# Origin of exact and closed differential expressions

In differential geometry and other fields, an expression involving differentials can be closed or exact. In $$\mathbb R^2\setminus\{0\}$$ for example, $$dr$$ is exact whereas $$d\theta$$ is closed but not exact. What is the origin of these terms? Note that I deliberately avoided the use of the term "form" because the use of open and exact in reference to expressions involving differentials may have preceded the theory of differential forms.

From Bottazzini U., Gray J., Hidden Harmony-Geometric Fantasies. The rise of Complex Functions, Springer, 2013 (from which it seems that the terms 'exact' and 'complete' for differentials were used interchangeably):

In the opening pages of his (1739)$$^1$$, read to the French Académie on March 4, 1739, Clairaut stated that if $$Adx+Bdy$$ is the differential of any function of $$x$$, $$y$$ and constants, then $$\frac{\partial A}{\partial y}=\frac{\partial B}{\partial x}$$; and vice versa, if $$\frac{\partial A}{\partial y}=\frac{\partial B}{\partial x}$$ then $$Adx+Bdy$$ is a 'complete' differential, as he called it [...].

In a subsequent Mémoire Clairaut used both the terms 'complete' and 'exact' differential (1740).$$^2$$

[...] In his (1768)$$^3$$[...] d'Alembert [...] picked up the example mentioned by Clairaut [...]concluded that not only has $$Pdx+Qdy$$ to be a complete differential...

[...] mathematicians of the eighteenth century thought of the integral of a differential form such as $$Pdx+Qdy$$ as a function whose differential is the given form, whereas nowadays the expression $$\int{Pdx+Qdy}$$ is understood as a line integral. [...]

Formal manipulations of exact differential and complex functions similar to d'Alembert can also be found in papers by Euler and by Lagrange on fluid dynamics, as well as in Euler's papers on orthogonal trajectory and conformal mappings.[...] (pp. 84-87).

I found the use of exact differential also in Lacroix, An Elementary Treatise On Differential and Integral Calculus, Cambridge, 1816 (engl. transl. of Traité Élémentaire du Calcul Différentiel et du Calcul Intégral, 1802):

Although the equation $$(B)$$ should become identical by the substitution of values of $$P, Q, R$$, it does not follow, as a necessary consequence, that the equation $$Pdx+Qdy+Rdx=0$$

is a exact differential; but at least it may be rendered such by multiplication by some factor. (p. 412, emphasis mine).

$$^1$$ Clairaut A.C. , 1739, Recherches générales sur le calcul integral. Hist. Acad. Sci., Paris (1740), 425-436.

$$^2$$ Clairaut A.C. , 1740, Sur l'intégration ou la construction des équations differentielles du premier ordre. Hist. Acad. Sci., Paris (1742), 293-322.

$$^3$$ Alembert, J. le Rond d', 1768. Sur l'équilibre des fluides. Opusc. math. 5, 1-40

From Mathwords

Exact differential is found in 1824 in First Principles of the Differential and Integral Calculus, or the Doctrine of Fluxions, Intended as an Introduction to the Physico-Mathematical Sciences; Taken Chiefly from the Mathematics of Bézout, And translated from the French For the Use of the Students of the University at Cambridge, New England: "But when the numerator is the differential of the denominator, multiplied or divided by a constant number, the proposed differential must be decomposed into two factors, of which one shall be a fraction having for its numerator the exact differential of the denominator, and the other a constant number."

The Oxford English Dictionary (subscription required) has this from 1825:

As there are many differentials of two variables which are not exact differentials, so also there are many differential equations which are not the immediate differentials of any primitive equation.
D. Lardner, Elem. Treat. Differential & Integral Calculus ii. xvii. 284

• This looks interesting but is it possible that here "the exact differential of the denominator" means merely "precisely the differential of the denominator"? If so, then "exact" is being used in a generic sense rather than the technical sense (it could be that the generic sense is the origin of the technical sense). Also, I can't really follow the math. What is this 1824 textbook saying exactly? Nov 20 at 14:57
• the word "exact" is used in a technical sense in the 1824 book; paragraph 152 on page 149 says: These explanations being thus given, let $A d x + B dy$ be an exact differential, and $M$ its integral; we shall therefore have... Nov 20 at 16:49