R2 (rts2) RI (I +q2)r2Pgs+(I +p2)t),/ (1 +p2+q2)
+(I+p2+q2)2=o.
44. The problem of change of variables was first considered by Brook Taylor in his Methodus incrementorum. In the case conChange of sidered by Taylor y is expressed as a function of z, and z variables. as a function of x, and it is desired to express the differ
ential 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
dydy dz
dx dz dx'
d2ydy.d2z d2y (dz)2
dx2 de, dx2 dz2 dx
d3ydy.d2z+ d ,dz d2z d3y (dz)
dx' dz3dz2 dx dx2+dz3 '
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=fi(u, v), y=f2(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
( ~at'2 av / (Li+ i (lv ) dx au av du au av 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, 18881893). (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
aV aV au aV av
ax au ax+av ax'
aVaVau aVav
ay au ay. av ay'
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
a_aua ava
Ox= ax au+ax av
When x, y are given functions of u, v, . . . we have, instead of the above, such equations as
aV__ aV ax aV ay
au ax auay au'
and aV/ax, aV/ay can be found by solving these equations, provided the Jacobian a(x, y)/a(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 a(x, y)/a(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 ''jcurvilinear 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. Threedimensional systems of such coordinates have important applications in mathematical physics. Reference may be made to G. Lame, Leyans sur les coordonnees curvilignes (Paris, 1859), and to G. Darboux, Lecons sur les coordonnees curvilignes et systemes 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) =o 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) =o is intersected by a curve f(x, y, u+Au) =o tend to limiting positions as Du 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 a(x, y)/a(u, v) vanishes at all points on these envelopes. It is easy to see also that at any point where the reciprocal Jacobian a(u, v)/a(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) =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
az _ L of az of . f
_
ay/ ex ax/ az' ay= az'
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.
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'Taylor's most simply in the form theorem.
f(a+h, b+k) =f (a, b) +df (a, b) + 2 d2f(a, b) + .. .
+ n I I)id'''f(a, b)+dnf(a+Oh,b+Bk),
d'f(a, b) = [(hdx+kay)'f(x, y')]y_a
in which
n
and d"f (a+6h, b+6k) _ (h —+kayl ) f (x, )] = +en, vb+Px
The last expression is the remainder after n terms, and in it 0 denotes some particular number between o and i. 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 t 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
End of Article: R2 (rts2) 

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