# Supplemental: Vector-Valued Differential Forms

This short-but-important supplemental lecture introduces some language we’ll need for describing geometry (curves, surfaces, etc.) in terms of differential forms. So far, we’ve said that a differential $k$-form produces a scalar measurement. But when talking about geometry, we often care about quantities that are vector-valued rather than scalar-valued. For instance, positions in $\mathbb{R}^n$, tangents, and normals are all vector-valued quantities. For the most part, all of our operations look pretty much the same as before. The one exception is the wedge product, which in $\mathbb{R}^3$ we now define in terms of the cross product.