= Exterior derivative
{title2=$d$}
{wiki}
The exterior derivative is the unique graded derivation $d:\Omega^p(M)\to\Omega^{p+1}(M)$ that agrees with the differential on functions and satisfies $d^2=0$.
On a smooth planar <one-form> $\alpha=P\,dx+Q\,dy$, the <product rule> and antisymmetry of the <wedge product of differential forms> give
$$
d\alpha=\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dx\wedge dy.
$$
For $\alpha=x\,dy$, the coefficient is $1$: there is a nonzero integral around the boundary of every positively oriented rectangle. For $\beta=d(xy)$, the coefficient is $\partial_x x-\partial_y y=0$. More generally, the identity $d^2=0$ makes every <exact differential form> a <closed differential form>. It expresses cancellation between mixed <partial derivatives> locally; it does not imply that every <closed differential form> has a global potential.
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