If the two coefficients a" and b" are not both zero, we will set X\ = a" X -\- b" Y + c" ; and since we cannot have at the same time ab" — ba" = 0 and a' b" — b' a" = 0, we will also set YI — a' X + b' Y + c', on the supposition that a' b" — b' a" is not zero. The preceding formulae may then be written, replacing X and Y by their values in terms of Xi and Fl5 in the form
YI a Xi + /3 FI + 7 YI 7
A! Xi AI Xi
It follows that the general projective transformation can be reduced to a succession of integral transformations of the form
x — aX + bY + c, y - a' X + b' Y + c', combined with the particular transformation
1 Y
x = — , y = — .
X X
Performing this latter transformation, we find
and
dy _-.-_
~ dx ~ '~~
y" = ~y~ = - XY"(- X*) = Xs Y". dx
Likewise, performing an integral projective transformation, we have d y a' + b' Y'
y -
a + bY'
„ _ dt/ _ (ab'-ba')Y" ~ dx (a + bY')3
In each case the equation y" — 0 goes over into Y" = 0.
We shall now consider functions of several independent variables, and, for definiteness, we shall give the argument for a function of two variables.
38. Problem III. Let w = f(x, y) be a function of the two independ ent variables x and y, and let u and v be two new variables connected with the old ones by the relations
It is required to express the partial derivatives of u with respect to the variables x and y in terms of u, v, and the partial derivatives of u> with respect to u and v.
II,
TRANSFORMATIONS 71
Let w = F(u, v) be the function which results from/(x, y) by the substitution. Then the rule for the differentiation of composite
functions gives
c oj 8 a) 8 <J> d o» d\ff
cu dx cu dy du
Cd) C w C (jt d ta C\}/
dv dx cv dy dv
whence we may find d<a/dx and du/dy; for, if the determinant D(<f>, \ji)/D(u, v) vanished, the change of variables performed would have no meaning. Hence we obtain the equations
(25)
dw d(a d d>
o~ = A a — r B ~z~' ex cu cv
Cu) __ Cd) Coi
— — — c -^ + u ~^~~>
cy cu cv
where A, B, C, D are determinate functions of u and v ; and these formulae solve the problem for derivatives of the first order. They show that the derivative of a function with respect to x is the sum of the two products formed by multiplying the two derivatives with respect to u and v by A and B, respectively. The derivative with respect to y is obtained in like manner, using C and D instead of A and B, respectively. In order to calculate the second derivatives we need only apply to the first derivatives the rule expressed by the preced ing formulae ; doing so, we find
£2<o d /8u>\ d I t d& i 575 ==2~\'"-~/==2\p — r •» •
C/X Q<Kf \ t/JC / GOT/ \ C/it
\ / \
(31 C o> v to \ .- w . _ _ _
= A — (A - + B „- + B y- (A r- + B -5- cu\ Cu cv / dv\ cu
or, performing the operations indicated,
^= A[A-z£+Br-%:+-^-- + -z
^-^2 v ^"^ CU CV CU CU CU CV
-B t-^ +
« My P cy </« 0v cv
and we could find 82(a/dxdy, 32<a/di/2 and the following derivatives in like manner. In all differentiations which are to be carried out we need only replace the operations d /dx and d /dy by the operations
'd 8 d d
A-Z-+B-Z-* C— + D^-,
du Cv cu cv
72
FUNCTIONAL RELATIONS
[II, §38
respectively. Hence everything depends upon the calculation of the coefficients A, B, C, D.
Example I. Let us consider the equation
(26)
C CO , 0 (i3 dJ
a — +26 -- + c — = 0,
ex ci/ cy*
where the coefficients a, 6, c are constants ; and let us try to reduce this equa tion to as simple a form as possible. We observe first that if a = c = 0, it would be superfluous to try to simplify the equation. We may then suppose that c, for example, does not vanish. Let us take two new independent variables u and v, defined by the equations
u = x + ay, v = x + py, where a and /3 are constants. Then we have
c u < <jj cu
cx
d<a
-
8y
du I w
a— + P — ,
du cv
and hence, in this case, A = B = 1, C = a, D = p. The general formulae then give
dx?
~dii? cucv "ai?2"'
ducv
eu2
and the given equation becomes
(a + 26a + ca2)^ + 2 [a + b(a + ft) + ca/3]-^- + (a -(-
au-^ au a» CB
It remains to distinguish several cases.
First case. Let 62 — ac> 0. Taking for a and £ the two roots of the equation a .)- 2 6r + cr2 = 0, the given equation takes the simple form
cudv
= 0.
Since this may be written
we see that dta/Su must be a function of the single variable, w, say/(w). Let F(u) denote a function of u such that F'(u) =f(u). Then, since the derivative of w — F(u) with respect to u is zero, this difference must be independent of w, and, accordingly, u = F(u) + *(«). The converse is apparent. Returning to the variables x and y, it follows that all the functions w which satisfy the equation (26) are of the form
II, §38] TRANSFORMATIONS 73
where F and $ are arbitrary functions. For example, the general integral of
the equation
c2w Oc2w
— = a2 — , cy2 dx2
which occurs in the theory of the stretched string, is w =f(x + ay) + <f> (x - ay).
Second case. Let b'1 — ac = 0. Taking a equal to the double root of the equa tion a -f 26r -f cr2 = 0, and |3 some other number, the coefficient of d^w/dudv becomes zero, for it is equal to a + ba + p(b + car). Hence the given equation reduces to 52w/cz>2 = 0. It is evident that w must be a linear function of t>, w = vf(u) + <f> (u), where f(u) and <f> (u) are arbitrary functions. Returning to the variables x and y, the expression for w becomes
w = (x + Py)f(x + ay) + <f>(x + ay), which may be written
w = [x + ay + (p - a)y]f(x + ay) + <f>(x + ay), or. finally,
w = yF(x + ay) + <l>(x + ay).
Third case. If 62 — ac < 0, the preceding transformation cannot be applied without the introduction of imaginary variables. The quantities a and /3 may then be determined by the equations
a + 26a + c a2 = a + b(a + p)
which give
2b 2&2-ac
a + /3= , a/3=
c c2
The equation of the second degree,
26 262-ac
r2 H r H = 0,
c c2
whose roots are a and £, has, in fact, real roots. The given equation then
becomes
a2w c2w
Aw = H = 0.
du2 c«2
This equation Aw = 0, which is known as Laplace's Equation, is of fundamental importance in many branches of mathematics and mathematical physics.
Example II. Let us see what form the preceding equation assumes when we set x = p cos <£, y = p sin 0. For the first derivatives we find
8 u du Su
— = — cos^ H
8p dx dy
i u ("ui d u
— = p sin <b -\ p cos <z>,
p *
74 FUNCTIONAL RELATIONS [H,§a9
or, solving for dw/dx and du/cy,
du du sin rf> du
- — COS <p i
dx dp p dip
du du cosrf) du
- =sm0— +
dy dp p dip
Hence
a I du sin0 du\ sind> 8 t — ( costf> --- - — ) -- — ( dp\ dp p 8<f>/ p d<t>\
du simp du
--- - dp p
d'2u sin20a2w 2 sin </> cos 0 S2w 2sin0cos<£dw sin20cw
= COS2 <t> -- 1 ------- 1 ---- 1 --- i
a/?2 p2 dtf>2 p dp dip p2 d<f> p dp
and the expression for d2u/dy2 is analogous to this. Adding the two, we find
39. Another method. The preceding method is the most practical when the function whose partial derivatives are sought is unknown. But in certain cases it is more advantageous to use the following method.
Let z =f(x, y) be a function of the two independent variables x and y. If x, y, and z are supposed expressed in terms of two aux iliary variables u and v, the total differentials dx, dy, dz satisfy the relation
O /> o /•
dz = -^- dx + -£- dy. ex cy '
which is equivalent to the two distinct equations
_ du dx du dy du
dz _d_fdx
dv dx dv dy dv
whence df/dx and df/dy may be found as functions of u, v, dz/du, dz/dv, as in the preceding method. But to find the succeeding derivatives we will continue to apply the same rule. Thus, to find d2f/dx2 and d2f/dxdy, we start with the identity
>
dx2 dxcy
which is equivalent to the two equations
d(dx) = d2fdx | ay dy}
du dx2 du dx dy du
II, §39] TRANSFORMATIONS 75
: | }
dec2 ov dx dy dv
where it is supposed that df/dx has been replaced by its value cal culated above. Likewise, we should find the values of d2f/dx dy and 2 by starting with the identity
df\ a2/ 82f
a I — Q ?T *** ' ~cT~a *%• dy/ dxdy dy2
The work may be checked by the fact that the two values of c2f/8x dy found must agree. Derivatives of higher order may be calculated in like manner.
Application to surfaces. The preceding method is used in the study of surfaces. Suppose that the coordinates of a point of a surface S are given as functions of two variable parameters u and v by means of the formulae
(27) x=f(u,v), y = $(u,v), z = f(u,v).
The equation of the surface may be found by eliminating the vari ables u and v between the three equations (27); but we may also study the properties of the surface S directly from these equations themselves, without carrying out the elimination, which might be practically impossible. It should be noticed that the three Jacobians
D(«, -y) D(u, v)
cannot all vanish identically, for then the elimination of w and v would lead to two distinct relations between x, y, z, and the point whose coordinates are (x, y, %) would map out a curve, and not a sur face. Let us suppose, for definiteness, that the first of these does not vanish : D(f, <j>)/D(u, v) =£ 0. Then the first two of equations (27) may be solved for u and v, and the substitution of these values in the third would give the equation of the surface in the form z = F(x, y). In order to study this surface in the neighborhood of a point we need to know the partial derivatives p, q, r,s,t, ••• of this function F(x, y) in terms of the parameters u and v. The first derivatives p and q are given by the equation
dz = p dx -f- q dy, which is equivalent to the two equations
76 FUNCTIONAL RELATIONS [II, §40
^ - n^4-n^
TT— = p T; -- r q r~
du f du du
Q a
cv cv cv
from which p and q may be found. The equation of the tangent plane is found by substituting these values of p and q in the equation
Z - z = p(X - *) + q(Y - y), and doing so we find the equation
The equations (28) have a geometrical meaning which is easily remembered. They express the fact that the tangent plane to the surface contains the tangents to those two curves on the surface which are obtained by keeping v constant while u varies, and vice versa*
Having found p and q, p =/i(w, v), q = f2(u, v), we may proceed to find r, s, t by means of the equations
dp = r dx + s dy, \
dq = s dx + t dy,
each of which is equivalent to two equations ; and so forth.
40. Problem IV. To every relation between x, y, z there corresponds by means of the equations
(30) x =/(w, v, w), y = <£ (M, v, w), z = \j/(u, v, w),
a new relation between u, v, w. It is required to express the partial derivatives of z with respect to the variables x and y in terms of u, v, w, and the partial derivatives of iv with respect to the variables u and v.
This problem can be made to depend upon the preceding. For, if we suppose that w has been replaced in the formulae (30) by a function of u and v, we have x, y, z expressed as functions of the
* The equation of the tangent plane may also be found directly. Every curve on the surface is defined by a relation between u and w, say v = U (u) ; and the equations of the tangent to this curve are
X-x Y-y Z-z
df df ~ dd> d4> ~ d4> d&
f- + -f n'(«) ^ + ~ n'(«) -^- + -— H'(M)
du dv du dv du dv
The elimination of IT(a) leads to the equation (29) of the tangent plane.
II, §41] TRANSFORMATIONS 77
two parameters u and v; and we need only follow the preceding method, considering /, <£, ^ as composite functions of u and v, and w as an auxiliary function of u and v. In order to calculate the first derivatives p and y, for instance, we have the two equations
_ , ,
P a Q — T5 I cT~ i T~ ~o — 1+7 "^ -- P ^ -- o
* cu ow 8u ' du dw cu
_ a^ a_w
dv dw dv \d» 8w dv \8v + 8w
The succeeding derivatives may be calculated in a similar manner.
In geometrical language the above problem may be stated as fol lows : To every point m of space, whose coordinates are (x, y, z), there corresponds, by a given construction, another point M, whose coordinates are X, Y, Z. When the point m maps out a surface S, the point M maps out another surface 2, whose properties it is pro posed to deduce from those of the given surface S.
The formulae which define the transformation are of the form
x =f(x> y> «), y = <t> (*, y, *),
Let
Y)
be the equations of the two surfaces S and 2, respectively. The problem is to express the partial derivatives P, Q, R, S,T, ••• of the function $(A", Y) in terms of x, y, z and the partial derivatives p, q, r, s, t, • • • of the function F(x, y). But this is precisely the above problem, except for the notation.
The first derivatives P and Q depend only on x, y, z, p, q ; and hence the transformation carries tangent surfaces into tangent sur faces. But this is not the most general transformation which enjoys this property, as we shall see in the following example.
41. Legendre's transformation. Let z =f(x, y) be the equation of a surface S, and let any point m (x, ?/, z) of this surface be carried into a point M, whose coordinates are X, Y, Z, by the transformation
X=p, Y = q, Z — px + qy — z.
Let Z = $ (X, Y) be the equation of the surface 2 described by the point M. If we imagine z, p, q replaced by /, df/dx, df/dy, respec tively, we have the three coordinates of the point M expressed as functions of the two independent variables x and y.
78 FUNCTIONAL RELATIONS [II, §41
Let P, Q, R, S, T denote the partial derivatives of the function $>(X, Y). Then the relation
dZ = PdX+ QdY becomes
p dx + q dy + x dp + y dq — dz = P dp + Q, dq, or
x dp + y dq = P dp -f Q dq.
Let us suppose that p and q, for the surface S, are not functions of each other, in which case there exists no identity of the form \dp + p.dq=. 0, unless X = fi — 0. Then, from the preceding equation, it follows that
In order to find R, S, T we may start with the analogous relations
dP = RdX + SdY, dQ = SdX+ TdY,
which, when X, Y, P, Q are replaced by their values, become
dx = R (r dx + s dy) + S (s dx + t dy) , dy = S (r dx + s dy~) + T(sdx + t dy} ; whence
and consequently
t — s r
rt — s2 rt — s2 rt — s2
From the preceding formulae we find, conversely,
x = P, y=Q, z = PX+QY-Z, p = X, q = Y,
T - S R
t —
RT- ^ RT- .S'2 RT - S'2
which proves that the transformation is involutory. Moreover, it is a contact transformation, since X, Y, Z, P, Q depend only on x, y, z, p, q. These properties become self-explanatory, if we notice that the formulae define a transformation of reciprocal polars with respect to the paraboloid
x2 + y2 - 2 z = 0.
Note. The expressions for R, S, T become infinite, if the relation rt — s2 = 0 holds at every point of the surface S. In this case the point M describes a curve, and not a surface, for we have
II, §42] TRANSFORMATIONS 79
= ,, = 0
*,y) D*,y
and likewise
D(X, Z) = Jfo ;>* + gy - *) = _ ^ = D(*,y) £(*, ?/)
This is precisely the case which we had not considered.
42. Ampfere's transformation. Retaining the notation of the preceding article, let us consider the transformation
X — x, Y = q, Z — qy — z. The relation
dZ = PdX + QdY becomes
qdy + ydq - dz = Pdx + Qdq, or
y dq — p dx = Pdx + Qdq. Hence
P=-P, Q = y ;
and conversely we find
x = X, y=Q, z=QY-Z, p = - P, q = Y.
It follows that this transformation also is an involutory contact transformation.
The relation
dP = EdX+ SdY next becomes
— r dx — s dy = R dx + S (s dx + t dy) ; that is,
R -f Ss = - r, St = - s,
whence
Starting with the relation dQ = SdX + TdY, we find, in like manner,
r=l. t
As an application of these formulae, let vis try to find all the functions /(x, y) which satisfy the equation rt — s2 = 0. Let S be the surface represented by the equation z =f(x, y), S the transformed surface, and Z = 4>(X, Y) the equation of S. From the formulae for R it is clear that we must have
R - ^ - 0 ~ S~ ~ °'
and * must be a linear function of X :
where 0 and ^ are arbitrary functions of F. It follows that
80 FUNCTIONAL RELATIONS [II, §43
and, conversely, the coordinates (a;, y, z) of a point of the surface S are given as functions of the two variables X and Y by the formulae
x = X, y = Xt'(Y) + y(Y), z = Y[X<t>'(Y) + f'(Y)] - X<f,(Y) - t(Y).
The equation of the surface may be obtained by eliminating X and Y ; or, what amounts to the same thing, by eliminating a between the equations
z = ay -x<t>(a)-t (a), 0 = y — x 0'(a) — ^'(a).
The first of these equations represents a moving plane which depends upon the parameter a, while the second is found by differentiating the first with respect to this parameter. The surfaces defined by the two equations are the so-called developable surfaces, which we shall study later.
43. The potential equation in curvilinear coordinates. The calculation to which a change of variable leads may be simplified in very many cases by various devices. We shall take as an example the potential equation in orthogonal curvilinear coordinates.* Let
F (x, y, z) - p,
FI(X, y, Z)=PI,
F*(X, y, «)=P2i
be the equations of three families of surfaces which form a triply orthogonal system, such that any two surfaces belonging to two different families intersect at right angles. Solving these equations for x, y, z as functions of the parame ters p, pi, PQ, ^6 obtain equations of the form
fx-<t>(p, PI, pa),
(31) j y = <Pi(p, Pi, Pa),
l*= 02 (p, Pi, pa);
and we may take p, pt , p^ as a system of orthogonal curvilinear coordinates.
Since the three given surfaces are orthogonal, the taagents to their curves of intersection must form a trirectangular trihedron. It follows that the equations
must be satisfied where the symbol x> indicates that we are to replace 0 by <£i, then by 02, and add. These conditions for orthogonalism may be written in the following form, which is equivalent to the above :
|
dp dpi 8p dpi dp dpi _ |
0, |
|
|
(33) |
£x dx By dy dz cz dp dpz , n dpi Spy — -H • • • — \j. — — -+- . . |
V7 • = 0. |
|
. Bx dx dx dx |
* Lame", TraiU des coordonnees curvilignes. See also Bertrand, Traitt de Calcul differ entiel, Vol. I, p. 181.
H, § 43] TRANSFORMATIONS 81
Let us then see what form the potential equation
ax2 dy* az2 assumes in the variables p, p1? p2. First of all, we find
dv _ dv dp 8V dpi aF apa
ax dp dx dpi dx dpz dx ' and then
a2F_ a^F/aA2 a2F ap a^. aF av
ax2 ~ a/»2 \ax/ apapi ax "aaT "a7 ax2
| 2 a2F aPl aP2 |
p^ \dx/ dpidpz ax ax api ax2
,
-- \ax/ a/>ap2 ax ax apa ax2
Adding the three analogous equations, the terms containing derivatives of the second order like a2 V / dp dpi fall out, by reason of the relations (33), and we have
a2F . . , v a2F
(34)
A2(p) ^ + A2(P1) ~ + A2(p2) pT, op cpi apa
where AI and A2 denote Lam&s differential parameters :
The differential parameters of the first order Ai(p), Ai(pi), Aj(p2) are easily calculated. From the equations (31) we have
a^ ap a^ api aj^ apg _
ap ax api ax ap2 ax
a 01 ap a 01 api a 01 ap2 _
ap ax api ax apa ax
a02 a_p a02 api a02 ap2 _ . _
ap ax api ax ap2 ax
whence, multiplying by — , — -, — -, respectively, and adding, we find dp dp dp
a0
ap _
ax ~
/a0\2 /a0A2 /a02 \dp) +\dp/ +\dpj
Then, calculating dp/dy and dp/dz in like manner, it is easy to see that i2 /ap\2 1
«*/ \dy/ \dz/ (d<f>\ , ..„. +
82 FUNCTIONAL RELATIONS [II, §43
Let us now set
a =
dp
where the symbol $ indicates, as before, that we are to replace <f> by 0i, then by 02) and add. Then the preceding equation and the two analogous equations may be written
A!(P) = — , Ai(pi) = — , A!(p2) = — • H HI HZ
Lame* obtained the expressions for A2(p), A2(pi), A2(p2) as functions of p, pi, pa by a rather long calculation, which we may condense in the following form. In the identity (34)
1 a2F 1 &V , 1 a2F , .8V , .8V , . , ,aF
A2 F = -—- + — —T + — -^ + A2 (p) •— + A2 (pi) — + A2(p2) — i H dpz HI dpi H2 epjj dp dpi dpz
let us set successively V — x, V = y, V = z. This gives the three equations
i a20 i 52^ i c>20 a0
2 -- =0,
dp* HI cpj HZ cpz dp
. , • — -TT + W VF + A2 (p) iF + **(pl) -^ + ^(pz)1-— = 0, H Cfr HI Cpj .a 2 cp2 op cpi cpz
which we need only solve for Aa(p), A2(pi), A2(p2). For instance, multiplying by d<p/dp, d<j>i/dp, dfa/dp, respectively, and adding, we find
Moreover, we have
oe>4> 620 _ 1 ag *-> ~d~p Up* ~ 2 lp~'
and differentiating the first of equations (32) with respect to pi, we find
~dp ~dp\ ~~ V dpi dp dpi ~ 2 In like manner we have
o£0 ev _ _ i a g2
^ ap" "ap| ~ 2 ap
and consequently
A2(p)= - -+ --
2HHi dp 2HH* dp 2H£p_ °\H1H2
H. EXS.] EXERCISES 83
Setting
*=i, zr,= », *,= £,
this formula becomes
A2(p) = ^
and in like manner we find
A2 (p!) = h\ A (log A.) , A2 (p2) = A" A (log A api \ hhzj cpz \ fifii
Hence the formula (34) finally becomes
(35)
^ ' ^ * ll*z
ax2 a«2 az2
ra2F , a /. A \aF~l
-- h — I log - I — I
L V <>P \ *!*§/ ^/» J
*£
a
or, in condensed form,
. Fa / h dV\ , a / ftj aF\ a / &2 dF\~| 2 * 2 1 — \ / — \ »nr — / — I ^T^ s — / I '
Let us apply this formula to polar coordinates. The formulae of transforma tion are
x = p sin0cos</>, y
where 0 and <f replace pi and p2, and the coefficients ft, fti, hq have the following values :
& = 1 hi = -» h% = — : — -• p p sin 0
Hence the general formula becomes
i Fa / aF\ a / . aF\ a / i aF\~|
A2F= _ - _ — Ip2sin(?— - J + — (sin^ — ) + — (- -J h
P2sine \_dp \ dp] de\ c0J a0 \sin^ a^ / J
or, expanding,
a2F i a2F i a2F 2 aF cot0aF
Ao F = -- 1 ---- h - -- h --- 1 --- '
ap2 p2 a*?2 p2sin20 a^>2 p ap p2 a<?
which is susceptible of direct verification.
EXERCISES
1. Setting u = x2 + y2 + z2, v = x + y + z, w = xy + yz + zx, the functional determinant D (u, t>, w>) /D («, y, 2) vanishes identically. Find the relation which exists between M, v, w.
Generalize the problem.
84 FUNCTIONAL RELATIONS pi, EXS.
2. Let
«i = —==. 1 =» •••, un -
Vi _ -r2 — a-2
1 xl — *n
Derive the equation
i, W2, • • •, M,,) . • 1
3. Using the notation
Xi = COS 0i,
X2 = sin 0i cos 02 ,
x3 = sin 0i sin 02 cos03,
Xn = sin 0i sin 02 • • • sin0n_1 cos0n, show that
(Xl, Xg, •••, Xn) _ ^_ 1)nsinn^lSJnn-l^2Smn-2^g. . . 8in2 0n _ ! sin 0n .
4. Prove directly that the function z = F(x, y} defined by the two equations
z = ax+ yf(a) + 0(or), 0 = x + yf'(a) + 0'(a),
where a. is an auxiliary variable, satisfies the equation rt — s2 = 0, where /(a) and 0 (a) are arbitrary functions.
5. Show in like manner that any implicit function z = F(x, y) defined by an equation of the form
where 0 (z) and ^ (z) are arbitrary functions, satisfies the equation rg2 - 2pqs + tp2 = 0.
6. Prove that the function z = F(x, y) defined by the two equations
z 0'(a) = [y - 0 (a)] 2, (x + a) 0'(a) = y - 0 (a),
where a is an auxiliary variable and 0 (a) an arbitrary function, satisfies the equation pq = z.
7. Prove that the function z — F(x, y) defined by the two equations
[Z - 0 («)] 2 = Z2 (y2 _ a2), [z _ 0 ((r)] 0'(Q,) = aa.2
satisfies in like manner the equation pq = xy.
8*. Lagrange's formulae. Let y be an implicit function of the two variables x and a, defined by the relation y = a + x<j>(y); and let u =f(y) be any func tion of y whatever. Show that, in general,
[LAPLACE.]
II, EM.] EXERCISES 85
Note. The proof is based upon the two formulas
d F, x du~\ d F 7 \dM~l du ±i \cu
— F(u)~ = — F(u) — . — — <(,(y) — ,
da |_ dx J dx ]_ da_\ dx da
where u is any function of y whatever, and F(u) is an arbitrary function of u. It is shown that if the formula holds for any value of n, it must hold for the value n + I.
Setting x = 0, y reduces to a and M to /(a); and the nth derivative of u with respect to x becomes
ff/*\ ^dxn
9. If x =f(u, v), y = (J>(u, v) are two functions which satisfy the equations
dj d 0 dj d <j>
du dv dv du
show that the following equation is satisfied identically :
\ -!« i r~ ~i
M \ S" — • ,., I
:»/o da»-l\_ J
,
10. If the function F(x, y, 2) satisfies the equation
show that the function
satisfies the same equation, where A; is a constant and r2 = x2 + yz + z2.
[LORD KELVIN.]
11. If V(x, y, z) and Vi(x, y, z) are two solutions of the equation A^V = 0, show that the function
U = F(z, y, z) + (x2 + y2 + z2) FI (x, y, z) satisfies the equation
12. What form does the equation
(x - x8)7/"+ (1 - Sx2)?/'- xy = 0 assume when we make the transformation x = Vl — t- ?
13. What form does the equation
2
= 0
Sx2 dx dy
assume when we make the transformation x = u», y = l/v?
14*. Let 0(xi, x2, • • •, xn; MI, MS, • • •, un) be a function of the 2 n independent variables Xi, x2, • • •, xn, MI, u2, • • •, Mn, homogeneous and of the second degree with respect to the variables MI, M2, •••,«„. If we set
86 FUNCTIONAL RELATIONS [II, Exs.
S 0 C(t> cd>
-=pi, -=pz, •••, ~^-=pn,
CUi CUz CUn
and then take p\ , pz , • • • , pn as independent variables in the place of Ui , uz , • • • , un, the function <f> goes over into a function of the form
d<f>
i, X2,
Derive the formulae :
15. Let N be the point of intersection of a fixed plane P with the normal MN erected at any point M of a given surface S. Lay off on the perpendicular to the plane P at the point N a length Nm = NM. Find the tangent plane to the surface described by the point m, as M describes the surface S.
The preceding transformation is a contact transformation. Study the inverse transformation.
16. Starting from each point of a given surface S, lay off on the normal to the surface a constant length I. Find the tangent plane to the surface 2 (the parallel surface) which is the locus of the end points.
Solve the analogous problem for a plane curve.
17*. Given a surface S and a fixed point O ; join the point 0 to any point M of the surface S, and pass a plane OM N through OM and the normal MN to the surface S at the point M. In this plane OMN draw through the point O a per pendicular to the line OM, and lay off on it a length OP = OM. The point P describes a surface 2, which is called the apsidal surface to the given surface S. Find the tangent plane to this surface.
The transformation is a contact transformation, and the relation between the surfaces <S and 2 is a reciprocal one. When the given surface S is an ellipsoid and the point 0 is its center, the surface 2 is Fresnel's wave surface.
18*. Halphen's differential invariants. Show that the differential equation
dx2/ da* dx2 dx» dx* \dx*
remains unchanged when the variables x, y undergo any projective transfor mation (§ 37).
19. If in the expression Pdx + Qdy + fidz, where P, Q, R are any functions of x, y, z, we set
x=/(u, B, 10), y = <f> (u, v, w) , Z = ^(M, v, «>),
where «, v, w are new variables, it goes over into an expression of the form Pidu + Q\dv + Ridw,
where PI, Qi, RI are functions of «, v, w. Show that the following equation is satisfied identically:
gi = -P(Si ^ z)g> I) (M, v, w)
II, Exs.] EXERCISES 87
where
/»* _ •»
\dv du
20*. Bilinear covariants. Let 0</ be a linear differential form :
where JTx, X2, • • •, -^n are functions of the n variables xlt x2, • • •, xn. Let us consider the expression
where
and where there are two systems of differentials, d and 5. If we make any transformation
Xi = 4>i(yi, y2, • • •, 2/n), (i = 1, 2, • • ., n),
the expression Qj goes over into an expression of the same form
Q'd= Yldyl + ••• + Yndyn, where FI, F2, • • •, Yn are functions of yi, y2, • • •, yn. Let us also set
and
Show that H = H', identically, provided that we replace dx, and dxt, respec tively, by the expressions
— Syi -\ 5j/2 + • • • H Syn .
The expression // is called a bilinear covariant of Qj.
21*. Beltrami's differential parameters. If in a given expression of the form Edx* + 2Fdxdy + Gdy*,
where E, F, G are functions of the variables x and y, we make a transformation z =/(M, v), y = <f>(u, «), we obtain an expression of the same form:
EI du2 + 2Fidudv+ Gj dv2,
88
FUNCTIONAL RELATIONS
[II, Exs.
where EI, FI, G\ are functions of u and v. Let 0(x, y) be any function of the variables x and y, and 0i(u, v) the transformed function. Then we have, iden tically,
ax
- 2 F + *
ex dy \dy
du dv
dv
|
EG- F2 |
El Gl - F? |
||||||
|
IG ™ _ F —\ |
(E — — F — |
||||||
|
i |
8 |
dx |
gyl | |
1 |
dy |
ax |
|
|
VF.G - F2 |
dx \ |
^EG- |
"F2 / |
^EG - F2 |
^EG- |
F"2 |
|
|
i |
'-( |
J1au |
?i-~\ 4- |
1 |
d |
^~&o~ |
1 du |
ffj G! - Ff
- F
I
22. Schwarzian. Setting y — (ax + b) / (ex +<8), where x is a function of t and a, 6, c, d are arbitrary constants, show that the relation
y' 2 yy
is identically satisfied, where x', x", x'", y', T/", y'" denote the derivatives with respect to the variable t.
23*. Let u and v be any two functions of the two independent variables x and y, and let us set
U =
au + bv + c
F =
a'w + 6'» + c'
a" « + 6" u + c" a" w + 6" u + c"
where a, &, c, • • • , c" are constants. Prove the formulae :
c*udv_G*vdu d*U dV _ cPV SU dx2 dx gx2 dx ax2 dx dx* dx
(u, v)
d*u dv d*v du /dv
dx2 dy ex2 dy \dx dxdy dx dxdy/
(u, v)
dV a2 V d U ax2 ~dy ~ ~d& Hy
dx dxdy ex dxdy/
(ff.F)
and the analogous formulae obtained by interchanging x and y, where
du dv du dv r dU dV dV dU
dx dy dy dx dx dy dx dy
[GOURSAT and PAINLEVE, Comptes rendus, 1887.]
CHAPTER III
TAYLOR'S SERIES ELEMENTARY APPLICATIONS MAXIMA AND MINIMA
I. TAYLOR'S SERIES WITH A REMAINDER TAYLOR'S SERIES
44. Taylor's series with a remainder. In elementary texts on the Calculus it is shown that, if f(x) is an integral polynomial of degree n, the following formula holds for all values of a and h :
This development stops of itself, since all the derivatives past the (n + l)th vanish. If we try to apply this formula to a function /(x) which is not a polynomial, the second member contains an infinite number of terms. In order to find the proper value to assign to this development, we will first try to find an expression for the difference
f> li 2 J)n
f(a + h) -f(a) - 2 f(a) - f^ /"(a) ___/W(a),
with the hypotheses that the function /(#), together with its first n derivatives /'(a:), f"(x), • • • , f^n)(x), is continuous when x lies in the interval (a, a -f A), and that f(n\x) itself possesses a derivative /(" + J)(x) in the same interval. The numbers a and a -f h being given, let us set
(2)
where p is any positive integer, and where P is a number which is defined by this equation itself. Let us then consider the auxiliary function
89
90 TAYLOR'S SERIES [in, §44
=f(a + A) -/(») - =f,(x) _ ~
_ O + h ~ *)" /-A _ (a + h-x "
J
1.2- "ii 1.2..-
It is clear from equation (2), which defines the number P, that
and it results from the hypotheses regarding f(x) that the func tion <£(x) possesses a derivative throughout the interval (a, a -f A). Hence, by Rolle's theorem, the equation </>'(#) = 0 must have a root a + Oh which lies in that interval, where 0 is a positive number which lies between zero and unity. The value of <t>'(x), after some easy reductions, turns out to be
The first factor (a -f h — x~)p~l cannot vanish for any value of x other than a + h. Hence we must have
P= hn-p + l(l - 0)"-* + !/(» + J> (a + Ofy, where 0<^<1; whence, substituting this value for P in equation (2), we find
(3) /I where
JL =
0 . 2 • • • n .p
We shall call this formula Taylor's scries with a remainder, and the last term or Rn the remainder. This remainder depends upon the positive integer p, which we have left undetermined. In practice, about the only values which are ever given to p are p = n + 1 and p = 1. Setting p = n + 1, we find the following expression for the remainder, which is due to Lagrange :
setting p — 1, we find
Ill, §44] TAYLOR'S SERIES WITH A REMAINDER 91
an expression for the remainder which is due to Cauchy. It is clear, moreover, that the number 0 will not be the same, in general, in these two special formulae. If we assume further that /(n + 1)(a:) is continuous when x = a, the remainder may be written in the form
where e approaches zero with h.
Let us consider, for definiteness, Lagrange's form. If, in the gen eral formula (3), n be taken equal to 2, 3, 4, • • • , successively, we get a succession of distinct formulae which give closer and closer approximations for f(a -f- A) for small values of h. Thus for n = 1 we find
1 1.2
which shows that the difference
/(* + *) -/(a) -*/*(«)
is an infinitesimal of at least the second order with respect to h, provided that /" is finite near x = a. Likewise, the difference
_/•/ , 7 \ _/»/ \ '*" _/»l / \ *^ /»/// \
f(a + A) — f(a) — - f '(a) — - — - f (a)
«/\ / «/ \ / 1 V \ / 1 O */ \ /
J. JL » *«
is an infinitesimal of the third order ; and, in general, the expression
A a + h) — f(a) — - f'(a) — • • • : f(n)(a) J v \ / ~1 v \ / -w! \ f
is an infinitesimal of order n + 1. But, in order to have an exact idea of the approximation obtained by neglecting R, we need to know an upper limit of this remainder. Let us denote by M* an upper limit of the absolute value of y(" + 1)(#) in the neighborhood of x = a, say in the interval (a — 17, a -f rj). Then we evidently have
Kti< i^r1 M)
provided that | h | < 77.
* That is, 3/>|/(« + J)(z) I when |z — a\<ij. The expression " the upper limit," defined in § 68, must be carefully distinguished from the expression " an upper limit," which is used here to denote a number greater than or equal to the absolute value of the function at any point in a certain interval. In this paragraph and in the next /(» + i)(a;) is supposed to have an upper limit near x — a. — TRANS.
92 TAYLOR'S SERIES [III, §45
45. Application to curves. This result may be interpreted geomet rically. Suppose that we wished to study a curve C, whose equa tion is y =f(x), in the neighborhood of a point A, whose abscissa is a. Let us consider at the same time an auxiliary curve C", whose equation is
A line x = a -f- h, parallel to the axis of y, meets these two curves in two points M and M', which are near A. The difference of their ordinates, by the general formula, is equal to
This difference is an infinitesimal of order not less than n -+- 1 ; and consequently, restricting ourselves to a small interval (a — 77, a + rj), the curve C sensibly coincides with the curve C'. By taking larger and larger values of n we may obtain in this way curves which differ less and less from the given curve C; and this gives us a more and more exact idea of the appearance of the curve near the point A.
Let us first set n = \. Then the curve C" is the tangent to the curve C at the point A :
and the difference between the ordinates of the points M and M ' of the curve and its tangent, respectively, which have the same abscissa a -f h, is
Let us suppose that /"(«) =£ 0, which is the case in general. The preceding formula may be written in the form
where c approaches zero with h. Since f"(a) =£ 0, a positive num ber rj can be found such that e | < | /"(a) | , when h lies between — rj and + 77. For such values of h the quantity /"(«) + e will have the same sign as /"(a)> an(i hence y — Y will also have the same sign as /"(a). If /"(a) is positive, the ordinate y of the curve is
Ill, § 46] TAYLOR'S SERIES WITH A REMAINDER 93
greater than the ordinate F of the tangent, whatever the sign of h ; and the curve C lies wholly above the tangent, near the point A. On the other hand, if /"(a) is negative, y is less than Y, and the curve lies entirely below the tangent, near the point of tangency. If f"(a) = 0, let /(p)(«) be the first succeeding derivative which does not vanish for x = a. Then we have, as before, if f(p)(x) is continuous when x = a,
and it can be shown, as above, that in a sufficiently small interval (a — ^ a -f- 17) the difference y — Y has the same sign as the product Ap/(p)(a). When p is even, this difference does not change sign with h, and the curve lies entirely on the same side of the tangent, near the point of tangency. But if p be odd, the difference y — Y changes sign with h, and the curve C crosses its tangent at the point of tangency. In the latter case the point A is called a point of inflection ; it occurs, for example, if f'"(a) =£ 0.
Let us now take n = 2. The curve C' is in this case a parabola :
Y =f(a) + (x
whose axis is parallel to the axis of y\ and the difference of the ordinates is
If /'"(a) does not vanish, y — Y has the same sign as A8/'"(«) for sufficiently small values of h, and the curve C crosses the parabola C' at the point A. This parabola is called the osculatory parabola to the curve C ; for, of the parabolas of the family
Y = mxz + nx + p,
this one comes nearest to coincidence with the curve C near the point A (see § 213).
46. General method of development. The formula (3) affords a method for the development of the infinitesimal f(a + K) ~f(a} according to ascending powers of h. But, still more generally, let x be a principal infinitesimal, which, to avoid any ambiguity, we
94 TAYLOR'S SERIES [III, §46
will suppose positive ; and let y be another infinitesimal of the form
(4) y = AlX»i + A2x»* + ...+x»P (Ap + c),
where nl} nz, •• • , np are ascending positive numbers, not necessarily integers, Al} At, ••-, Ap are constants different from zero, andc is another infinitesimal. The numbers HI, Al, n2, A2, ••• may be cal culated successively by the following process. First of all, it is clear that HI is equal to the order of the infinitesimal y with respect to x, and that Av is equal to the limit of the ratio y/xni when x approaches zero. Next we have
y — A:xn^ = ui = Azx"* -| ---- + (Ap + c) x"p,
which shows that nz is equal to the order of the infinitesimal MI? and A2 to the limit of the ratio u^/xn-i. A continuation of this process gives the succeeding terms. It is then clear that an infini tesimal y does not admit of two essentially different developments of the form (4). If the developments have the same number of terms, they coincide ; while if one of them has p terms and the other p + q terms, the terms of the first occur also in the second. This method applies, in particular, to the development of f(a + A) —f(a) according to powers of h ; and it is not necessary to have obtained the general expression for the successive derivatives of the func tion f(x) in advance. On the contrary, this method furnishes us a practical means of calculating the values of the derivatives
Examples. Let us consider the equation
(5) F(x, y) = Axn + By + xy<S>(x, y) + Cxn + l + • • • + Dy2 + - - - = 0,
where 4> (x, y) is an integral polynomial in x and y, and where the terms not written down consist of two polynomials P(x) and Q(y), which are divisible, respectively, by xn + 1 and y2. The coefficients A and B are each supposed to be different from zero. As x approaches zero there is one and only one root of the equation (5) which ap proaches zero (§ 20). In order to apply Taylor's series with a remainder to this root, we should have to know the successive deriv atives, which could be calculated by means of the general rules. But we may proceed more directly by employing the preceding method. For this purpose we first observe that the principal part
Ill, §46] TAYLOR'S SERIES WITH A REMAINDER 95
of the infinitesimal root is equal to — (.4 /E)xn. For if in the equa tion (5) we make the substitution
y = •
and then divide by xn, we obtain an equation of the same form :
(cc, yi)
which has only one term in ylt namely By^. As x approaches zero the equation (6) possesses an infinitesimal root in ylt and conse quently the infinitesimal root of the equation (5) has the principal part — (A/B)xn, as stated above. Likewise, the principal part of ?/! is — (A±l B)x**] and we may set
where yz is another infinitesimal whose principal part may be found by making the substitution
in the equation (6).
Continuing in this way, we may obtain for this root y an expres sion of the form
(ap + e)x
.n + MJ H 1- n
n
which we may carry out as far as we wish. All the numbers H!, nz, •••, np are indeed positive integers, as they should be, since we are working under conditions where the general formula (3) is applicable. In fact the development thus obtained is precisely the same as that which we should find by applying Taylor's series with a remainder, where a = 0 and h = x.
Let us consider a second example where the exponents are not necessarily positive integers. Let us set
y
96 TAYLOR'S SERIES
[III, §46
where a, ft, y, • • • and fa, yu • • • are two ascending series of positive numbers, and the coefficient A is not zero. It is clear that the prin cipal part of y is Axa, and that we have
tt _ + Cx* + • • • - A xa(Blx^ + Cix* + •••)
y — J]_ 00 — ' — - • — '- • ••— •— y
1 + B^ + dx* H ----
which is an expression of the same form as the original, and whose principal part is simply the term of least degree in the numerator. It is evident that we might go on to find by the same process as many terms of the development as we wished.
Let / (x) be a function which possesses n + 1 successive derivatives. Then replacing a by x in the formula (3), we find
f(x + h) -/(x) + -f(x) + ~f"(x) + ..-+ *" [/00(x) + e] , l . fi 1 . 2 • • • n
where e approaches zero with h. Let us suppose, on the other hand, that we had obtained by any process whatever another expression of the same form for
f(x + h) = /(x) + hfa (x) + A202 (X) + . . . + hn faty + e/] .
These two developments must coincide term by term, and hence the coefficients 0i> 02> • • •, <t>n are equal, save for certain numerical factors, to the successive derivatives of /(x) :
, -_.
1.2 1. 2- • -n
This remark is sometimes