Special cases of Stoke’s theorem:

Ambient Space, $V$				Metric signature:
Differential Form, $ω$				Hodge dual:

Adjust the parameters above to see the associated Stokes theorem. The metric signature is relevant to the Hodge dual operation (which requires the notion of a metric). Divergence-type theorems arise naturally when $ω$ is the Hodge dual of a $1$-form.