half-plane, we can write , but the angle function is neither smooth nor continuous over (as is any choice of angle function). Because has vanishing derivative, we say that it is closed.

On the other hand, for the one-form
:.Thus is not even closed, never mind exact.

The form generates the de Rham cohomology group meaning that any closed form is the sum of an exact form and a multiple of $d\theta$ : $\omega = df + k\ d\theta$ , where accounts for a non-trivial contour integral around the origin,

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