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
| ∂ | = | ∂u | ∂ | + | ∂v | ∂ | . | ||
| ∂x | ∂x | ∂u | ∂x | ∂v |
When x, y are given functions of u, v, ... we have, instead of the above, such equations as
| ∂V | = | ∂V | ∂x | + | ∂V | ∂y | ; | ||
| ∂u | ∂x | ∂u | ∂y | ∂u |
and ∂V/∂x, ∂V/∂y can be found by solving these equations, provided the Jacobian ∂(x, y)/∂(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 ∂(x, y)/∂(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 “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, Leçons sur les coordonnées curvilignes (Paris, 1859), and to G. Darboux, Leçons sur les coordonnées curvilignes et systèmes orthogonaux (Paris, 1898).
When such a coordinate as u is connected with x and y by a functional relation of the form ƒ(x, y, u) = 0 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 ƒ(x, y, u) = 0 is intersected by a curve ƒ(x, y, u + Δu) = 0 tend to limiting positions as Δu 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 ∂(x, y)/∂(u, v) vanishes at all points on these envelopes. It is easy to see also that at any point where the reciprocal Jacobian ∂(u, v)/∂(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 ƒ(x, y, z) = 0, 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
| ∂z | = − | ∂ƒ | / | ∂ƒ | , | ∂z | = − | ∂ƒ | / | ∂ƒ | ; |
| ∂x | ∂x | ∂z | ∂y | ∂y | ∂z |
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.