Differential Forms, the Early Days; or the Stories of Deahna's Theorem and of Volterra's Theorem

This is a short informal history of the beginning of differential forms, up to the time of de Rham's work. It started with my being curious about how Poincare actually stated Poincare's Lemma. There were some surprises, mainly that Poincare's Lemma is due to another person, and the same for the well known Frobenius Theorem. I did look up everything that I quote. But of course I haven't looked up everything; when I say that something appeared first at such and such a place, I mean that I haven't come across an earlier reference. Let us recall briefly what differential forms are and do: They are generalizations of the well known expressions Pdx + Qdy and Adydz + Bdzdx + Cdxdy that function as integrands of line or surface integrals and represents things like work done moving along a curve in a force field or the flux of a vector field through a surface. Thus a differential form co, of degree r, defined in some open set in n-space R n (coordinates xl, x2, . ., xn), is an expression of the form E3 aili2.ir dxi1dxi2 . dXir, where the "coefficients" a. are (suitably differentiable) functions, and where the "differentials" dxi are symbols associated with the coordinates xi (they are actually 1-forms, with coefficient 1). The basic rule for multiplication of the dxi is skew-symmetry: dxidxj = -dxjdxi. In particular one has dxidxi = 0. (This makes the appearance of Jacobians automatic: e.g., if x and y are functions of u and v, then dxdy = (x d u + xvd v) (y d u + yvd v) = xuyududu + xuyvdudv + xvyudvdu + xvyvdvdv = 0 + xuyvdudv xvy,dudv + 0 _(x_y) dudv. The algebraic context is exterior algebra. ) In line with this the a. are usually taken skew-symmetric in their indices; for the above example Adydz + this would mean rewriting it as 1/2Adydz 1/2Adzdy + *. For a function f of the xi one defines the "differential" df as the 1-form E3 fx dxi. (Thus the dxi are the differentials of the functions xi.) For integration of an r-form over any oriented r-manifold in the domain of definition of co one represents the manifold locally parametrically by writing the xi as functions of r variables ua that are adapted to the orientation of the manifold. The differentials dxi now become 1-forms in the dua, and the form co reduces (by "multiplying out") to a single term A du 1 du2 . dUr, where A is a function of the u,. (This is the "restriction" of the form to the manifold.) One forms the usual integral of this over the appropriate region in ua-space and combines the local contributions (using a partition of unity) to get the integral of co over the whole manifold. (Parenthetically, there is a slightly different way of integrating something over a manifold, namely when the manifold carries some kind of distribution-matter or electricity or . -with a density p. On the manifold an area-element dA is given, and finds the total mass or charge as f p dA. This reduces of course again to local integrals, using local parametrizations. An orientation of the manifold is here usually not required.) The next thing is to extend the operation d (as in df) to all forms: One forms the "exterior derivative" dco by replacing each coefficient a. by its differential da. (and again "multiplying out"), resulting in an (r + 1)-form. It turns out, quite formally, that the operation dd (d applied twice) is always 0 (from fxy = fyx).