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The Visualization of Differential Forms (2021)

post talks about visualization of differential forms on differential manifolds. It is aimed at a reader who is in the process of learning about differential forms (or who already has), but who has not seen any visual intuition for what they are.

Ordinarily, there would be no reason to believe that there would be a way to simplify this computation, but the hypothesis that $Q$ splits the space in two implies a very important fact: any line which enters $Q$ must either exit it again or stop inside $Q$. With only a small set of rules, it is possible to do all kinds of manipulations and calculations in coordinates, which is part of what makes tensors so useful and applicable in engineering and physics. However, a bit of mental gymnastics should convince you of the following alternative definition: To evaluate $f^*\omega(w)$, pull the surfaces composing $\omega$ back into $M$ (taking their preimage in $f$), and count how many of those $w$ crosses.

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