Similar considerations are necessary ill the general case of n variables. The first consider able contribution to this theory is due to Pfaff. For this reason such an equation is known as a Pfaftan equation, and the problem of its inte gration as Nap problem. The problem leads to a system of no more than n integral equa tions when the number of variables is 2n or 2n 1. If the equations are of higher than the first degree in the differentials, Lie spealcs of them as Monge equations. Many problems of differential geometry, especially in relation to the theory of complexes, are connected with Pfaffian and Monge equations.
Partial afferent:al Frequently functions of several vanables are defined by relations between those functions and their partial derivatives. Such equations are called partial differential equations. Foi the sake of simplicity we will confine ourselves to the case of a single unknown function, and for the most part to the case of two independent variables. As in the case of ordinary differential equations, it will be instructive to see first how such equa tions may arise as the result of elimination of arbitrary elements from equations which do not involve the derivatives. Let z be given as a function of x, y and of the two arbitrary con stants a, b by the equation (21) f y, z; a, b)=0.
as ds iLet q represent and respect vely, ax ay Then differentiation will give af af at. af Os ax ay Between the three equations (21) and (22) a and b may be eliminated. Let (23) F(p, q; x, y, 2)=0.
be the result of this elimination. It is the partial differential equation which corresponds to (21) ; (21) is called the comp/ete integral of (23).
But a and b in (21) may be functions of x, y and still the result of the elimination may be the same equation (23). In fact we find from (21), assuming that a and b are functions of x and y.
af af af aa af ab P az Ox aa ax ab ax af af aa kf ab + + =- ay Oa ahoy 0, which equations will reduce to (22), and there fore give rise to the same equation (23), if as , of ab = u, -r of as , of ab =v. Oa ax ab ax Oa ay ib ay Let the determinant of these equations be de noted by 4, so that aadb aa ab 4; ax ay ay ax then we may write, in place of (24), the equiva lent equations Of (24a) =0, A-- =O.
aa ab If 0, we must therefore have o o. as ab From these equations a and b may be obtained as functions of x and y; if these values are substituted in (21), a function a of x and y is obtained, independent of any arbitrary con stants, but still a solution of the partial differ ential equation (23). This solution is called a
singular integral of (23). It may or may not be a special case of the complete integral.
Equations (24a) are also satisfied if v 0, i.e., if (25) b= where denotes an arbitrary function of a, If we multiply the left members of (24) by dx and dy respectively, and add, we find af da db=--0, b whence, since dt11/(a)da, 3f af , (26) , as I. ta= v.
If we eliminate a and b from the equations (21), (25) and (26), we find a as a' function of x and y, the expression of which depends upon the arbitrary function 4'. Moreover this func tion a will again be a solution of (23). It is known as the general integral and involves an arbitrary function. It may be shown that every integral of such a partial differential equation belongs to one of these three classes.
Geometrical interpretation will again render the matter perfectly clear. Let x, y, z be co ordinates of a point in space; (21) will repre sent a two-parameter family of surfaces, or, as we may say, a family of surfaces. The equation of the plane tangent to one of these surfaces at a point (x, y, z) will be 1z=gfx) For a fixed value of x, y, z, (23) gives therefore an infinity of planes through that point (en veloping a cone) ; any integral surface of (21), which passes through that point must have one of these planes as its tangent plane. In other words, the differential equation deter mines a certain cone corresponding to every point of space, and with this point as vertex; an integral surface must be tangent at each of its points to the corresponding cone. Now let a complete solution of the equation be given, so that we know a family of surfaces each of which fulfils the requirements of the problem. If we put b= 4(a), where 0(a) is any function of a, we obtain a one-parameter family of sur faces included among the co' surfaces just mentioned. The envelope of this one-parameter family is given by the general integral. The singular integral is the envelope of all of the co 2 surfaces of the complete integrals, provided that such an envelope exists.