{"id":3242,"date":"2022-02-17T01:02:38","date_gmt":"2022-02-17T06:02:38","guid":{"rendered":"http:\/\/brickisland.net\/DDGSpring2022\/?p=3242"},"modified":"2022-01-17T16:13:29","modified_gmt":"2022-01-17T21:13:29","slug":"supplemental-vector-valued-differential-forms","status":"publish","type":"post","link":"https:\/\/brickisland.net\/DDGSpring2022\/2022\/02\/17\/supplemental-vector-valued-differential-forms\/","title":{"rendered":"Supplemental: Vector-Valued Differential Forms"},"content":{"rendered":"

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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<\/em> 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.<\/p>\n