Page:EB1911 - Volume 08.djvu/246

involves that there exists a functional relation connecting the three functions F, u, v, may be proved somewhat roughly as follows:—

The corresponding theorem is true for any number of variables. Consider first the case of two functions p, q, of two variables x, y. The function p, not being constant, must contain one of the variables, say x; we can then suppose x expressed in terms of y and the function p; thus the function q can be expressed in terms of y and the function p, say q = Q(p, y). This is clear enough in the simplest cases which arise, when the functions are rational. Hence we have

${\frac {\partial q}{\partial x}}{=}{\frac {\partial \mathrm {Q} }{\partial p}}{\frac {\partial p}{\partial x}}$ and ${\frac {\partial q}{\partial y}}{=}{\frac {\partial \mathrm {Q} }{\partial p}}{\frac {\partial p}{\partial y}}+{\frac {\partial \mathrm {Q} }{\partial y}};$

these give

${\frac {\partial p}{\partial x}}{\frac {\partial q}{\partial y}}-{\frac {\partial p}{\partial y}}{\frac {\partial q}{\partial x}}{=}{\frac {\partial p}{\partial x}}{\frac {\partial \mathrm {Q} }{\partial y}};$

by hypothesis ∂p/∂x is not identically zero; therefore if the Jacobian determinant of p and q in regard to x and y is zero identically, so is ∂Q/∂y, or Q does not contain y, so that q is expressible as a function of p only. Conversely, such an expression can be seen at once to make the Jacobian of p and q vanish identically.

Passing now to the case of three variables, suppose that the Jacobian determinant of the three functions F, u, v in regard to x, y, z is identically zero. We prove that if u, v are not themselves functionally connected, F is expressible as a function of u and v. Suppose first that the minors of the elements of ∂F/∂x, ∂F/∂y, ∂F/∂z in the determinant are all identically zero, namely the three determinants such as

${\frac {\partial u}{\partial y}}{\frac {\partial v}{\partial z}}-{\frac {\partial u}{\partial z}}{\frac {\partial v}{\partial y}};$

then by the case of two variables considered above there exist three functional relations. ψ₁(u, v, x) = 0, ψ₂(u, v, y) = 0, ψ₃(u, v, z) = 0, of which the first, for example, follows from the vanishing of

${\frac {\partial u}{\partial y}}{\frac {\partial v}{\partial z}}-{\frac {\partial u}{\partial z}}{\frac {\partial v}{\partial y}}.$

We cannot assume that x is absent from ψ₁, or y from ψ₂, or z from ψ₃; but conversely we cannot simultaneously have x entering in ψ₁, and y in ψ₂, and z in ψ₃, or else by elimination of u and v from the three equations ψ₁ = 0, ψ₂ = 0, ψ₃ = 0, we should find a necessary relation connecting the three independent quantities x, y, z; which is absurd. Thus when the three minors of ∂F/∂x, ∂F/∂y, ∂F/∂z in the Jacobian determinant are all zero, there exists a functional relation connecting u and v only. Suppose no such relation to exist; we can then suppose, for example, that

${\frac {\partial u}{\partial y}}{\frac {\partial v}{\partial z}}-{\frac {\partial u}{\partial z}}{\frac {\partial v}{\partial y}}$

is not zero. Then from the equations u(x, y, z) = u, v(x, y, z) = v we can express y and z in terms of u, v, and x (the attempt to do this could only fail by leading to a relation connecting u, v and x, and the existence of such a relation would involve that the determinant

${\frac {\partial u}{\partial y}}{\frac {\partial v}{\partial z}}-{\frac {\partial u}{\partial z}}{\frac {\partial v}{\partial y}}$

was zero), and so write F in the form F(x, y, z) = Φ(u, v, x). We then have

${\frac {\partial \mathrm {F} }{\partial x}}{=}{\frac {\partial \Phi }{\partial u}}{\frac {\partial u}{\partial x}}+{\frac {\partial \Phi }{\partial v}}{\frac {\partial v}{\partial x}}+{\frac {\partial \Phi }{\partial x}},{\frac {\partial \mathrm {F} }{\partial y}}{=}{\frac {\partial \Phi }{\partial u}}{\frac {\partial u}{\partial y}}+{\frac {\partial \Phi }{\partial v}}{\frac {\partial v}{\partial y}},{\frac {\partial \mathrm {F} }{\partial z}}{=}{\frac {\partial \Phi }{\partial u}}{\frac {\partial u}{\partial z}}+{\frac {\partial \Phi }{\partial v}}{\frac {\partial v}{\partial z}};$

thereby the Jacobian determinant of F, u, v is reduced to

${\frac {\partial \Phi }{\partial x}}\left({\frac {\partial u}{\partial y}}{\frac {\partial v}{\partial z}}-{\frac {\partial u}{\partial z}}{\frac {\partial v}{\partial y}}\right);$

by hypothesis the second factor of this does not vanish identically; hence ∂Φ/∂x = 0 identically, and Φ does not contain x; so that F is expressible in terms of u, v only; as was to be proved.

Part II.—General Theory.

Differential equations arise in the expression of the relations between quantities by the elimination of details, either unknown or regarded as unessential to the formulation of the relations in question. They give rise, therefore, to the two closely connected problems of determining what arrangement of details is consistent with them, and of developing, apart from these details, the general properties expressed by them. Very roughly, two methods of study can be distinguished, with the names Transformation-theories, Function-theories; the former is concerned with the reduction of the algebraical relations to the fewest and simplest forms, eventually with the hope of obtaining explicit expressions of the dependent variables in terms of the independent variables; the latter is concerned with the determination of the general descriptive relations among the quantities which are involved by the differential equations, with as little use of algebraical calculations as may be possible. Under the former heading we may, with the assumption of a few theorems belonging to the latter, arrange the theory of partial differential equations and Pfaff’s problem, with their geometrical interpretations, as at present developed, and the applications of Lie’s theory of transformation-groups to partial and to ordinary equations; under the latter, the study of linear differential equations in the manner initiated by Riemann, the applications of discontinuous groups, the theory of the singularities of integrals, and the study of potential equations with existence-theorems arising therefrom. In order to be clear we shall enter into some detail in regard to partial differential equations of the first order, both those which are linear in any number of variables and those not linear in two independent variables, and also in regard to the function-theory of linear differential equations of the second order. Space renders impossible anything further than the briefest account of many other matters; in particular, the theories of partial equations of higher than the first order, the function-theory of the singularities of ordinary equations not linear and the applications to differential geometry, are taken account of only in the bibliography. It is believed that on the whole the article will be more useful to the reader than if explanations of method had been further curtailed to include more facts.

When we speak of a function without qualification, it is to be understood that in the immediate neighbourhood of a particular set x₀, y₀, ... of values of the independent variables x, y, ... of the function, at whatever point of the range of values for x, y, ... under consideration x₀, y₀, ... may be chosen, the function can be expressed as a series of positive integral powers of the differences x − x₀, y − y₀, ..., convergent when these are sufficiently small (see Function: Functions of Complex Variables). Without this condition, which we express by saying that the function is developable about x₀, y₀, ..., many results provisionally stated in the transformation theories would be unmeaning or incorrect. If, then, we have a set of k functions, ƒ₁ ... ƒ_k of n independent variables x₁ ... x_n, we say that they are independent when n ≥ k and not every determinant of k rows and columns vanishes of the matrix of k rows and n columns whose r-th row has the constituents dƒ_r/dx₁, ... dƒ_r/dx_n; the justification being in the theorem, which we assume, that if the determinant involving, for instance, the first k columns be not zero for x₁ = xº₁ ... x_n = xº_n, and the functions be developable about this point, then from the equations ƒ₁ = c₁, ... ƒ_k = c_k we can express x₁, ... x_k by convergent power series in the differences x_k+1 − xº_k+1, ... x_n − x_nº, and so regard x₁, ... x_k as functions of the remaining variables. This we often express by saying that the equations ƒ₁ = c₁, ... ƒ_k = c_k can be solved for x₁, ... x_k. The explanation is given as a type of explanation often understood in what follows.

We may conveniently begin by stating the theorem: If each of the n functions φ₁, ... φ_n of the (n + 1) variables x₁, ... x_nt be developable Ordinary equations of the first order. about the values xº₁, ... x_n⁰t⁰, the n differential equations of the form dx₁/dt = φ₁(tx₁, ... x_n) are satisfied by convergent power series

x_r = xº_r + (t − t⁰) A_r1 + (t − t₀)² A_r2 + ...

reducing respectively to xº₁, ... xº_n when t = t⁰; and the only functions satisfying the equations and reducing respectively to xº₁, ... xº_n when t = t⁰, are those determined by continuation of these series. If the result of solving these n equations for xº₁, ... xº_n be written in the form ω₁(x₁, ... x_nt) = xº₁, ... ω_n(x₁, ... x_nt) = xº_n, Single homogeneous partial equation of the first order. it is at once evident that the differential equation

dƒ/dt + φ₁dƒ/dx₁ + ... + φ_ndƒ/dx_n = 0

possesses n integrals, namely, the functions ω₁, ... ω_n, which are developable about the values (xº₁ ... x_n⁰t⁰) and reduce respectively to x₁, ... x_n when t = t⁰. And in fact it has no other integrals so reducing. Thus this equation also possesses a unique integral reducing when t = t⁰ to an arbitrary function ψ(x₁, ... x_n), this integral being. ψ(ω₁, ... ω_n). Conversely the existence of these principal integrals ω₁, ... ω_n of the partial equation establishes the existence of the specified solutions of the ordinary equations dx_i/dt = φ_i. The following sketch of the proof of the existence of these principal integrals for the case n = 2 will show the character of more general investigations. Put x for x − x⁰, &c., and consider the equation a(xyt) dƒ/dx + b(xyt) dƒ/dy = dƒ/dt, wherein the functions a, b are developable about x = 0, y = 0, t = 0; say

a(xyt) = a₀ + ta₁ + t²a₂/2! + ..., b(xyt) = b₀ + tb₁ + t²b₂/2! + ...,

so that

ad/dx + bd/dy = δ₀ + tδ₁ + ½t²δ₂ + ...,

where δ = a_rd/dx + b_rd/dy. In order that

ƒ = p₀ + tp₁ + t²p₂/2! + ...