A Course in Mathematical Analysis

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 EXPANSION IN SERIES

DEFINITE INTEGRALS

APPLICATIONS TO GEOMETRY

GINN AND COMPANY BOSTON

ATLANTA

NEW YORK DALLAS

CHICAGO LONDON SAN FKANCISCO

COLUMBUS

STAT.

LIBRARY

ENTERED AT STATIONERS HALL 1

COPYRIGHT,

1904,

BY

EARLE RAYMOND HEDRICK ALL RIGHTS RESERVED PRINTED IN THE UNITED STATES OF AMERICA 426.6

jgregg

GINN AND COMPANY PRIETORS BOSTON

PRO 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

that has to do with functions of real variables, except

all

the theory of differential equations.

being treated in the

"

The

differential notation not

Classe de Mathematiques

treated this notation from the beginning,

speciales,"

* I

have

and have presupposed only

a knowledge of the formal rules for calculating derivatives. Since mathematical analysis

tinuum,

it

logically,

is

essentially the science of the con

would seem that every course

in analysis should begin,

with the study of irrational numbers.

however, that the student

is

theory of incommensurable

well-known works f that a discussion.

As

basis of analysis,

double integral,

I

I

have supposed,

already familiar with that subject.

numbers

is

have thought

for the other

treated in so it useless to

many

The

excellent

enter upon such

fundamental notions which

lie at

the

such as the upper limit, the definite integral, the 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

TRANS. 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 (1900), p. 225. t

functions. iii

7814G2

AUTHOR S PREFACE

iv

supplementary matter which the reader ing without inconvenience.

may omit

Each chapter

is

at the first read

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

and M. Jean

old students at the Ecole Normale,

I take this occasion to tender

JANUARY

M. Emile Cotton

Clairin, have kindly assisted in the correction of proofs

them

my

hearty thanks. E.

27, 1902

GOURSAT

;

TRANSLATOR The

PREFACE

S

was undertaken at the suggestion

translation of this Course

whose review of the original appeared of Professor W. in the July number of the Bulletin of the American Mathematical The lack of standard texts on mathematical sub Society in 1903. F. Osgood,

too well

known

jects in the

English language

I earnestly

hope that this book will help to

felt

is

fill

to require insistence.

the need so generally

throughout the American mathematical world.

conveniently in our in calculus,

It

may

be used

system of instruction as a text for a second course

and as a book of reference

it

will be

found valuable

an American student throughout his work. Few alterations have been made from the French

text.

to

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,

work

in the

of translation.

who

has very kindly interested himself

To him

is

due

all

the additional matter

not to be found in the French text, except the footnotes which are signed,

and even

edited by him.

these,

though not of his

initiative,

I take this opportunity to express

were always

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

make

publishers, Messrs.

Ginn

the typography excellent.

& Company, have spared

no pains to

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

AUGUST, 1904

America.

E R HEDRICK

CONTENTS PAGE

CHAPTER I.

DERIVATIVES AND DIFFERENTIALS

1

Functions of a Single Variable Functions of Several Variables

11

The

19

I.

II.

III.

II.

Differential Notation

IMPLICIT FUNCTIONS. FUNCTIONAL DETERMINANTS.

I.

II.

III.

Implicit Functions

II.

Functional Determinants

52 61

Taylor

s

Series with a Remainder.

Singular Points.

II.

Taylor

Maxima and Minima

DEFINITE INTEGRALS I. Special Methods of Quadrature III.

.

s

Series

.

.

.

.

.

.

Definite Integrals. Allied Geometrical Concepts Change of Variable. Integration by Parts .

Integrals.

Line Integrals

V. INDEFINITE INTEGRALS

III.

VI.

Integration of Rational Functions

Double

.110

.

Methods

of

.

.

.

Evaluation.

Green

III.

Change

192

208

208 226

.236 250

s

250

Area of a Surface Double Integrals. Improper

of Variables.

Generalizations of

175

.196

Theorem II.

140

.166

........

Integrals.

134

.134

..... ....

and Hyperelliptic Integrals Integration of Transcendental Functions Elliptic

DOUBLE INTEGRALS I.

.

89

89

Improper

V. Functions defined by Definite Integrals VI. Approximate Evaluation of Definite Integrals

II.

.

.

...... ....

IV. Generalizations of the Idea of an Integral.

I.

35

35

Transformations

TAYLOR S SERIES. ELEMENTARY APPLICATIONS. MAXIMA AND MINIMA I.

IV.

CHANGE

........ .... ......... ........ ........

OF VARIABLE

III.

1

Surface Integrals

.

.

.

.......

IV. Analytical and Geometrical Applications

.

264

Integrals.

.

.

277 284

CONTENTS

viii

PAGE

CHAPTER VII.

MULTIPLE INTEGRALS.

INTEGRATION OF TOTAL DIFFER 296

ENTIALS I.

II.

VIII.

of Variables

Multiple Integrals. Change Integration of Total Differentials

.

.

INFINITE SERIES I. Series of Real Constant Terms.

296

.

.

.

.

.

.313 327

.

.

General Properties.

327

Tests for Convergence II.

Series of

III. Series of

IX.

POWER I.

II.

Complex Terms. Multiple Series Variable Terms. Uniform Convergence

SERIES.

Power Power

.

TRIGONOMETRIC SERIES

Series of a Single Variable Series in Several Variables

.

350

.

.

360

....

375

..... .

.

.

.

Analytic Curves and Surfaces IV. Trigonometric Series. Miscellaneous Series

III.

.

Implicit Functions.

.

375

.

S94 399

.

.

.411 426

X. PLANE CURVES I.

II.

III.

XI.

Envelopes Curvature

426

Contact of Plane Curves

443

SKEW CURVES I.

II.

433

........ .... ........

Osculating Plane Envelopes of Surfaces

.

.

.

.

.

.

.

Curvature and Torsion of Skew Curves IV. Contact between Skew Curves. Contact between Curves

III.

and Surfaces XII.

II.

III.

Curvature of Curves drawn on a Surface Asymptotic Lines. Conjugate Lines

459 468 486

....

Lines of Curvature

.

.

.

.

.

497

.

.

.

.

.

.514

506

526

IV. Families of Straight Lines

INDEX

453

497

SURFACES I.

453

.

541

CHAPTER

I

DERIVATIVES AND DIFFERENTIALS

FUNCTIONS OF A SINGLE VARIABLE

I.

1.

Limits.

When

the successive values of a variable x approach

nearer and nearer a constant quantity a, in such a way that the a finally becomes and remains absolute value of the difference x less

than any preassigned number, the constant a is called the This definition furnishes a criterion for

limit of the variable x.

determining whether a sary and

is

the limit of the variable

sufficient condition that it

should

x.

The neces

is

be, that, given any no matter how small, the absolute value of x a should remain less than e for all values which the variable x can

positive

number

e,

assume, after a certain instant.

Numerous examples

of limits are to be found in Geometry For example, the limit of the variable quantity x = (a 2 m 2 ) / (a m), as m approaches a, is 2 a for x 2 a will a is taken less than e. Likewise, the be less than e whenever m variable x = a where n is a positive integer, approaches the 1/n, limit a when n increases indefinitely for a x is less than e when

and Algebra.

;

;

ever n

It is apparent from these examples that greater than 1/e. the successive values of the variable x, as it approaches its limit, may is

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 selfevident, enables us, in

many

cases, to establish the existence of a limit.

variable quantity which never decreases, and which ahvays less than a constant quantity L, approaches a limit I, which

Any

remains 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 1

to

L

.

DERIVATIVES AND DIFFERENTIALS

2.

For example,

[I,

2

each of an infinite series of positive terms is 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 2 n of the first n terms evidently increases with n, and this sum is constantly less than the total sum if

less, respectively,

;

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

Let y be another variable, such that to each value of x b). (a 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


h where h

From

is

positive, it follows in the

these two results

same manner that f\x\) ^

evident that/

it is

^) =

0,

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,

Then

b).

m-f(a) = (b-a)f(c-),

(1)

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




y + kf) + kfy (x + ht,

y

-f-

kt)

;

is

hence the pre

be written in the form

We

to a surface.

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. z

(2)

. F(x,

Let

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 Let z be the corresponding value of z, (^o? yo) of the xy plane. and AT (cr 7/0 the corresponding point on the surface S. If ) ,

>

the equations

*=/(*),

(3)

z/

=

=

*

^(9

M

the represent a curve C on the surface S through the point three functions f(f), which we shall suppose continuous <j>(t),

and

differentiable,

must reduce to x y t. The tangent

,

,

of the parameter

value

t

M

given by the equations

is

x Since the curve

must hold

z

C lies on the surface t;

z , respectively, for some to this curve at the point

5)

(

Y

x

for all values of

,

"A(0>

that

S,

is,

*

the equation \j/(t)=F[f(t~),

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 =f(x + ht,y + k) +f(x, y + kt) <j>(t)

gives

;

operations, 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 :

+

where p

+

q

r

AX

=

A

=

AX

A*p A*

1

A^/(z,

and where the increments

y, z),

be either equal or This increment may be expressed in terms of a unequal. partial derivative of order n, being equal to the product hihy

hpki

n,

may

I

lr

kgl\

+

h, k,

+ d,,hp y + eiki + + Oq kq z + ffi li + + Kir), where every 6 lies between 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 x

fx p*z

(x

+

*i Ai

,

=

(X, y, 2)

,

AX

Ah/ Ajt

A**

1

f.

Then, by hypothesis, $(x,y,z)

=

h z ---hp ki-- -k q li--

fxp-i

Ir

if, i

r(x + 0sh2 +

----\-6 ---P hp, y-\

,H----

).

But the nth increment considered is equal to 0(x + hi, y, z) y, z); and if we apply the law of the mean to this increment, we finally obtain the formula sought. (x,

Conversely, the partial derivative fxT^ zr

AX

hp ki k 2

hi h?

as all the increments h, k, It is interesting to

the usual definition.

function of x and y,

no

first

I

approach

kg

---

A/-/ lr

li

zero.

notice that this definition

sometimes more general than + ^(y) is a Then u also has

is

Suppose, for example, that w =/(x, y) where neither nor ^ has a derivative.

(x)

and consequently second derivatives are out

derivative,

the ordinary sense. Nevertheless, tive fxy is the limit of the fraction in

/(x

the limit of the ratio

is

.-AX-.. -AX

-

+

h,

y

+

-/(x +

k)

if

we adopt

h, y)

the

-/(x, y

+

of the question,

new definition, k)

+/(x,

the deriva

y)

hk which

is

equal to h)

+

t( V

+

k)

(x

+

h)

hk

But the numerator of this ratio is identically zero as a limit, and we find/xy = 0.* *

A

similar

zero.

Hence the

remark may be made regarding functions of a = xs cosl/x has the derivative

ratio

approaches

single variable.

example, the f unction /(K)

f and f

(x)

(x)

has no derivative for x

=

3 x 2 cos -

0.

+

But the

xsin-i 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.

For

THE DIFFERENTIAL NOTATION

)14 ]

l

19

THE DIFFERENTIAL NOTATION

III.

which has been in use longer than any it is by no means indispensable, Although other,* it possesses certain advantages of symmetry and of generality which

The

differential notation, is

due to Leibniz.

are convenient, especially in the study of functions of several varia This notation is founded upon the use of infinitesimals. bles.

which approaches zero as small a limit is called an infinitely quantity, or simply an infinitesi the that The condition mal. quantity be variable is essential, for not an infinitesimal unless it is zero. is a constant, however small,

Any

14. Differentials.

variable quantity

which approach zero Ordinarily several quantities are considered standard of compari as the is chosen them of One simultaneously. Let called the principal infinitesimal. Then infinitesimal. another infinitesimal, and ft son,

and

be the principal said to be an

is

is

infinitesimal of higher order with respect to a, if the ratio ft/a On the other hand, ft is called an infini zero with a.

approaches of

tesimal

the

order with respect to a,

first

approaches a limit this case

K

different

^ where

=K+

ft=a(K + c)= Ka

if

a.

Hence

+ at,

Ka

The complementary term is called the principal part of ft. an infinitesimal of higher order with respect to a. In general, such that ft we can find a positive power of a, say

and at

the ratio ft/a In zero.

e,

another infinitesimal with respect to

c is

if

from zero as a approaches

is

a",

zero,

ft

is

K

/a"

from zero as a approaches Then called an infinitesimal of order n with respect to a.

approaches a

finite

limit

different

we have

=K+ 4 a ;

e,

or ft

The term

=

an (K

-f e)

= Ka* +

".

again called the principal part of ft. these definitions, let us consider a continuous func Having given Let Aa; be an tion y=f(x), which possesses a derivative (x). Ka"

is

f

*

With the

possible exception of

Newton

s

notation.

TRANS.

DERIVATIVES AND DIFFERENTIALS

20 increment of

From

x,

and

let A?/

14

[I,

denote the corresponding increment of

the very definition of a derivative,

y.

we have

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

where

c

denoted by dy.

dy=f(x)&x.

When /(x)

reduces to x

itself, the above formula becomes dx and hence we shall write, for symmetry,

= Ax

;

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

functions of x which

may

dependent

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, and whose abscissae are x and x -f dx, In the triangle MTN we have respectively.

M

M

,

NT = MN tan Z TMN = dxf (x). Hence NT represents the differential

while

Ay

is

equal to

NM

.

It

is

evident from the figure that

dy,

MT

an infinitesimal of higher order, in general, with respect to NT, is parallel to the x axis. approaches M, unless Successive differentials may be defined, as were successive deriv atives, each in terms of the preceding. Thus we call the differ is

as

M

MT

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 2 above. It is denoted by d

y:

d*y

= d (dy) =

[/"(x)

Similarly, the third differential

d*y

dx] dx

=

f"(x)

= d(d*y) = [_f(x)dx*]dx

=f"(x)(dx)*,

* Strictly speaking, we should here exclude the case ever, convenient to retain the same definition of dy

even though

it is

(dx}*.

is

not the principal part of Ay.

where f

=f

TRANS.

(x)

= 0.

It is,

(x)&x in this case

how also.

THE DIFFERENTIAL NOTATION

14]

I,

and so (n

1)

The

21

In general, the differential of the differential of order

on. is

/ (or),

derivatives

/"(a),

-,

f

(n

...

\x),

the other hand, in terms of differentials, and tion for the derivatives

can be expressed, on a new nota

we have

:

~ dy

t

y

_ ~

,,

dx

dx

To each of the

M



rules for the calculation of a derivative corresponds

a rule for the calculation of a differential.

= mx m dx d log x = d xm

,

dn y ~dtf

For example, we have

da x =

l

dx,

,

d sin x

j

=

a x log a dx cos

x dx

SC ,

.

aarcsmcc

dx

=

Vl -

darctanx

>

a;

=-

2

1

;

;

;

dx

+x

2

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

yx dx =/(M) X ux dx; that

is,

dy =f(u)du.

The formula

for dy is therefore the same as if u were the inde variable. This is one of the advantages of the differential pendent notation. In the derivative notation there are two distinct formulae,

yx=f(u)uxy

&=/(*)>

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 to represent the derivative of

same formula applies in each case.* If y = f(u, v, w) is a composite function, we have Vx at least

if

fu ,f ,fw v

= U xfu + Vx f + Wx fn v

,

are continuous, or, multiplying

yx dx =

u x dxfu

+

v x dxfv

+ wx dxfw

by dx,

;

* This particular advantage is slight, however for the last formula ahove well a general one and covers both the cases mentioned. TRANS. ;

is

equally

DERIVATIVES AND DIFFERENTIALS

22 that

[I,

15

is,

dll

Thus we have,

= f du + f dv +fw dw. u

v

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),

We

for instance.

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 du, where u (u)

We

and du are the auxiliary functions.

To

thus find

calculate d*y, we must consider d*y as a composite function, with d2 u as auxiliary functions, which leads to the

u, du,

expression

d*y

and so

=f

8

+

"(u)du

+f (u)d*u

3f"(u)dud*u

;

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, d z u, etc.*

A

on.

similar notation

is

of several variables.

f(x, is

y, s),

which

is

used for the partial derivatives of a function

Thus the

f

xf>flzr

n of

in our previous notation,

represented by

____

in the differential notation.f

in

partial derivative of order

represented by

This notation is purely symbolic, and no sense represents a quotient, as it does in the case of functions

of a single variable.

.

Let w =f(x,

15. Total differentials.

three independent variables x, o

du * This disadvantage

=

o

/

dx ^ex

+

y,

z)

be a function of the

The expression

y, z.

Q

/

/

^ dy + dz -^ dz dy -

would seem completely

to offset the

advantage mentioned

TRANS. Strictly speaking, we should distinguish between d^y and d?uy, etc. 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

above.

.

the derivatives of a function of a single variable.

I,

THE DIFFERENTIAL NOTATION

15]

23

where dx, dy, dz are three fixed called the total differential of otherwise which are arbitrary, assigned to the three increments, is

o>,

independent variables

The three products

x, y, z.

8f TT-

df j ~ dz

df j dy,

7

dx.

ex

cz

dy

are called partial differentials. The total differential of the second order ential of

the"

dx, dy, dz remaining the

=

d2 u

same

d*

=

Let w F(u, v, w~) being themselves functions of the The partial derivatives may then be

16. Successive differentials of composite functions.

be a composite function, u, v, independent variables x, y, z, t.

d_F_d_v

_

dt

dt

dz

dt

If these four equations be multiplied by dx, dy, dz, dt, respectively, and added, the left-hand side becomes d(

+

3- dx dx

that

is, do*

;

and the

,