Abstract
Let x, y be two independent variables, f(x, y) a function of these two variables, and \(x_0, X\) two particular values of x. We will find, by setting \(\varDelta y=\alpha \, dy, \) and employing the notations adopted in the thirteenth lecture,
then, in dividing by \(\alpha \, dy, \) and letting \(\alpha \) converge toward the limit zero,
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Notes
- 1.
This equation will be referenced in Lecture Thirty-Five assuming \(f(x, \, y)\) is a well-behaved function.
- 2.
As differentiation and integration are each limit processes, Cauchy is swapping the order in which these limits are taken, a process which is not always allowed.
- 3.
The expression included here is that of the original 1823 edition. The 1899 reprint reverses the order of the two final differentials.
- 4.
The 1899 reprint contains another small change by once again reversing the order of the two ending differentials within the first integral. The expression included here is that of Cauchy’s original 1823 edition.
- 5.
The equations included here are those appearing in the original 1823 edition. The 1899 reprint reverses the order of dx and dy in each of the four integrals.
- 6.
Recall \(\varGamma (\mu ) = \int _{0}^{\infty }{z^{\mu -1}e^{-z}dz}.\)
- 7.
It is somewhat unusual to find this pronoun employed within a mathematics text. This is not the only time Cauchy has used the first person within his Calcul infinitésimal text, perhaps illustrating how personal this effort was for Cauchy at the time.
- 8.
The third expression included here is that of the 1899 reprint. The original 1823 edition contains errors which have been corrected. The original reads \(\frac{d\varphi (x, \, y, z)}{dx}=\frac{d\chi (x, \, y, \, z)}{dy}. \)
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Cates, D.M. (2019). DIFFERENTIATION AND INTEGRATION UNDER THE \(\int \) SIGN. INTEGRATION OF DIFFERENTIAL FORMULAS WHICH CONTAIN SEVERAL INDEPENDENT VARIABLES.. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_33
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DOI: https://doi.org/10.1007/978-3-030-11036-9_33
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