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R2(rt-s2)

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Originally appearing in Volume V14, Page 548 of the 1911 Encyclopedia Britannica.
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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 of variables was first considered by See also:Brook See also:Taylor in his Methodus incrementorum. In the See also:case See also:con-Change of sidered by Taylor y is expressed as a See also:function of z, and z variables. as a function of x, and it is desired to See also: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 coefficients enables us to See also:deal with more See also: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(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 of the See also: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 of transformations, has been studied exhaustively by Sophus See also:Lie (see, in particular, his Theorie der Transformationsgruppen, See also: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 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 of (x, y) in which the Jacobian a(x, y)/a(u, v) does not vanish or become See also: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 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 coordonnees curvilignes (See also: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 f(x, y, u) =o the curves u=const. are a See also:family of curves, and this family may be such that no two curves of the family have a See also:common point. When this is not the case the points in which a See also: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 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 See also:mula, with a See also: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)'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 (1797); the form written above is due to See also: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 how high the See also:order, are continuous in such a region, the theorem leads to an expansion of the function in a multiple See also:power See also:series. Such expansions are just as important in See also:analysis, See also:geometry.and See also:mechanics as expansions of functions of one variable.

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