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**HallsofIvy** The term is from "exact differential".

While any thing of the form f(x,y,z)dx+ g(x,y,z)dy+ h(x,y,z)dz **looks** like the differentiall of some F(x,y,z), it may not be. Given any function F(x,y,z), its differential is $\displaystyle dF(x,y,z)= \frac{\partial F}{\partial x}dx+ \frac{\partial F}{\partial y}dy+ \frac{\partial F}{\partial z}dz$. that comes from the chain rule: $\displaystyle \frac{dF(x,y,z)}{dt}= \frac{\partial F}{\partial x}\frac{dx}{dt}+ \frac{\partial F}{\partial y}\frac{dy}{dt}+ \frac{\partial F}{\partial z}\frac{dz}{dt}$ for any parameter, t.

In order that f(x,y)dx+ g(x,y)dy+ h(x,y)dz be a "real" (or "exact") differential, we must have and the same for the other mixed derivatives.

Back in Calculus III such a thing was called an "exact differential" (some texts use the term "conservative" in analogy with "conservative force fields" in physics but I dislike using physics terms in mathematics). In the theory of "differential forms" on manifolds, a differential form, $\displaystyle \omega$, of order n, is called "exact" if there exist some differential form [itex]\alpha[/itex], of order n-1, such that [itex]\omega= d\alpha[/itex].