Smooth vector area

In an immersed surface

f : U \subseteq R^{2} \to R^{3}

you can compute the average normal vector on a region $Ω$ by means of

\bar{N} := \int_{Ω} N d A

But it turns out that

\bar{N} := \int_{Ω} N d A = \int_{Ω} \frac{1}{2} d f \land d f = \int_{\partial Ω} \frac{1}{2} f \land d f

That is to say, this average normal vector to the region $Ω$ only depends on the frontier of the region. This is relevant for the definition of the "discrete normal vector" in Discrete Differential Geometry.