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Who and when was it realized that the classical operators of vector calculus (grad, rot, div) can be expressed in a unified form using the exterior differential? I have searched a little bit on the web, and I can find some accounts of the history of vector calculus, and some accounts of the history of differential forms, but the first involves mainly physicists while the second involves mainly mathematicians, and I can find no hint of when the synthesis was made.

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    $\begingroup$ I recall that Kaehler invented the notation $d$ for exterior derivative; Cartan had used $\omega'$ instead of $d\omega$. Pfaff realized that a line subbundle of the cotangent bundle, locally written as the zero locus of a 1-form $\omega$, has an invariant, which we can think of as $d\omega$ restricted to the hyperplane $\omega=0$, but defined only up to rescaling. So roughly, Pfaff knew about $d\omega$ for $1$-forms, but not quite. I know Poincare played some role in the story, as did Cartan, but I don't know exactly what. $\endgroup$ – Ben McKay Nov 17 '16 at 9:33
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V.J. Katz in History of Topology:

Although Cartan realized in 1899 [1] that the three theorems of vector calculus (Gauss, Green, Stokes) could be easily stated using differential forms, it was Edouard Goursat (1858-1936) who in 1917 [2] first noted that these three theorems were all special cases of a generalised Stokes theorem for differential forms, $$\int_M d\omega=\int_{\partial M}\omega,$$ first stated in coordinate free form by Volterra in 1889 [3].

[1] E. Cartan, Sur certaines expressions différentielles et sur le problème de Pfaff, Ann. École Normale 16 (1899) 230-332.

[2] E. Goursat, Sur certaines systèmes d'équations aux différentielles totales et sur une généralisation du problème de Pfaff, Ann. Fac. Sci. Toulouse (3) 7 (1917), 1-58.

[3] V. Volterra, Sulle funzioni coniugate, Rendiconti Accademia dei Licei (4) 5 (1889), 599-611.

Victor Katz remarks elsewhere that the connection between differential forms and the big three theorems of vector calculus, as expressed by the generalized Stokes theorem, did not appear in textbooks until the second half of the 20th century, the first occurrence probably being in a 1959 Advanced Calculus text.

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  • $\begingroup$ Does anyone know if this history of the differential forms had anything to do with the complex version, where Abel and Riemann were integrating a holomorphic 1-form? Or is the "Abelian differential of the first/second/third kind" later creation? $\endgroup$ – liuyao Apr 18 '18 at 14:32
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You have the history backwards. Differential forms came first; and the general integrability theorem actually preceded differential forms, going back to Clairaut, 1739-1740.

For an equation of the form $A dx + B dy = dC$, Clairaut used Taylor expansions to prove the necessity of $∂A/∂y = ∂B/∂x$ and indefinite integrals to prove sufficiency. Cauchy used definite integrals (1823): $C(x,y) = \int_0^x A(X,y) dX + \int_0^y B(0,Y) dY + C(0,0)$ for sufficiency. It might be possible to use the 2nd order Taylor's Theorem with remainder to directly establish necessity, eliminating the need to use any infinite series expansion. This requires only $C^2$-ness, which is what is already required for the theorem.

The exterior algebra (that is: where differentials anti-commute with each other) is from Grassmann in the 1840's. Before Maxwell stripped down Hamilton's quaternions to a vector algebra and applied it in his treatise in the 1870's, he used differential forms in his papers in the 1850's and 1860's; and also made much more use of them in the treatise than he did vectors. But he only made sparing use of Grassmann's anti-commuting rule, just one place in the treatise, as far as I know.

Here: this will clear things up. This is what you were really asking and looking for. https://sites.math.washington.edu/~morrow/335_17/diff%20forms%20history%20katz.pdf

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