The differential coefficient

ani

axpayvazf

in which p+q+r=n, is formed b differentia tin times with Y . g P

reseaaect to x, q times with respect to y, r times with respect to z, the d i erentiations being performed in any order. Abbreviated notations are sometimes used in such forms as

f» .)

fffyczf 07 LZ;-

Difcrenlials of higher orders are introduced by the defining equation

6 8 ' "

f1"f- (11165-lr*'55,) f
*

*3" " 0"f
*

*= (dx)"£+n(dx) 1dyax, , 1ay+ . . .
*

*in which the expression dx%c+dy(% n is developed by the binomial
theorem in the same way as if dxéggc and dy% were numbers, and
0
*

*B-rf is replaced by 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)=o is the
equation defining y as a function of x, we have
ei: (aff (er) =<n .Q1;.Qt. av: + at) air E
dx' By <9y 6x2 6x ay Bxdy 6x 6312
*

*The differential expression Xdx-l-Ydy, in which both X and Y are
functions of the two variables x and y, is a total deferential if there
exists a function f of x and y which is such that
df/6x =X, df/dy = Y.
*

*When this is the case we have the relation
BY/6x = 6X/dy. (ii.)
*

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

*dj=Xdx-I-Ydy.
*

*The expression Xdx-l-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 %n(n-I) 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
*

*p rstfor%»Q1&»iyi»&
*

*9' ' ' ax 0y ax= axay ayf
*

*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<1+q')S-Pail (ds/)'+{(1+q')f-(1+i>”)l}dxdy
~i<1+p2>.=-pqf1<dx>==so.
*

*and the principal radu of curvature are the values of R which
satisfy the equation
*

*R'(ff-82) -Rlfl +Q”)f-2P9S+(I +l>')llw/ (I +P”-l-Qi)
+(I+P“+<1'>'=<>.
*

*44. The problem of change of variables was first considered by
Brook Taylor in his Methodus lncremenlofum. In the case concbmn
of sidered by Taylor y is expressed as a function of z, and z
"Hawes as a function of x, and it is desired to express the differential
coefficients of y with respect to xwithout eliminating
z. The result can be obtained at once by the rules for differentiating
a product and a function of a function. We have
&' Q'&
*

*dx dz'dx
*

*4a a»e+<1f i. (ay,
*

*dx2"dz dx' dz* dx
*

*¢l3y dy d3z dz>3
*

*z%° ¢E ¢IF+3d22 dx dx2+dz“ (EEC,
*

*The introduction of partial differential coefficients enables us to
deal with more general cases of change of variables than that considered
above. If u, 11 are new variables, and x, y are connected with
them by equations of the type
*

*xzflcui v)v y=f2(u1 U):
*

*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
*

*fi % 0 2531 in 3 Q)
*

*Exec- (6u+5£ du>/ (614 +55 du
*

*547
*

*by the rule of the total differential. In the same way, by means of
differentials of higher orders, we may express d2y/dxz, and so on.
Equations such as (i.) ma be interpreted as effecting a transformation
by which a point (u, vlyis 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 So hus Lie (see, in particular,
his Theorie der Transformalionsgruppen, lbeipzig, 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,11, .,
where u, v, . . . are connected with x, y, . by any functional
relations. Wllen there are
*

*functions of x, y, we have
*

*two variables x, y, and u, 'v are given
a-@va»+@ya, I
*

*6x -El 6x 311 élx
*

*6y du dy. dv dy
*

*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 »
*

*2. = as +a 1.
*

*6x '6x du 6x 61:
*

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

*n ne+na.
*

*6u dx Bu 6y Hu
*

*and 6V/élx, 6V/By can be found by solving these equations, provided
the Jacobian 6(x, y)/6(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, with u, v as effecting a
point transformation, but to consider the lyoci 'u=const., v=const.
as two “families” of curves. Then in any region of the plane of
(x, y) in which the Iacobian 6(x, y)/6(u, 11) does not vanish or become
infinite, any point (x, y) is uniquely determined by the values of u
and 'zz 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, Legons sur les coordonnées cur-vllignes (Paris, 1859), and
to G. Darboux, Legans sur les coordonnées curvllignes el systémes
orlhogonaux (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) =o the curves u=c0nst.
are a family of curves, and this family may be such that no two
curves of the family have a common point. lVhen this is not the
case the points in which a curve j(x, y, u) =o is intersected by a
curve f(x, y, u+Au) =o tend to limiting positions as Au 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 envelo s, the Jacobian 6(x, y)/6(u, v) vanishes
at all points on these envelifpes. It is easy to see also that at any
point wheie the reciprocal Jacobian 6(u, "v)/6(x, y) vanishes, a curve
of the family n touches a curve of the family 11.
If three variables x, y, z are connected by a functional relation
f(x, y, z)=o, 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
*

*<E= '2i/Q '?E= Q'/QI.
*

*0x 6x dz' 6y éy dz
*

*and there is no difficulty in proceeding to express the higher differential
coefficients. There arises the problem of expressing the artial
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. "
*

*45. Taylor's theorem can be extended to functions of several
variables. In the case of two variables the general for- Extension
mula, with a remainder after n terms, can be written of 7-, ,, ,|o, .»,
most simply in the form
*

*f<a+h, z>+k> =f<a, b>+df<a, b>+ édwa., b>+ . .
theorem.
*

*+(@d"-val, b>+;, %d"f<f»+@h, b+@1=>.
in which
*

*
dv°<a.1»>= [(h§ +k§)'f<x, y> M H;*

*
*