Chapter XI: Act 1890: , the effect of which is explained in the article Insanity. Any (7)

It is easy to prove that this is less than e^-1 when x lies between 0
and 1, and also that f(x) is greater than e^-l when x = 1/[root]3.
Hence the sum of the series (i.) is not equal to the sum of the series
(ii.).

The particular case of Taylor's theorem in which a = 0 is often called
Maclaurin's theorem, because it was first explicitly stated by Colin
Maclaurin in his _Treatise of Fluxions_ (1742). Maclaurin like Taylor
worked exclusively with the fluxional calculus.

Expansions in power series.

Examples of expansions in series had been known for some time. The
series for log (1 + x) was obtained by Nicolaus Mercator (1668) by
expanding (1 + x)^-1 by the method of algebraic division, and
integrating the series term by term. He regarded his result as a
"quadrature of the hyperbola." Newton (1669) obtained the expansion of
sin^-1 x by expanding (l - x²)^-½ by the binomial theorem and
integrating the series term by term. James Gregory (1671) gave the
series for tan^-1 x. Newton also obtained the series for sin x, cos x,
and e^x by reversion of series (1669). The symbol e for the base of
the Napierian logarithms was introduced by Euler (1739). All these
series can be obtained at once by Taylor's theorem. James Gregory
found also the first few terms of the series for tan x and sec x; the
terms of these series may be found successively by Taylor's theorem,
but the numerical coefficient of the general term cannot be obtained
in this way.

Taylor's theorem for the expansion of a function in a power series was
the basis of Lagrange's theory of functions, and it is fundamental
also in the theory of analytic functions of a complex variable as
developed later by Karl Weierstrass. It has also numerous applications
to problems of maxima and minima and to analytical geometry. These
matters are treated in the appropriate articles.

The forms of the coefficients in the series for tan x and sec x can be
expressed most simply in terms of a set of numbers introduced by James
Bernoulli in his treatise on probability entitled _Ars Conjectandi_
(1713). These numbers B1, B2, ... called Bernoulli's numbers, are the
coefficients so denoted in the formula

x x B1 B2 B3
------- = 1 - --- + -- x² - -- x^4 + -- x^6 - ...,
e^x - 1 2 2! 4! 6!

and they are connected with the sums of powers of the reciprocals of
the natural numbers by equations of the type

(2n)! / 1 1 1 \
B_n = ------------------ ( ------ + ------ + ------ + ... ).
2^(2n-1) [pi]^(2n) \ 1^(2n) 2^(2n) 3^(2n) /

The function

m m·m - 1
x^m - --- x^(m-1) + ------- B1 x^(m-2) - ...
2 2!

has been called Bernoulli's function of the mth order by J. L. Raabe
(Crelle's _J. f. Math._ Bd. xlii., 1851). Bernoulli's numbers and
functions are of especial importance in the calculus of finite
differences (see the article by D. Seliwanoff in _Ency. d. math.
Wiss._ Bd. i., E., 1901).

When x is given in terms of y by means of a power series of the form

x = y(C0 + C1y + C2y² + ...) (C0 [not eq.] 0) = y [f]0(y), say,

there arises the problem of expressing y as a power series in x. This
problem is that of _reversion of series_. It can be shown that
provided the absolute value of x is not too great,

__n=[oo] _ _
x \ | x^n d^(n-1) 1 |
y = ------ + ) | --- -------- ----------- |
[f](0) /__n=2 |_ n! dy^(n-1) {[f]0(y)}^n _| y=0

To this problem is reducible that of expanding y in powers of x when x
and y are connected by an equation of the form

y = a + x[f](y),

for which problem Lagrange (1770) obtained the formula

__n=[oo] _ _
\ | x^n d^(n-1) |
y = a + x[f](a) + ) | --- · -------- {[f](a)}^n |.
/__n=2 |_ n! da^(n-1) _|

For the history of the problem and the generalizations of Lagrange's
result reference may be made to O. Stolz, _Grundzüge d. Diff. u. Int.
Rechnung_, T. 2 (Leipzig, 1896).

Indeterminate forms.

38. An important application of the theorem of intermediate value and
its generalization can be made to the problem of evaluating certain
limits. If two functions [phi](x) and [psi](x) both vanish at x = a,
the fraction [phi](x)/[psi](x) may have a finite limit at a. This
limit is described as the limit of an "indeterminate form." Such
indeterminate forms were considered first by de l'Hospital (1696) to
whom the problem of evaluating the limit presented itself in the form
of tracing the curve y = [phi](x)/[psi](x) near the ordinate x = a,
when the curves y = [phi](x) and y = [psi](x) both cross the axis of x
at the same point as this ordinate. In fig. 10 PA and QA represent
short arcs of the curves [phi], [psi], chosen so that P and Q have the
same abscissa. The value of the ordinate of the corresponding point R
of the compound curve is given by the ratio of the ordinates PM, QM.
De l'Hospital treated PM and QM as "infinitesimal," so that the
equations PM : AM =[phi]´(a) and QM : AM = [psi]´(a) could be assumed
to hold, and he arrived at the result that the "true value" of
[phi](a)/[psi](a) is [phi]´(a)/[psi]´(a). It can be proved rigorously
that, if [psi]´(x) does not vanish at x = a, while [phi](a) = 0 and
[psi](a) = 0, then

[phi](x) [phi]´(a)
lim. -------- = ---------.
x=a [psi](x) [psi]´(a)

It can be proved further if that [phi]^m (x) and [psi]^n (x) are the
differential coefficients of lowest order of [phi](x) and [psi](x)
which do not vanish at x = a, and if m = n, then

[phi](x) [phi]^n(a)
lim. -------- = ----------.
x=a [psi](x) [psi]^n(a)

If m > n the limit is zero; but if m < n the function represented by
the quotient [phi](x)/[psi](x) "becomes infinite" at x = a. If the
value of the function at x = a is not assigned by the definition of
the function, the function does not exist at x = a, and the meaning of
the statement that it "becomes infinite" is that it has no finite
limit. The statement does not mean that the function has a value which
we call infinity. There is no such value (see FUNCTION).

Such indeterminate forms as that described above are said to be of the
form 0/0. Other indeterminate forms are presented in the form 0 ×
[oo], or 1^[oo], or [oo]/[oo], or [oo] - [oo]. The most notable of the
forms 1^[oo] is lim.(x=0) (1 + x)^(1/x), which is e. The case in which
[phi](x) and [psi](x) both tend to become infinite at x = a is
reducible to the case in which both the functions tend to become
infinite when x is increased indefinitely. If [phi]´(x) and [psi]´(x)
have determinate finite limits when x is increased indefinitely, while
[phi](x) and [psi](x) are determinately (positively or negatively)
infinite, we have the result expressed by the equation

[phi](x) lim.x=[oo] [psi]´(x)
lim. -------- = --------------------.
x=[oo] [psi](x) lim.x=[oo] [psi](x)

For the meaning of the statement that [phi](x) and [psi](x) are
determinately infinite reference may be made to the article FUNCTION.
The evaluation of forms of the type [oo]/[oo] leads to a scale of
increasing "infinities," each being infinite in comparison with the
preceding. Such a scale is

log x,...x, x²,...x^n,...e^x,...x^x;

each of the limits expressed by such forms as lim.x=[oo]
[phi](x)/[psi](x), where [phi](x) precedes [psi](x) in the scale, is
zero. The construction of such scales, along with the problem of
constructing a complete scale was discussed in numerous writings by
Paul du Bois-Reymond (see in particular, _Math. Ann._ Bd. xi., 1877).
For the general problem of indeterminate forms reference may be made
to the article by A. Pringsheim in _Ency. d. math. Wiss._ Bd. ii., A.
1 (1899). Forms of the type 0/0 presented themselves to early writers
on analytical geometry in connexion with the determination of the
tangents at a double point of a curve; forms of the type [oo]/[oo]
presented themselves in like manner in connexion with the
determination of asymptotes of curves. The evaluation of limits has
innumerable applications in all parts of analysis. Cauchy's _Analyse
algébrique_ (1821) was an epoch-making treatise on limits.

If a function [phi](x) becomes infinite at x = a, and another function
[psi](x) also becomes infinite at x = a in such a way that
[phi](x)/[psi](x) has a finite limit C, we say that [phi](x) and
[psi](x) become "infinite of the same order." We may write [phi](x) =
C[psi](x) + [phi]1(x), where lim. x=a [phi]1(x)/[psi](x) = 0, and thus
[phi]1(x) is of a lower order than [phi](x); it may be finite or
infinite at x = a. If it is finite, we describe C[psi](x) as the
"infinite part" of [phi](x). The resolution of a function which
becomes infinite into an infinite part and a finite part can often be
effected by taking the infinite part to be infinite of the same order
as one of the functions in the scale written above, or in some more
comprehensive scale. This resolution is the inverse of the process of
evaluating an indeterminate form of the type [oo] - [oo].

For example lim.x=0 {(e^x - 1)^-1 - x^-1} is finite and equal to =
½, and the function (e^x - 1)^-1 - x^-1 can be expanded in a power
series in x.

Functions of several variables.

39. The nature of a function of two or more variables, and the meaning
to be attached to continuity and limits in respect of such functions,
have been explained under FUNCTION. The theorems of differential
calculus which relate to such functions are in general the same
whether the number of variables is two or any greater number, and it
will generally be convenient to state the theorems for two variables.

Partial differentiation.

40. Let u or [f](x, y) denote a function of two variables x and y. If
we regard y as constant, u or f becomes a function of one variable x,
and we may seek to differentiate it with respect to x. If the function
of x is differentiable, the differential coefficient which is formed
in this way is called the "partial differential coefficient" of u or f
with respect to x, and is denoted by ðu/ðx or ð[f]/ðx. The symbol "ð"
was appropriated for partial differentiation by C. G. J. Jacobi
(1841). It had before been written indifferently with "d" as a symbol
of differentiation. Euler had written (df/dx) for the partial
differential coefficient of f with respect to x. Sometimes it is
desirable to put in evidence the variable which is treated as
constant, and then the partial differential coefficient is written
"(df/dx)_y" or "(ð[f]/ðx)_y". This course is often adopted by writers
on Thermodynamics. Sometimes the symbols d or ð are dropped, and the
partial differential coefficient is denoted by u_x or [f]_x. As a
definition of the partial differential coefficient we have the formula

ð[f] [f](x + h, y) - f(x, y)
---- = lim. -----------------------.
ðx h=0 h

In the same way we may form the partial differential coefficient with
respect to y by treating x as a constant.

The introduction of partial differential coefficients enables us to
solve at once for a surface a problem analogous to the problem of
tangents for a curve; and it also enables us to take the first step in
the solution of the problem of maxima and minima for a function of
several variables. If the equation of a surface is expressed in the
form z = [f](x, y), the direction cosines of the normal to the surface
at any point are in the ratios ð[f]/ðx : ð[f]/ðy : = 1. If f is a maximum
or a minimum at (x, y), then ð[f]/ðx and ð[f]/ðy vanish at that point.

In applications of the differential calculus to mathematical physics
we are in general concerned with functions of three variables x, y, z,
which represent the coordinates of a point; and then considerable
importance attaches to partial differential coefficients which are
formed by a particular rule. Let F(x, y, z) be the function, P a point
(x, y, z), P´ a neighbouring point (x + [Delta]x, y + [Delta]y, z +
[Delta]z), and let [Delta]s be the length of PP´. The value of F(x, y,
z) at P may be denoted shortly by F(P). A limit of the same nature as
a partial differential coefficient is expressed by the formula

F(P´) = F(P)
lim. ------------,
[Delta]s=0 [Delta]s

in which [Delta]s is diminished indefinitely by bringing P´ up to P,
and P´ is supposed to approach P along a straight line, for example,
the tangent to a curve or the normal to a surface. The limit in
question is denoted by ðF/ðh, in which it is understood that h
indicates a direction, that of PP´. If l, m, n are the direction
cosines of the limiting direction of the line PP´, supposed drawn from
P to P´, then

ðF ðF ðF ðF
-- = l -- + m -- + n --.
ðh ðx ðy ðz

The operation of forming ðF/ðh is called "differentiation with respect
to an axis" or "vector differentiation."

Theorem of the Total Differential.

41. The most important theorem in regard to partial differential
coefficients is the _theorem of the total differential_. We may write
down the equation

[f](a + h, b + k) - [f](a, b) = [f](a + h, b + k) - [f](a, b + k)
+ [f](a, b + k) - [f](a, b).

If [f]x is a continuous function of x when x lies between a and a + h
and y = b + k, and if further [f]y is a continuous function of y when
y lies between b and d + k, there exist values of [Theta] and [eta]
which lie between 0 and 1 and have the properties expressed by the
equations

[f](a + h, b + k) - [f](a, b + k) = h[f]_x (a + [Theta]h, b + k),
[f](a, b + k) - [f](a, b) = k[f]_y (a, b + [eta]k).

Further, [f]x(a + [Theta]h, b + k) and [f]_y (a, b + [eta]k) tend to
the limits [f]_x (a, b) and [f]_y (a, b) when h and k tend to zero,
provided the differential coefficients [f]_x, [f]_y, are continuous
at the point (a, b). Hence in this case the above equation can be
written

[f](a + h, b + k) - [f](a, b) = h[f]_x (a, b) + k[f]_y (a, b) + R,

where

R R
lim. --- = 0 and lim. --- = 0.
h=0, k=0 h h=0, k=0 k

In accordance with the notation of differentials this equation gives

ð[f] ðy
d[f] = ---- dx + -- dy.
ðx ðy

Just as in the case of functions of one variable, dx and dy are
arbitrary finite differences, and d[f] is not the difference of two
values of [f], but is so much of this difference as need be retained
for the purpose of forming differential coefficients.

The theorem of the total differential is immediately applicable to the
differentiation of _implicit functions_. When y is a function of x
which is given by an equation of the form [f](x, y) = 0, and it is
either impossible or inconvenient to solve this equation so as to
express y as an explicit function of x, the differential coefficient
dy/dx can be formed without solving the equation. We have at once

dy ð[f] / ðf
-- = - ---- / --.
dx ðx / ðy

This rule was known, in all essentials, to Fermat and de Sluse before
the invention of the algorithm, of the differential calculus.

An important theorem, first proved by Euler, is immediately deducible
from the theorem of the total differential. If [f](x, y) is a
homogeneous function of degree n then

ð[f] ð[f]
x ---- + y ---- = n[f](x, y).
ðx ðy

The theorem is applicable to functions of any number of variables and
is generally known as _Euler's theorem of homogeneous functions_.

Jacobians.

42. Many problems in which partial differential coefficients occur are
simplified by the introduction of certain determinants called
"Jacobians" or "functional determinants." They were introduced into
Analysis by C. G. J. Jacobi (_J. f. Math._, Crelle, Bd. 22, 1841, p.
319). The Jacobian of u1, u2, ... u_n with respect to x1, x2, ... x_n
is the determinant

| ðu1 ðu1 ðu1 |
| --- --- ... ---- |
| ðx1 ðx2 ðx_n |
| |
| ðu2 ðu2 ðu2 |
| --- --- ... ---- |
| ðx1 ðx2 ðx_n |
| . |
| . |
| . |
| ðu_n ðu_n ðu_n |
| ---- ---- ... ----- |
| ðx1 ðx2 ðx_n |

in which the constituents of the rth row are the n partial
differential coefficients of u_r, with respect to the n variables x.
This determinant is expressed shortly by

ð(u1, u2, ..., u_n)
-------------------.
ð(x1, x2, ..., x_n)

Jacobians possess many properties analogous to those of ordinary
differential coefficients, for example, the following:--

ð(u1, u2, ..., u_n) ð(x1, x2, ..., x_n)
------------------- × ------------------- = 1,
ð(x1, x2, ..., x_n) ð(u1, u2, ..., u_n)

ð(u1, u2, ..., u_n) ð(y1, y2, ..., y_n) ð(u1, u2, ..., u_n)
------------------- × ------------------- = -------------------.
ð(y1, y2, ..., y_n) ð(x1, x2, ..., x_n) ð(x1, x2, ..., x_n)

If n functions (u1, u2, ... u_n) of n variables (x1, x2, ..., x_n) are
not independent, but are connected by a relation [f](u1, u2, ... u_n)
= 0, then

ð(u1, u2, ..., u_n)
------------------- = 0;
ð(x1, x2, ..., x_n)

and, conversely, when this condition is satisfied identically the
functions u1, u2 ..., u_n are not independent.

Interchange of order of differentiations.

43. Partial differential coefficients of the second and higher orders
can be formed in the same way as those of the first order. For
example, when there are two variables x, y, the first partial
derivatives ð[f]/ðx and ð[f]/ðy are functions of x and y, which we may
seek to differentiate partially with respect to x or y. The most
important theorem in relation to partial differential coefficients of
orders higher than the first is the theorem that the values of such
coefficients do not depend upon the order in which the
differentiations are performed. For example, we have the equation

ð /ð[f]\ ð /ð[f]\
-- ( ---- ) = -- ( ---- ) (i.)
ðx \ ðy / ðy \ ðx /

This theorem is not true without limitation. The conditions for its
validity have been investigated very completely by H. A. Schwarz (see
his _Ges. math. Abhandlungen_, Bd. 2, Berlin, 1890, p. 275). It is a
sufficient, though not a necessary, condition that all the
differential coefficients concerned should be continuous functions of
x, y. In consequence of the relation (i.) the differential
coefficients expressed in the two members of this relation are written

ð²f ð²f
---- or ----.
ðxðy ðyðx

The differential coefficient

ð^_n [f]
--------------,
ðx^p ðy^q ðz^r

in which p + g + r = n, is formed by differentiating p times with
respect to x, q times with respect to y, r times with respect to z,
the differentiations being performed in any order. Abbreviated
notations are sometimes used in such forms as

(p, q, r)
[f] or [f] .
x^p y^q z^r x, y, z

_Differentials_ of higher orders are introduced by the defining
equation

/ ð ð \ n
d^n [f] = ( dx -- + dy -- ) [f]
\ ðx ðy/

ð^n [f] ð^n [f]
= (dx)^n ------- + n(dx)^(n-1) dy ----------- + ...
ðx^n ðx^(n-1) ðy

in which the expression (dx·ð/ðx + dy·ð/ðy)^n is developed by the
binomial theorem in the same way as if dx·ð/ðx and dy·ð/ðy were
numbers, and (ð/ðx)^r·(ð/ðy)^(n-r) [f] is replaced by ð^n [f]/[ðx^r
ðy^(n-r)]. When there are more than two variables the multinomial
theorem must be used instead of the binomial theorem.

The problem of forming the second and higher differential coefficients
of _implicit functions_ can be solved at once by means of partial
differential coefficients, for example, if [f](x, y) = 0 is the
equation defining y as a function of x, we have
_ _
d²y /ð[f]\ -3 | /ð[f]\² ð²[f] ð[f] ð[f] ð²[f] /ð[f]\² ð²[f] |
--- = ( ---- ) | ( ---- ) ----- - 2 ---- · ---- · ----- + ( ---- ) ----- |.
dx² \ ðy / |_ \ ðy / ðx² ðx ðy ðxðy \ ðx / ðy² _|

The differential expression Xdx + Ydy, in which both X and Y are
functions of the two variables x and y, is a _total differential_ if
there exists a function [f] of x and y which is such that

ð[f]/ðx = X, ð[f]/ðy = Y.

When this is the case we have the relation

ðY/ðx = ðX/ðy. (ii.)

Conversely, when this equation is satisfied there exists a function
[f] which is such that

d[f] = Xdx + Ydy.

The expression Xdx + Ydy in which X and Y are connected by the
relation (ii.) is often described as a "perfect differential." The
theory of the perfect differential can be extended to functions of n
variables, and in this case there are ½n(n - 1) such relations as
(ii.).

In the case of a function of two variables x, y an abbreviated
notation is often adopted for differential coefficients. The function
being denoted by z, we write

ðz ðz ð²z ð²z ð²z
p, q, r, s, t for --, --, ---, ----, ---.
ðx ðy ðx² ðxðy ðy²

Partial differential coefficients of the second order are important in
geometry as expressing the curvature of surfaces. When a surface is
given by an equation of the form z = [f](x, y), the lines of curvature
are determined by the equation

{(l + q²)s - pqt} (dy)² + {(1 + q²)r - (1 + p²)t} dx dy
- {(1 + p²)s - pqr} (dx)² = 0,

and the principal radii of curvature are the values of R which satisfy
the equation

R²(rt - s²) - R{(1 + q²)r - 2pqs + (1 + p²)t} [root](1 + p² + q²)
+ (1 + p² + q²)² = 0.

Change of variables.

44. The problem of change of variables was first considered by Brook
Taylor in his _Methodus incrementorum_. In the case considered by
Taylor y is expressed as a function of z, and z as a function of x,
and it is desired to express the differential coefficients of y with
respect to x without eliminating z. The result can be obtained at once
by the rules for differentiating a product and a function of a
function. We have

dy dy dz
-- = -- · --,
dx dz dx

d²y dy d²z d²y /dz\²
--- = -- · --- + --- · ( -- ),
dx² dz dx² dz² \dx/

d³y dy d³z, d²y dz d²z, d³y /dz\³
--- = -- · --- + 3 --- · -- · --- + --- · ( -- ) ,
dx³ dz dx³ dz² dx dx² dz³ \dx/

. . . . . . .

The introduction of partial differential coefficients enables us to
deal with more general cases of change of variables than that
considered above. If u, v are new variables, and x, y are connected
with them by equations of the type

x = [f]1(u, v), y = [f]2(u, v), (i.)

while y is either an explicit or an implicit function of x, we have
the problem of expressing the differential coefficients of various
orders of y with respect to x in terms of the differential
coefficients of v with respect to u. We have

dy /ð[f]2 ð[f]2 dv \ / /ð[f]1 ð[f]1 dv \
-- = ( ----- + ----- -- ) / ( ----- + ----- -- )
dx \ ðu ðv du / / \ ðu ðv du /

by the rule of the total differential. In the same way, by means of
differentials of higher orders, we may express d²y/dx², and so on.

Equations such as (i.) may be interpreted as effecting a
_transformation_ by which a point (u, v) is made to correspond to a
point (x, y). The whole theory of transformations, and of functions,
or differential expressions, which remain invariant under groups of
transformations, has been studied exhaustively by Sophus Lie (see, in
particular, his _Theorie der Transformationsgruppen_, Leipzig,
1888-1893). (See also DIFFERENTIAL EQUATIONS and GROUPS).

A more general problem of change of variables is presented when it is
desired to express the partial differential coefficients of a function
V with respect to x, y, ... in terms of those with respect to u, v,
..., where u, v, ... are connected with x, y, ... by any functional
relations. When there are two variables x, y, and u, v are given
functions of x, y, we have

ðV ðV ðu ðV ðv
-- = -- -- + -- --,
ðx ðu ðx ðv ðx

ðV ðV ðu ðV ðv
-- = -- -- + -- --,
ðy ðu ðy ðv ðy

and the differential coefficients of higher orders are to be formed by
repeated applications of the rule for differentiating a product and
the rules of the type

ð ðu ð ðv ð
-- = -- -- + -- --.
ðx ðx ðu ðx ðx

When x, y are given functions of u, v, ... we have, instead of the
above, such equations as

ðV ðV ðx ðV ðy
-- = -- -- + -- --;
ðu ðx ðu ðy ðu

and ðV/ðx, ðV/ðy can be found by solving these equations, provided the
Jacobian ð(x, y) / ð(u, v) is not zero. The generalization of this
method for the case of more than two variables need not detain us.

In cases like that here considered it is sometimes more convenient not
to regard the equations connecting x, y with u, v as effecting a point
transformation, but to consider the loci u = const., v = const. as two
"families" of curves. Then in any region of the plane of (x, y) in
which the Jacobian ð(x, y) / d(u, v) does not vanish or become
infinite, any point (x, y) is uniquely determined by the values of u
and v which belong to the curves of the two families that pass through
the point. Such variables as u, v are then described as "curvilinear
coordinates" of the point. This method is applicable to any number of
variables. When the loci u = const., ... intersect each other at right
angles, the variables are "orthogonal" curvilinear coordinates.
Three-dimensional systems of such coordinates have important
applications in mathematical physics. Reference may be made to G.
Lamé, _Leçons sur les coordonnées curvilignes_ (Paris, 1859), and to
G. Darboux, _Leçons sur les coordonnées curvilignes et systèmes
orthogonaux_ (Paris, 1898).

When such a coordinate as u is connected with x and y by a functional
relation of the form [f](x, y, u) = 0 the curves u = const. are a
family of curves, and this family may be such that no two curves of
the family have a common point. When this is not the case the points
in which a curve [f](x, y, u) = 0 is intersected by a curve [f](x, y,
u + [Delta]u) = 0 tend to limiting positions as [Delta]u is diminished
indefinitely. The locus of these limiting positions is the "envelope"
of the family, and in general it touches all the curves of the family.
It is easy to see that, if u, v are the parameters of two families of
curves which have envelopes, the Jacobian ð(x, y) / ð(u, v) vanishes
at all points on these envelopes. It is easy to see also that at any
point where the reciprocal Jacobian ð(u, v) / ð(x, y) vanishes, a
curve of the family u touches a curve of the family v.

If three variables x, y, z are connected by a functional relation
[f](x, y, z) = 0, one of them, z say, may be regarded as an _implicit
function_ of the other two, and the partial differential coefficients
of z with respect to x and y can be formed by the rule of the total
differential. We have

ðz ð[f] / ð[f] ðz ð[f] / ð[f]
-- = - ---- / ----, -- = - ---- / ----;
ðx ðx / ðz ðy ðy / ðz

and there is no difficulty in proceeding to express the higher
differential coefficients. There arises the problem of expressing the
partial differential coefficients of x with respect to y and z in
terms of those of z with respect to x and y. The problem is known as
that of "changing the dependent variable." It is solved by applying
the rule of the total differential. Similar considerations are
applicable to all cases in which n variables are connected by fewer
than n equations.

Extension of Taylor's theorem.

45. Taylor's theorem can be extended to functions of several
variables. In the case of two variables the general formula, with a
remainder after n terms, can be written most simply in the form

[f](a + h, b + k) = [f](a, b) + d[f](a, b) + (1/2!) d²[f](a, b) + ...

1 1
+ -------- d^(n-1) [f](a, b) + -- d^n [f](a+[Theta]h, b + [theta]k),
(n - 1)! n!

in which
_ _
| / ð ð \r |
d^r [f](a, b) = | ( h -- + k -- ) [f](x, y) | ,
|_ \ ðx ðy / _| x=a, y=b

and

d^n [f](a + [Theta]h, b + [Theta]k) =
_ _
| / ð ð \n |
| ( h -- + k -- ) [f](x, y) |.
|_ \ ðx ðy/ _| x=a+[Theta]h, y=b+[Theta]k

The last expression is the remainder after n terms, and in it [Theta]
denotes some particular number between 0 and 1. The results for three
or more variables can be written in the same form. The extension of
Taylor's theorem was given by Lagrange (1797); the form written above
is due to Cauchy (1823). For the validity of the theorem in this form
it is necessary that all the differential coefficients up to the nth
should be continuous in a region bounded by x = a ± h, y = b ± k. When
all the differential coefficients, no matter how high the order, are
continuous in such a region, the theorem leads to an expansion of the
function in a multiple power series. Such expansions are just as
important in analysis, geometry and mechanics as expansions of
functions of one variable. Among the problems which are solved by
means of such expansions are the problem of maxima and minima for
functions of more than one variable (see MAXIMA and MINIMA).

Plane curves.

46. In treatises on the differential calculus much space is usually
devoted to the differential geometry of curves and surfaces. A few
remarks and results relating to the differential geometry of plane
curves are set down here.

(i.) If [psi] denotes the angle which the radius vector drawn from the
origin makes with the tangent to a curve at a point whose polar
coordinates are r, [Theta] and if p denotes the perpendicular from the
origin to the tangent, then

cos [psi] = dr/ds, sin [psi] = r d[Theta]/ds = p/r,

where ds denotes the element of arc. The curve may be determined by an
equation connecting p with r.

(ii.) The locus of the foot of the perpendicular let fall from the
origin upon the tangent to a curve at a point is called the _pedal_ of
the curve with respect to the origin. The angle [psi] for the pedal is
the same as the angle [psi] for the curve. Hence the (p, r) equation
of the pedal can be deduced. If the pedal is regarded as the primary
curve, the curve of which it is the pedal is the "negative pedal" of
the primary. We may have pedals of pedals and so on, also negative
pedals of negative pedals and so on. Negative pedals are usually
determined as envelopes.

(iii.) If [phi] denotes the angle which the tangent at any point makes
with a fixed line, we have

r² = p² + (dp/d[phi])².

(iv.) The "average curvature" of the arc [Delta]s of a curve between
two points is measured by the quotient

| [Delta][phi] |
| ------------ |
| [Delta]s |

where the upright lines denote, as usual, that the absolute value of
the included expression is to be taken, and [phi] is the angle which
the tangent makes with a fixed line, so that [Delta][phi] is the angle
between the tangents (or normals) at the points. As one of the points
moves up to coincidence with the other this average curvature tends to
a limit which is the "curvature" of the curve at the point. It is
denoted by

| d[phi] |
| ------ |
| ds |

Sometimes the upright lines are omitted and a rule of signs is
given:--Let the arc s of the curve be measured from some point along
the curve in a chosen sense, and let the normal be drawn towards that
side to which the curve is concave; if the normal is directed towards
the left of an observer looking along the tangent in the chosen sense
of description the curvature is reckoned positive, in the contrary
case negative. The differential d[phi] is often called the "angle of
contingence." In the 14th century the size of the angle between a
curve and its tangent seems to have been seriously debated, and the
name "angle of contingence" was then given to the supposed angle.

(v.) The curvature of a curve at a point is the same as that of a
certain circle which touches the curve at the point, and the "radius
of curvature" [rho] is the radius of this circle. We have 1/[rho] =
|d[phi]/ds|. The centre of the circle is called the "centre of
curvature"; it is the limiting position of the point of intersection
of the normal at the point and the normal at a neighbouring point,
when the second point moves up to coincidence with the first. If a
circle is described to intersect the curve at the point P and at two
other points, and one of these two points is moved up to coincidence
with P, the circle touches the curve at the point P and meets it in
another point; the centre of the circle is then on the normal. As the
third point now moves up to coincidence with P, the centre of the
circle moves to the centre of curvature. The circle is then said to
"osculate" the curve, or to have "contact of the second order" with it
at P.

(vi.) The following are formulae for the radius of curvature:--

1 | { /dy\² }-3/2 d²y |
----- = | { 1 + ( -- ) } --- |,
[rho] | { \dx/ } dx² |

| dr | | d²p |
[rho] = | r -- | = | p + ------- |.
| dp | | d[phi]² |

(vii.) The points at which the curvature vanishes are "points of
inflection." If P is a point of inflection and Q a neighbouring point,
then, as Q moves up to coincidence with P, the distance from P to the
point of intersection of the normals at P and Q becomes greater than
any distance that can be assigned. The equation which gives the
abscissae of the points in which a straight line meets the curve being
expressed in the form [f](x) = 0, the function [f](x) has a factor (x
- x0)³, where x0 is the abscissa of the point of inflection P, and the
line is the tangent at P. When the factor (x - x0) occurs (n + 1)
times in [f](x), the curve is said to have "contact of the nth order"
with the line. There is an obvious modification when the line is
parallel to the axis of y.

(viii.) The locus of the centres of curvature, or envelope of the
normals, of a curve is called the "evolute." A curve which has a given
curve as evolute is called an "involute" of the given curve. All the
involutes are "parallel" curves, that is to say, they are such that
one is derived from another by marking off a constant distance along
the normal. The involutes are "orthogonal trajectories" of the
tangents to the common evolute.

(ix.) The equation of an algebraic curve of the nth degree can be
expressed in the form u0 + u1 + u2 + ... + u_n = 0, where u0 is a
constant, and u_r is a homogeneous rational integral function of x, y
of the rth degree. When the origin is on the curve, u0 vanishes, and
u1 = 0 represents the tangent at the origin. If u1 also vanishes, the
origin is a double point and u2 = o represents the tangents at the
origin. If u2 has distinct factors, or is of the form a(y - p1x)(y -
p2x), the value of y on either branch of the curve can be expressed
(for points sufficiently near the origin) in a power series, which is
either

p1x + ½ q1x² + ..., or p2x + ½ q2X² + ...,

where q1, ... and q2, ... are determined without ambiguity. If p1 and
p2 are real the two branches have radii of curvature [rho]1, [rho]2
determined by the formulae

1 | | 1 | |
------ = |(1 + p1²)^{-3/2} q1 |, ------ = |(1 + p2²)^{-3/2} q2 |.
[rho]1 | | [rho]2 | |

When p1 and p2 are imaginary the origin is the real point of
intersection of two imaginary branches. In the real figure of the
curve it is an _isolated point_. If u2 is a square, a(y - px)², the
origin is a _cusp_, and in general there is not a series for y in
integral powers of x, which is valid in the neighbourhood of the
origin. The further investigation of cusps and multiple points belongs
rather to analytical geometry and the theory of algebraic functions
than to differential calculus.

(x.) When the equation of a curve is given in the form u0 + u1 + ... +
u_(n-1) + u_n = 0 where the notation is the same as that in (ix.), the
factors of u_n determine the directions of the _asymptotes_. If these
factors are all real and distinct, there is an asymptote corresponding
to each factor. If u_n = L1 L2 ... L_n, where L1, ... are linear in x,
y, we may resolve u_(n-1)/u_n into partial fractions according to the
formula

u_(n-1) A1 A2 A_n
------- = -- + -- + ... + ---,
u{n} L1 L2 L_n

and then L1 + A1 = 0, L2 + A2 = 0, ... are the equations of the
asymptotes. When a real factor of u_n is repeated we may have two
parallel asymptotes or we may have a "parabolic asymptote." Sometimes
the parallel asymptotes coincide, as in the curve x²(x² + y² - a²) =
a^4, where x = 0 is the only real asymptote. The whole theory of
asymptotes belongs properly to analytical geometry and the theory of
algebraic functions.

Integral calculus.

47. The formal definition of an integral, the theorem of the existence
of the integral for certain classes of functions, a list of classes of
"integrable" functions, extensions of the notion of integration to
functions which become infinite or indeterminate, and to cases in
which the limits of integration become infinite, the definitions of
multiple integrals, and the possibility of defining functions by means
of definite integrals--all these matters have been considered in
FUNCTION. The definition of integration has been explained in § 5
above, and the results of some of the simplest integrations have been
given in § 12. A few theorems relating to integrations have been noted
in §§ 34, 35, 36 above.

Methods of integration.

48. The chief methods for the evaluation of indefinite integrals are
the method of integration by parts, and the introduction of new
variables.

From the equation d(uv) = udv + vdu we deduce the equation
_ _
/ dv / du
| u -- dx = uv - | v -- dx,
_/ dx _/ dx

or, as it may be written
_ _ _ _
/ / / du / / \
| uw dx = u | w dx - | -- ( | w dx ) dx.
_/ _/ _/ dx \ _/ /

This is the rule of "integration by parts."

As an example we have
_ _
/ e^(ax) / e^(ax) / x 1 \
| xe^(ax) dx = x ------ - | ------ dx = ( --- - -- ) e^(ax).
_/ a _/ a \ a a² /

When we introduce a new variable z in place of x, by means of an
equation giving x in terms of z, we express [f](x) in terms of z. Let
[phi](z) denote the function of z into which [f](x) is transformed.
Then from the equation

dx
dx = -- dz
dz

we deduce the equation
_ _
/ / dx
| [f](x) dx = | [phi](z) -- dz.
_/ _/ dz

As an example, in the integral
_
/
| [root](1 - x²) dx
_/

put x = sin z; the integral becomes

_ _
/ /
| cos z · cos zdz = | ½(1 + cos 2z)dz = ½(z + ½ sin 2z) = ½(z + sin z cos z).
_/ _/

Integration in terms of elementary functions.

49. The indefinite integrals of certain classes of functions can be
expressed by means of a finite number of operations of addition or
multiplication in terms of the so-called "elementary" functions. The
elementary functions are rational algebraic functions, implicit
algebraic functions, exponentials and logarithms, trigonometrical and
inverse circular functions. The following are among the classes of
functions whose integrals involve the elementary functions only: (i.)
all rational functions; (ii.) all irrational functions of the form
[f](x, y), where [f] denotes a rational algebraic function of x and y,
and y is connected with x by an algebraic equation of the second
degree; (iii.) all rational functions of sin x and cos x; (iv.) all
rational functions of e^x; (v.) all rational integral functions of the
variables x, e^(ax), e^(bx), ... sin mx, cos mx, sin nx, cos nx, ...
in which a, b, ... and m, n, ... are any constants. The integration of
a rational function is generally effected by resolving the function
into partial fractions, the function being first expressed as the
quotient of two rational integral functions. Corresponding to any
simple root of the denominator there is a logarithmic term in the
integral. If any of the roots of the denominator are repeated there
are rational algebraic terms in the integral. The operation of
resolving a fraction into partial fractions requires a knowledge of
the roots of the denominator, but the algebraic part of the integral
can always be found without obtaining all the roots of the
denominator. Reference may be made to C. Hermite, _Cours d'analyse_,
Paris, 1873. The integration of other functions, which can be
integrated in terms of the elementary functions, can usually be
effected by transforming the functions into rational functions,
possibly after preliminary integrations by parts. In the case of
rational functions of x and a radical of the form [root](ax² + bx + c)
the radical can be reduced by a linear substitution to one of the
forms [root](a² - x²), [root](x² - a²), [root](x² + a²). The
substitutions x = a sin [theta], x = a sec [theta], x = a tan [theta]
are then effective in the three cases. By these substitutions the
subject of integration becomes a rational function of sin [theta] and
cos [theta], and it can be reduced to a rational function of t by the
substitution tan ½[theta] = t. There are many other substitutions by
which such integrals can be determined. Sometimes we may have
information as to the functional character of the integral without
being able to determine it. For example, when the subject of
integration is of the form (ax^4 + bx³ + cx² + dx + e)^-½ the integral
cannot be expressed explicitly in terms of elementary functions. Such
integrals lead to new functions (see FUNCTION).

Methods of reduction and substitution for the evaluation of indefinite
integrals occupy a considerable space in text-books of the integral
calculus. In regard to the functional character of the integral
reference may be made to G. H. Hardy's tract, _The Integration of
Functions of a Single Variable_ (Cambridge, 1905), and to the memoirs
there quoted. A few results are added here
_
/
(i.) | (x² + a) - ½ dx = log {x + (x² + a)^½ }.
_/
_
/ dx
(ii.) | -----------------------------
_/ (x - p) [root](ax² + 2bx + c)

can be evaluated by the substitution x - p = 1/z, and
_
/ dx
| ---------------------------------
_/ (x - p)^{n} [root](ax² + 2bx + c)

can be deduced by differentiating (n - 1) times with respect to p.
_
/ (Hx + K)dx
(iii.) | ------------------------------------------------------
_/ ([alpha]x² + 2[beta]x + [gamma]) [root](ax² + 2bx + c)

can be reduced by the substitution y² = (ax² + 2bx + c)/([alpha]x² +
2[beta]x + [gamma]) to the form
_ _
/ dy / dy
A | ---------------------- + B | ----------------------
_/ [root]([lambda]1 - y²) _/ [root](y² - [lambda]2)

where A and B are constants, and [lambda]1 and [lambda]2 are the two
values of [lambda] for which (a - [lambda][alpha])x² + 2(b -
[lambda][beta])x + c - [lambda][gamma] is a perfect square (see A. G.
Greenhill, _A Chapter in the Integral Calculus_, London, 1888).

(iv.) [f]x^m (ax^n + b)^p dx, in which m, n, p are rational, can be
reduced, by putting ax^n = bt, to depend upon [f]t^q (1 + t)^p dt. If
p is an integer and q a fraction r/s, we put t = u^s. If q is an
integer and p = r/s we put 1 + t = u^s. If p + q is an integer and p =
r/s we put 1 + t = tu^s. These integrals, called "binomial integrals,"
were investigated by Newton (_De quadratura curvarum_).
_ _
/ dx x / dx
(v.) | ----- = log tan ---, (vi.) | ----- = log (tan x + sec x).
_/ sin x 2 _/ cos x

(vii.) [f] e^(ax) sin (bx + [alpha]) dx = (a² + b²)^-1 e^(ax){a sin
(bx + [alpha]) - b cos (bx + [alpha])}.

(viii.) [f] sin^m x cos^n x dx can be reduced by differentiating a
function of the form sin^p x cos^q x;

d sin x 1 q sin² x 1 - q q
e.g. -- ------- = ----------- + ----------- = ----------- + -----------.
dx cos^q x cos^(q-1) x cos^(q+1) x cos^(q-1) x cos^(q+1) x

Hence
_ _
/ dx sin x n - 2 / dx
| ------- = ------------------- + ----- | -----------.
_/ cos^n x (n - 1) cos^(n-1) x n - 1 _/ cos^(n-2) x
_ _
/ ½[pi] / ½[pi]
(ix.) | sin^(2n) x dx = | cos^(2n) x dx =
_/ 0 _/ 0

1·3 ... (2n - 1) [pi]
---------------- · ----, (n an integer).
2·4 ... 2n 2
_ _
/ ½[pi] / ½[pi]
(x.) | sin^(2n+1) x dx = | cos^(2n+1) x dx =
_/ 0 _/ 0

2·4 ... (2n)
--------------, (n an integer).
3·5 ... (2n+1)
_
/ dx
(xi.) | --------------- can be reduced by one of the substitutions
_/ (1 + e cos x)^n

e + cos x e + cos x
cos [phi] = -----------, cosh u = -----------,
1 + e cos x 1 + e cos x

of which the first or the second is to be employed according as e < or > 1.

New transcendents.

50. Among the integrals of transcendental functions which lead to new
transcendental functions we may notice
_ _
/ x dx / log x e^z
| ----- or | --- dz,
_/ 0 log x´ _/ -x z

called the "logarithmic integral," and denoted by "Li x," also the
integrals
_ _
/ x sin x / x cos x
| ----- dx and | ----- dx,
_/ 0 x _/ [oo] x

called the "sine integral" and the "cosine integral," and denoted by
"Si x" and "Ci x," also the integral
_
/ x
| e^-x² dx
_/ 0

called the "error-function integral," and denoted by "Erf x." All
these functions have been tabulated (see TABLES, MATHEMATICAL).

Eulerian integrals.

51. New functions can be introduced also by means of the definite
integrals of functions of two or more variables with respect to one of
the variables, the limits of integration being fixed. Prominent among
such functions are the Beta and Gamma functions expressed by the
equations
_
/ 1
B(l, m) = | x^(l-1) (1 - x)^(m-1) dx,
_/ 0
_
/ [oo]
[Gamma](n) = | e^-t t^(n-1) dt.
_/ 0

When n is a positive integer [Gamma](n + 1) = n!. The Beta function
(or "Eulerian integral of the first kind") is expressible in terms of
Gamma functions (or "Eulerian integrals of the second kind") by the
formula

B(l, m)·[Gamma](l+m) = [Gamma](l)·[Gamma](m).

The Gamma function satisfies the difference equation

[Gamma](x + 1) = x [Gamma](x),

and also the equation

[Gamma](x)·[Gamma](1-x) = [pi]/sin (x[pi]),

with the particular result

[Gamma](½)= [root][pi].

The number
_ _
| d |
- | -- {log [Gamma](1 + x)} | , or -[Gamma]´(1),
|_ dx _|x=0

is called "Euler's constant," and is equal to the limit
_ _
| / \ |
lim. | ( 1 + ½ + 1/3 + ... + 1/n ) - log n |;
n=[oo] |_ \ / _|

its value to 15 decimal places is 0.577 215 664 901 532.

The function log [Gamma](1 + x) can be expanded in the series

/ x[pi] \
log [Gamma](1 + x) = ½ log ( --------- )
\ sin x[pi] /
1 + x
- ½ log ----- + {1 + [Gamma]´(1)} x
1 - x

- 1/3 (S3 - 1)x³ - 1/5 (S5 - 1)x^5 - ...,

where

1 1
S_(2r+1) = 1 + -------- + -------- + ...,
2^(2r+1) 3^(2r+1)

and the series for log [Gamma](1 + x) converges when x lies between -
1 and 1.

Definite integrals.

52. Definite integrals can sometimes be evaluated when the limits of
integration are some particular numbers, although the corresponding
indefinite integrals cannot be found. For example, we have the result
_
/ 1
| (1 - x²)^-½ log x dx = -½ [pi] log 2,
_/ 0

although the indefinite integral of (1 - x²)^-½ log x cannot be found.
Numbers of definite integrals are expressible in terms of the
transcendental functions mentioned in § 50 or in terms of Gamma
functions. For the calculation of definite integrals we have the
following methods:--

(i.) Differentiation with respect to a parameter.
(ii.) Integration with respect to a parameter.
(iii.) Expansion in infinite series and integration term by term.
(iv.) Contour integration.

The first three methods involve an interchange of the order of two
limiting operations, and they are valid only when the functions
satisfy certain conditions of continuity, or, in case the limits of
integration are infinite, when the functions tend to zero at infinite
distances in a sufficiently high order (see FUNCTION). The method of
contour integration involves the introduction of complex variables
(see FUNCTION: § _Complex Variables_).

A few results are added
_
/ [oo] x^(a-1) [pi]
(i.) | ------- dx = ---------, (1 > a > 0),
_/ 0 1 + x sin a[pi]

_
/ [oo] x^(a-1) - x^(b-1)
(ii.) | ----------------- dx = [pi](cot a[pi] - cot b[pi]), (0 < a or b < 1),
_/ 0 1 - x

_
/ [oo] x^(a-1) log x [pi]²
(iii.) | ------------ dx = ----------, (a > 1),
_/ 0 x - 1 sin² a[pi]

_
/ [oo]
(iv.) | x²·cos 2x·e^-x² dx = -¼ e^-1 [root][pi],
_/ 0

_
/ 1 1 - x² dx [pi]
(v.) | ------- ----- = log tan ----,
_/ 0 1 + x^4 log x 8

_
/ [oo] sin mx / 1 1 1 \
(vi.) | -------------- dx = ½ ( ------- - --- + --- ),
_/ 0 e^(2[pi]x) - 1 \ e^m - 1 m 2 /

_
/ [pi]
(vii.) | log(1 - 2[alpha] cos x + [alpha]²) dx = 0
_/ 0

or 2[pi]log [alpha] according as [alpha] < or > 1,
_
/ [oo] sin x
(viii.) | ----- dx = ½[pi],
_/ 0 x

_
/ [oo] cos ax
(ix.) | ------- dx = ½[pi]b^-1 e^(-ab),
_/ 0 x² + b²

_
/ [oo] cos ax - cos bx
(x.) | --------------- dx = ½[pi](b - a),
_/ 0 x²

_
/ [oo] cos ax - cos bx b
(xi.) | --------------- dx = log ---,
_/ 0 x a

_
/ [oo] cos x - e ^(-mx)
(xii.) | ---------------- dx = log m,
_/ 0 x

_
/ [oo]
(xiii.) | e^(-x²+2ax) dx = [root][pi].e^(a2),
_/ -[oo]

_ _
/ [oo] / [oo]
(xiv.) | x^-½ sin x dx = | x^-½ cos x dx = [root](½[pi]),
_/ 0 _/ 0