Part 19

__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).

[Illustration: FIG. 10.]

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).