Abstract
Let \(u=f(x, y, z, \dots )\) always be a function of several independent variables \( x, y, z, \dots ; \) and, denote by
its first-order partial derivatives relative to x, to y, to z, \(\dots . \ \) If we make, as in the eighth lecture,
then, differentiate the two members of equation (1) with respect to the variable \(\alpha , \) we will find
If, in this last formula, we set \(\alpha =0, \) we will obtain the following
which is in accordance with equation (16) of the eighth lecture.
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Notes
- 1.
Both the original 1823 and the 1899 reprint texts share a typographical error in the last expression of this group of equations. The texts read
$$\begin{aligned} F^{(n)}(0)=\lim {\frac{F^{(n-1)}(\alpha )-F^{(n-1)}(0)}{\alpha }}=\lim {\frac{\Delta d^{n-1}u}{\alpha }}=d d^{n-1}=d^nu, \end{aligned}$$but this error has been corrected here for clarity.
- 2.
Cauchy investigates complex (or “imaginary” as he refers to them) expressions and functions at length in this and subsequent chapters within his 1821 Cours d’analyse.
- 3.
An error in equation (14) from the original 1823 edition is corrected in its ERRATA. The original has \( d_x^n u \) instead of \( v d_x^n u \) as its final term on the right.
- 4.
A typographical error in equation (15) occurs in both the original 1823 and the 1899 reprint editions. Both texts have \( \cdots + \frac{n}{1} dv d^{n-2}u + v d^nu \) as the final two terms in the equation. This error has been corrected here to avoid confusion.
- 5.
The equation Cauchy is referring to, equation (15), plays a prominent role in his subsequent Lecture Nineteen work.
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Cates, D.M. (2019). METHODS THAT WORK TO SIMPLIFY THE STUDY OF TOTAL DIFFERENTIALS FOR FUNCTIONS OF SEVERAL INDEPENDENT VARIABLES. SYMBOLIC VALUES OF THESE DIFFERENTIALS.. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_14
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