The Unapologetic Mathematician

The Algebra of Differential Forms

July 12, 2011 - Posted by | Differential Topology, Topology

1. […] just seen that smooth real-valued functions are differential forms with grade zero. We also know that functions pull back along smooth maps; if is a smooth function […]

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2. […] looks sort of familiar as a derivative, but we have another sort of derivative on the algebra of differential forms: the “exterior derivative”. But this one doesn’t really look like a derivative at […]

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3. […] really important thing about the exterior derivative is that it makes the algebra of differential forms into a “differential graded algebra”. We had the structure of a graded algebra before, […]

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4. […] forms” — top because is the highest degree possible for a differential form on a differential form — has rank over the algebra of smooth functions. That is, if is a top form then we can […]

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5. […] smooth maps, and homotopies form a 2-category, but it’s not the only 2-category around. The algebra of differential forms — together with the exterior derivative — gives us a chain complex. Since pullbacks of […]

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6. […] Armstrong: The algebra of differential forms, Pulling back forms, The Lie derivative on forms, The exterior derivative is a derivative, The […]