## The Algebra of Differential Forms

We’ve defined the exterior bundle over a manifold . Given any open we’ve also defined a -form over to be a section of this bundle: a function such that . We write for the collection of all such -forms over . It’s straightforward to see that this defines a sheaf on .

This isn’t just a sheaf of sets; it’s a sheaf of modules over the structure sheaf of smooth functions on . We define the necessary operations pointwise:

where the right hand sides are defined by the vector space structures on the respective .

We can go even further and define the sheaf of differential forms

This sheaf is not just a sheaf of modules over , it’s a sheaf of algebras. For an and a , we define their exterior product pointwise:

In fact, this is a graded algebra, and the multiplication has degree zero:

Even better, this is a unital algebra. We see this by considering the zero grade, since the unit must live in the zero grade. Indeed, , so sections of are simply functions on . That is, . Given a function we will just write instead of .

[…] 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 […]

Pingback by Pulling Back Forms « The Unapologetic Mathematician | July 13, 2011 |

[…] 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 […]

Pingback by The Exterior Derivative « The Unapologetic Mathematician | July 15, 2011 |

[…] 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, […]

Pingback by De Rham Cohomology « The Unapologetic Mathematician | July 20, 2011 |

[…] 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 […]

Pingback by Integration on the Standard Cube « The Unapologetic Mathematician | August 2, 2011 |

[…] 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 […]

Pingback by The Poincaré Lemma (setup) « The Unapologetic Mathematician | December 2, 2011 |

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

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