The generalised Stokes theorem has the remarkably compact form

\[\boxed{ \int_Ω \dd ω = \int_{∂Ω} ω }\]

where \(Ω\) is a \(k\)-dimensional manifold with boundary \(∂Ω\), and \(ω\) is a \((k - 1)\)-form on \(Ω\).

While it may seem alien at first when expressed in full generality, you may recognise some of the many special cases of Stokes’ theorem, especially from vector calculus.

Special cases of Stokes’ theorem:

Adjust the parameters below or select a preset to see the associated Stokes theorem.

Ambient Space:				Metric signature:
Differential Form:				Hodge dual:

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.