Abstract
In previous lectures, we have shown how we form the derivatives and the differentials of functions of a single variable. We now add new developments to the study that we have made to this subject. Let x always be the independent variable and \( \varDelta x=\alpha h=\alpha dx \) an infinitely small increment attributed to this variable. If we denote by \( s, u, \) \( v, w, \dots \) several functions of x, and by \( \varDelta s, \varDelta u, \) \( \varDelta v, \varDelta w, \dots \) the simultaneous increments that they receive while we allow x to grow by \( \varDelta x, \) the differentials \( ds, du, \) \( dv, dw, \dots \) will be, according to their own definitions, respectively, equal to the limits of the ratios
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- 1.
Cauchy uses this phrase throughout his text. His entire function is what we would call a polynomial with nonnegative integer powers. His usage is not to be confused with the modern definition generally used in complex analysis.
- 2.
Our common Power Rule of differentiation.
- 3.
Meaning \(u, v, w, \dots \) are positive or negative.
- 4.
Recall Cauchy has already shown in the previous lecture that \(d(\mathbf l x) = \frac{dx}{x}\!.\)
- 5.
Cauchy has just presented a clever derivation of what is essentially the multiple variable version of our Product Rule for differentiation in differential form.
- 6.
The differential form of our Quotient Rule for differentiation follows simply.
- 7.
Cauchy did not begin to use the symbol i to denote his \(\sqrt{-1}\) until the 1850s. Although it had been in print earlier, Carl Friedrich Gauss (1777–1855) is generally credited with the widespread use of the modern notation for i in use today.
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Cates, D.M. (2019). THE DIFFERENTIAL OF THE SUM OF SEVERAL FUNCTIONS IS THE SUM OF THEIR DIFFERENTIALS. CONSEQUENCES OF THIS PRINCIPLE. DIFFERENTIALS OF IMAGINARY FUNCTIONS.. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_5
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DOI: https://doi.org/10.1007/978-3-030-11036-9_5
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