Did Cauchy ever deal with double or triple integrals?

Question

Did Cauchy ever deal with double or triple integrals? Did he give rigorous proofs of multivariable integral calculus like what came to be called Stokes's theorem, the divergence theorem, etc.?

Answer 1

Yes, he did, multiple times.

Singular double integrals (1814)

In Mémoire sur les intégrales définies (1814) Cauchy studied why switching the order of integration in a double integral can sometimes lead to different results. This led him to introduce the notion of "singular integral". For $K = \frac{z}{x^2 + z^2}$ he showed that $\int_0^1\int_0^1\frac{\partial K}{\partial z}dzdx$ is $\pi/4$, but integrated in the reverse order it is $-\pi/4$. Good secondary sources are The entry of Cauchy: complex variables and differential equations, 1810–1822 by Grattan-Guinness and Gray's The Real and the Complex: A History of Analysis in the 19th Century:

"Mathematicians of the period, with their somewhat formal attitude to the processes of the calculus, not only tended to regard a double integral as a nested pair of single integrals but to be indifferent to the order of integration. This formal approach precluded interpreting double integrals in terms of line integrals along the boundaries of a rectangle in the plane of complex numbers; they were strictly about integrals on a rectangle in the real plane. However, it was known that in some cases this attitude was too naive and led to conflicting results, and it was this problem that Cauchy addressed in the second part of the Mémoire.

[...] Cauchy showed varying the order of integration of a double integral can yield two distinct definite values when the function becomes infinite or indeterminate for certain values of the variables within the domain of integration. To investigate the matter further, he introduced what he called singular integrals (“intégrales singulières”), which he defined (Cauchy 1814, 334, see Sect. 5.3) as limits as the end points of the integral tend to the same fixed value “without the integrals being zero.” Singular integrals are a profound idea. They were to crop up repeatedly in real analysis and, in an intriguing different way, in Cauchy’s later work on complex integrals."

Connection to repeated integration (1823)

Basic theory of double and repeated integrals, is laid out in Cauchy's Résumé des leçons données à l'École royale polytechnique sur le calcul (1823), a lecture course for students of École Polytechnique, lectures 34-35, see Landmark Writings in Western Mathematics by Grattan-Guinness. In particular, the Cauchy formula for repeated integration is derived there and is used to evaluate some elementary integrals.

Connection to the potential theory and complex integrals (1846)

Grattan-Guinness gives a panoramic view of the development of multiple integration in the context of the potential theory in Why did George Green Write His Essay of 1828 on Electricity and Magnetism? Lagrange (1762) was a pioneer, who formed volume integrals and integrated them by parts to form surface integrals. This was continued in Poisson's work on electrostatics (1812), and led to the divergence theorems such as Green's formula (1828).

For example, in Sur les intégrales qui s'étendent à tous les points d'une courbe fermée (1846) Cauchy gives a proof of the Cauchy integral theorem based on the application of the Green's formula (in modern notation): $$\int_C P dx + Q dy = \iint_R \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}dx\,dy.$$ On the earlier 1825 proof that did not use double integrals see What was the motivation for Cauchy's Integral Theorem? It is unclear if Cauchy was influenced by Green's 1828 work, see Cauchy Integral Theorem: a Histortical Development of its Proof by Scott:

"Whether Cauchy himself proved this result independently of George Green or actually utilized Green's work is not certain, for although Green's Theorem was published in a privately printed booklet in 1828, it was not published in a mathematical journal until 1850. So it is possible that Cauchy conceived of this result independently; however, there are indications that he was influenced by Green's work because he extended his integral theorem to areas on curved surfaces."

Volume integrals (1848)

Although generalities of the Résumé apply to volume integrals, Cauchy did not study them specifically. They are scarce even where one would expect to find them, in his extensive work on elasticity theory (1822ff) and the optics of elastic ether (1829ff). His best known contribution, Cauchy's tetrahedron argument, is typically presented in modern textbooks on continuous mechanics in terms of volume integrals and the divergence theorem, but Cauchy himself did not write any integrals. Everything is typically expressed in differential form, see e.g. Sur les équations qui expriment les conditions d'équilibre, ou les lois du mouvement intérieur d'un corps solide, élastique, ou non élastique (1828). A couple of triple integrals occur in passing in Mémoires sur les conditions relatives aux limites des corps et en particulier sur celles qui conduisent aux lois de la réflexion et de la réfraction de la lumiére (1848, p.265), but they are immediately reduced to a single integral using repeated integration in spherical coordinates. On Cauchy's elastic optics generally see Darrigol's History of Optics, ch.6.