| dy | = | dy | · | dz | , | ||||||||||
| dx | dz | dx | |||||||||||||
| d2y | = | dy | · | d2z | + | d2y | · ( | dz | )2, | ||||||
| dx2 | dz | dx2 | dz2 | dx | |||||||||||
| d3y | = | dy | · | d3z | + 3 | d2y | · | dz | · | d2z | + | d3y | · ( | dz | )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 = ƒ1(u, v), y = ƒ2(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 | = ( | ∂ƒ2 | + | ∂ƒ2 | dv | ) / ( | ∂ƒ1 | + | ∂ƒ1 | dv | ) | ||
| dx | ∂u | ∂v | du | ∂u | ∂v | 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
| ∂V | = | ∂V | ∂u | + | ∂V | ∂v | , | ||
| ∂x | ∂u | ∂x | ∂v | ∂x | |||||
| ∂V | = | ∂V | ∂u | + | ∂V | ∂v | , | ||
| ∂y | ∂u | ∂y | ∂v | ∂y |