Everybody knows that Leibniz and Newton (or Newton and Leibniz, if you wish) invented calculus, i.e. they developed the notion of differentiability for a function of one real variable. But who had for the first time the right idea about generalizing their notions to functions of several variables? I am talking in particular about total differentiability.



history of calculus of several variables
According to Ref. 1, see first screenshot below, Newton already had partial differentiation in his [*Method of Fluxions*](http://en.wikipedia.org/wiki/Method_of_Fluxions) (1671). (Whether or not he also introduced partial differential equations is a matter of debate.) 


According to Ref. 2, see second screenshot, the notion of an integrating factor goes back to Euler (1728), followed a decade later by Clairaut's more systematic treatment of total differentials.


1. Florian Cajori, [The Early History of
Partial Differential Equations and of Partial Differentiation and
Integration](http://www.jstor.org/stable/2298771?seq=1)
2. C. Edward Sandifer, [The Early Mathematics of Leonhard Euler](https://rads.stackoverflow.com/amzn/click/0883855593)




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**From Ref. 1, page 463:**


![](https://ilorentz.org/beenakker/MO/Newton_partial.png)


**From Ref. 2, page 22:**


![](https://ilorentz.org/beenakker/MO/Euler_partial.png)


