See also:

• R2 (rt-s2)

R2 (rt-s2)  -RI (I +q2)r-2Pgs+(I +p2)t),/ (1 +p2+q2) +(I+p2+q2)2=o . 44 . The problem of See also:

• CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. «a urmu', to bend)

change of variables was first considered by See also:

• BROOK

Brook See also:

• TAYLOR
• TAYLOR, ANN (1782-1866)
• TAYLOR, BAYARD (1825–1878)
• TAYLOR, BROOK (1685–1731)
• TAYLOR, ISAAC (1787-1865)
• TAYLOR, ISAAC (1829-1901)
• TAYLOR, JEREMY (1613-1667)
• TAYLOR, JOHN (158o-1653)
• TAYLOR, JOHN (1704-1766)
• TAYLOR, JOSEPH (c. 1586-c. 1653)
• TAYLOR, MICHAEL ANGELO (1757–1834)
• TAYLOR, NATHANIEL WILLIAM (1786-1858)
• TAYLOR, PHILIP MEADOWS (1808–1876)
• TAYLOR, ROWLAND (d. 1555)
• TAYLOR, SIR HENRY (1800-1886)
• TAYLOR, THOMAS (1758-1835)
• TAYLOR, TOM (1817-1880)
• TAYLOR, WILLIAM (1765-1836)
• TAYLOR, ZACHARY (1784-1850)

Taylor in his Methodus incrementorum . In the See also:

• CASE
• CASE, JOHN (d. 1600)

case See also:

• CON

con-Change of sidered by Taylor y is expressed as a See also:

• FUNCTION

function of z, and z variables. as a function of x, and it is desired to See also:

• EXPRESS (through the French from the past participle of the Lat. exprimere, to press out, by transference used of representing objects in painting or sculpture, or of thoughts, &c. in words)

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 See also:

• DIFFERENTIAL

differential coefficients enables us to See also:

• DEAL

deal with more See also:

• GENERAL
• GENERAL (Lat. generalis, of or relating to a genus, kind or class)

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=See also:

• F2(X2+Y2)

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 See also:

• RULE

rule of the See also:

• TOTAL

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 See also:

• GROUPS
• GROUPS, THEORY

groups of transformations, has been studied exhaustively by Sophus See also:

• LIE, JONAS LAURITZ EDEMIL (1833—1908)
• LIE, MARIUS SOPHUS (1842–1899)

Lie (see, in particular, his Theorie der Transformationsgruppen, See also:

• LEIPZIG

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 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 See also:

• AVA

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 See also:

• PLANE

plane of (x, y) in which the Jacobian a(x, y)/a(u, v) does not vanish or become See also:

• INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

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 " See also:

• CURVILINEAR

curvilinear coordinates .

Three-dimensional systems of such coordinates have important applications in mathematical physics . Reference may be made to G . Lame, Leyans sur See also:

• LES

les coordonnees curvilignes (See also:

• PARIS
• PARIS (also called ALEXANDROS)
• PARIS, ALEXIS PAULIN (1800-x881)
• PARIS, BRUNO PAULIN GASTON (1839-1903)
• PARIS, FRANCOIS DE (169o-1727)
• PARIS, LOUIS PHILIPPE ALBERT
• PARIS, TREATIES OF (1814-1815)

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 See also:

• FORM (Lat. forma)

form f(x, y, u) =o the curves u=const. are a See also:

• FAMILY

family of curves, and this family may be such that no two curves of the family have a See also:

• COMMON

common point . When this is not the case the points in which a See also:

• CURVE (Lat. curvus, bent)

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 See also:

• LOCUS (Lat. for " place "; in Gr. rinros)

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- See also:

• EXTENSION (Lat. ex, out ; tendere, to stretch)

Extension See also:

• MULA

mula, with a See also:

• REMAINDER, REVERSION

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+See also:

• KAY, JOHN (1742-1826)
• KAY, JOSEPH (1821-1878)

kay)'f(x, y')]y_a in which n and d"f (a+6h, b+6k) _ (h —+kayl ) f (x, )] = +en, v-b+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 ex-tension of Taylor's theorem was given by See also:

• LAGRANGE, JOSEPH LOUIS (1736-1813)

Lagrange (1797); the form written above is due to See also:

• CAUCHY, AUGUSTIN LOUIS, BARON (1789-1857)

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 See also:

• MATTER

matter how high the See also:

• ORDER
• ORDER (through Fr. ordre, for earlier ordene, from Lat. ordo, ordinis, rank, service, arrangement; the ultimate source is generally taken to be the root seen in Lat. oriri, rise, arise, begin; cf. " origin ")
• ORDER, HOLY

order, are continuous in such a region, the theorem leads to an expansion of the function in a multiple See also:

• POWER [WILLIAM GRATTAN] TYRONE (1797-1841)

power See also:

• SERIES (a Latin word from serere, to join)

series . Such expansions are just as important in See also:

• ANALYSIS (Gr. avci and Meta, to break up into parts)

analysis, See also:

• GEOMETRY

geometry.and See also:

• MECHANICS

mechanics as expansions of functions of one variable .