Differential Forms and Integration - Terence Tao. This is a masterful presentation on the topic and his explanations are as clear as any I’ve ever read in either book or paper. To why I’m hear, have any of you been incredibly satisfied after a paper illuminated a subject for you? If so, please post.

If you enjoyed that paper, you should check out *Calculus on Manifolds* by Spivak, and *Analysis on Manifolds* by Munkres. Probably my favourite part of 3rd year of my degree.
This is the article on differential forms in *The Princeton Companion to Mathematics*. A really good book to discover math concepts and disciplines as explained by the best minds in math.
I like Tao's exposition. It's very valuable to emphasize the importance of the signed measure in a world where measure theory is dominated by squashing the sign.

But if I'm completely honest there is rarely an introductory exposition to differential forms that I truly like. Perhaps Edwards' Advanced Calculus: A Differential Forms Approach comes closest though still too terse for my taste. One can also find some fun/snarky treatment in Arnold's ODE book. Part of the problem is that we usually study exterior algebra in a completely abstract, non-geometric setting, which hides the deep geometric reason why differential forms have this algebra/geometry. The benefits of keeping the sign around was already known to Grassmann, but it has not deeply permeated the mathematical collective conscience. (see Rota and Dieudonne for comments on this problem).

In my mind the most insightful and pedagogically helpful way to get there is compute volumes in a non-differential setting first using exterior algebra (multilinear "affine" geometry) and only after this is understood introduce the differential setting.

It's not unlike insisting that we introduce straight lines segment (aka vector) first before placing them in a tangent setting.

Then it becomes painfully obvious that the sign is simply a consequence of having the correct algebra of n-volumes (you can add and subtract n-volumes, and with the sign around they form an additive group (the thing that is lost if you work unsigned), plus allow the definition of the exterior product which in term constructs n-volumes). If anyone knows sources that do this (well) and sensibly modern please let me know! Certainly a noteworthy attempt is Burke's "Div, Grad & Curl are dead", which as best I know never was published.
Clearest intro to differential forms I've read. The physics department folks I've learnt from for some reason discuss forms in either a more abstract or a more hand-wavy way that always made it harder to appreciate their deeper significance in maths.
Excellent post, thank you for sharing.
Do note that this version is incomplete and is missing some of the most important material. If you want the full version you will need to get a copy of the *companion*.
How can I justify the definition of differential form on manifolds? Algebraically I'm ok with it but I'm not very geometrically motivated. I mean this definition:

> Given a differentiable manifold X, or even a generalized smooth space X for which this definition makes sense, a differential form on X is a section of the exterior algebra of the cotangent bundle over X.

I can understand what this is saying, but I can't understand why this is the case. Is there anything thoughtful about being a "section" here (maybe Tao's word can ring a bell, but I haven't found it yet)?

I'm ok with exterior algebrab of cotangent bundle.

* We care about cotangent spaces because they are dual to tangent spaces. A tangent space is spanned by differential operators (partial over partial x). A cotangent space, being dual to the corresponding tangent space, is spanned by dx, dy,... This amounts to integration.

* We need wedge products because we care about the order (signed), and wedge product gives rise to determinant, which means signed, oriented volume.

Correct me if I'm wrong but I think I get it right. But still, I find no way to reason being a section. Where should I set my mind to?
I really liked "A Geometric Approach to Differential Forms" by David Bachman
Thanks for sharing! I’ve been looking for something on this topic!
I think I finally understand differential forms, thank you for sharing this!

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