= Local potential of a closed differential one-form
{title2=$d\alpha=0\Longrightarrow\alpha=d\Phi\text{ locally}$}
For a smooth <differential one-form> $\alpha=\alpha_i(x)dx^i$ with $d\alpha=0$, take a <manifold chart> identifying a neighborhood with a ball centered at zero, and set $\Phi(x)=\int_0^1x^i\alpha_i(sx)\,ds$. Closedness means $\partial_j\alpha_i=\partial_i\alpha_j$. Consequently
$$
\partial_j\Phi=\int_0^1\bigl[\alpha_j(sx)+s x^i\partial_j\alpha_i(sx)\bigr]ds=\int_0^1\frac d{ds}\bigl[s\alpha_j(sx)\bigr]ds=\alpha_j(x).
$$
Thus $d\Phi=\alpha$ in any dimension. This is the one-form case of the <Poincaré lemma>, and applies to spacetime as well as three-dimensional <vector calculus>. A global potential need not exist on a domain with nonzero periods.
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