Later this week we’ll start talking about *exterior calculus*, which is a modern language used across differential geometry, mathematical physics, geometric computation… and the rest of our class! :-). Initially this language can look a bit daunting, but by making some connections with familiar ideas from vector calculus (like grad, div, and curl), we’ll see that it’s actually not so bad once you get down to concrete calculations. Slides here:

## 3 thoughts on “Slides—Exterior Calculus in \(R^n\)”

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For those who were curious about the statement that the “boundary of a boundary is empty,” the easiest way to state this fact is using the notions of a homeomorphism and a manifold, which we haven’t yet studied in class (but we will!). In particular, if \(\Omega \subset \mathbb{R}^n\) is an \(m\)-dimensional

submanifoldof \(\mathbb{R}^n\), a point \(p \in \Omega\) is on theinteriorif for a sufficiently small ball \(B_\epsilon(p)\) around \(p\), \(B_{\epsilon}(p) \cap S\) ishomeomorphicto the unit open \(m\)-dimensional ball; it is on theboundaryif this same intersection is homeomorphic to ahalf-ball, i.e., the intersection of the unit ball with a half-space.So the ‘boundary operation’ here is creating a sub-manifold in $\mathbb{R}^n$, instead of just extracting a sub-set in $\mathbb{R}^n$, which means the second boundary operation works on the sub-manifold instead of $\mathbb{R}^n$. Am I correct?

Also by $S$ you actually mean $\Omega$, right?

In class we said that we don’t make a distinction between 0-vectors and 0-forms (covectors). Is this because these objects are identical (at least up to isomorphism) no matter what the underlying vector space is? (whereas the set of 1-vectors might not be the same (isomorphic to) as 1-forms in more exotic contexts like function spaces.)