DIFFERENTIAL FORMS. The theory of differential forms is a branch of mathematics which presupposes several other branches, including differential calculus, algebra, and theory of functions. In essence, it is a theory of transformations of co ordinates, such as x = f (x', y', z') y=g(x',y',z') z=h(x', y', z') which are analytic and have single-valued inverses. This trans formation carries any set of values of the variables x, y, z into a definite set of values of x', y', z'. It also induces the transforma tion of differentials dx = of aJ of dz', ax az dx'+ ag dy'+ ag dz', (2) ax ay az dz= ah ady'+ dz'.
ax az The differentials, it should be understood, are merely a second set of independent variables associated with the first set x, y, z.
Now consider a function of these two sets of variables, for example, P(x, y, z)dx+Q(x, y, z)dy+R(x, y, z)dz, (3) in which P, Q, and R represent analytic functions of any sort of x, y, and z. This function has a definite numerical value whenever definite values are assigned to the variables x, y, z, dx, dy, dz. If we substitute for these variables according to the equations (I) and (2) we obtain a function P'(x', y', y', z')dy'+R'(x', y', z')dz', (4) which has the same value as (3) whenever x', y', z', dx', dy', dz' and x, y, z, dx, dy, dz are assigned values which are related by the equations (I) and (2). The two functions (3) and (4) are repre sentations or components in different co-ordinate systems of the same differential form. In this case, since the expressions (3) and (4) are both linear functions, the differential form is said to be linear.
The components of a differential form can also be quadratic in the differentials, for example, (5) in which E, F, and G are functions of two variables u and v. In general, the components of a differential form are required to be functions which are analytic in the variables x, y, etc., and analytic and homogeneous in the differentials dx, dy, etc. In the cases usually considered they are homogeneous polynomials in the differentials. They may also be functions of several sets of differentials. For example, the quadratic form (5) is intimately related to the bilinear form, E du bu+F(du bvd-dv ou)+G dv by.
It is obviously a fundamental problem to determine whether two differential expressions, such as (3) and (4) for example, are or are not components of the same differential form. This is known as the equivalence problem. The study of this problem, as well as of related problems, has led to the discovery of differ ential invariants of various kinds. The simplest of these are functions formed from the given form which are unchanged in value by transformations of coordinates. For example, an in variant of (3) is the bilinear form, (ap a.2 ac. aR — — (dx by — dy bx)+ (— — —) (dy bz —day) ay ax az ay aR ap +(— — —) ax az This example illustrates one of the uses to which differential invariants are put. For the vanishing of this bilinear form is a necessary and sufficient condition that (3) be a "complete differ ential." In other words, there exists a function F(x, y, z) such that if and only if this bilinear form vanishes. Other properties of the differential form (3) are expressed by the vanishing of other invariants. Indeed, the typical way of saying anything about a differential form is to assert that such and such an invariant vanishes—and a very large proportion of the theorems of geom etry and physics reduce to such statements.
Returning to the example of a linear differential form which has the components (3) and (4) in two coordinate systems we find, on carrying out the substitution of (I) and (2) in (3), that In these equations P stands for the function of x', y', z', obtained by substituting (I) in P(x, y, z) ; similarly Q and R. In the language of Tenson Analysis (q.v.) the equations (6) state that the coefficients of a linear differential form are the components of a covariant vector. In like manner we can work out the equa tions of transformation, analogous to (6), of the coefficients of a differential form of any degree. It comes out that whenever the differential form is a polynomial in the differentials, the co efficients are the components of a covariant tensor. The theory of these differential forms is therefore co-extensive with that of covariant tensors.
The theory even of linear differential forms is very extensive and has applications in a wide variety of fields of mathematics and physics. We need only mention line integrals, vector analysis (q.v.), and electricity and magnetism (qq.v.). The higher theory of linear differential forms and systems of linear differential forms is to be found in mathematical books usually under the heading -The problem of Pfaff," so called because the first investigations of the subject were made by Pfaff in 1814 and 1815. The further development of the subject is associated with the names of Gauss, Jacobi, Natani, Clebsch, Grassmann, Frobenius, Darboux and Cartan.
The theory of quadratic differential forms was initiated in 1827 by Gauss, who showed that the metric properties of surfaces de pend on forms of the type (5). This work of Gauss is also the foundation of modern differential geometry (q.v.). From the point of view of differential forms his chief contribution was the discovery of an invariant, called the curvature, which is a func tion of E, F, and G, and their first and second derivatives. which is unaltered by all analytic transformations of the variables u, v. The next important step was taken by Riemann, who in 1854 outlined the theory in its full generality and used it as the basis of what has come to be known as Riemannian geometry. He also showed that the curvature of Gauss must be replaced in the gen eral case by what is now called the curvature tensor. The work of Riemann was followed immediately by that of Christoffel and Lipschitz. The former introduced the functions often called Christoffel symbols or the components of affine connection, and gave a solution of the equivalence problem. Lipschitz developed the calculus of variations (q.v.) side of the subject and also the system of normal coordinates which had been sketched by Rie mann. This work was followed by a long series of researches by such mathematicians as Ricci, Voss, Lie, Levi-Chita, Zorawski, Wright and Haskins. Differential forms of degree higher than the second have been studied by Lipschitz, E. Noether, E. Pascal, and others.
The theory of quadratic differential forms has found many ap plications in geometry and physics, notably in dynamics. In recent years it has received a great deal of attention and been generalized in various directions because it is the foundation of Einstein's theory of relativity. In the researches of Einstein, as extended by Weyl and others, the phenomena of gravitation and electricity are described by means of a quadratic and a linear differential form restricted by the vanishing of certain invariants.
BIBLIOGRAPHY.-For the general theory of differential forms we may Bibliography.-For the general theory of differential forms we may refer the reader to: Weitzenbock, Invariantentheorie (1923) ; Pascal, Repertorium der Hoheren Mathematik, vol. i. (1927) ; and the Ency klopiidie der Mathematischen Wissenschaften, Band iii., Teil 3 (1927). For linear differential forms A. R. Forsyth, Theory of Differential Equations, part (189o) ; E. von Weber, Vorlesungen iiber das Pfaffsche Problem (i9oo) ; E. Goursat, Lecons sur le Probleme de Pfaff (1922). For quadratic differential forms: T. Levi-Civita, The Absolute Differential Calculus (1927) ; T. Y. Thomas and A. D. Michal, "Differential Invariants of Relative Quadratic Differential Forms," Annals of Mathematics (1927) ; L. P. Eisenhart, Riemannian Geometry (Princeton, '926) ; O. Veblen, Invariants of Quadratic Differential Forms (1927) ; and all the mathematical books on the Theory of Relativity. (O. V.)