In the study of manifolds up to a certain point, all the definitions feel natural in the sense that there's no other way to define them. I'm thinking of concepts like tangent vectors, one-forms, pushforward and pullback, vector fields, etc. In what sense is the exterior derivative a natural, unique concept? It feels like I could have defined a symmetric derivative, or a derivative that doesn't symmetrize or antisymmetrize, or a derivative of contravariant tensors. I'm aware of the "usefulness" of exterior derivative, but it's unclear to me where the usefulness comes from, and why other definitions don't have the same usefulness. It seems like the usefulness comes out of nowhere.
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I believe it's the unique operator satisfying "desirable" properties like $d^2=0$ and the anti-product rule. – Randall May 12 '18 at 01:34
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2Stokes' theorem is the motivation, and a good part of its usefulness. Take a look at the section underlying principle to see how its definition comes naturally: A multi-dimensional version of a telescoping sum. – May 12 '18 at 01:47
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1In $\mathbb{R}^3$, grad, curl and divergence are dual to exterior derivatives through the inner product. One can take this backwards and define exterior derivatives in $\mathbb{R}^3$ uniquely as the duals of the grad, curl, and div operators. In the generalization to other Euclidean spaces and manifolds, one would then seek to retain the essential properties of the exterior derivative in the case of $\mathbb{R}^3$. – Gyu Eun Lee May 12 '18 at 06:31
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The word natural has a strict meaning: determined only by the structure without making additional choices.
The only structure that we have on a manifold is its smooth structure.
In this sense, the tangent bundle is a natural vector bundle, defined on any manifold.
Next, we would probably look at sections, but no natural section is available in vector bundles.
Instead, we could make a canonical choice sometimes. Say, we always can find the zero section of a vector bundle.
Sections of vector bundles can be mapped to each other, and, recalling calculus, we would probably look at the mappings with the property of being local, that is only defined around a neighborhood at each point.
Surprisingly (Peetre theorem), all such mappings turn out to be differential operators! (I apologize for being very sloppy and informal here. Please see the link to references below).
The best operators are, indeed, linear, so we would look for a natural linear operator. Another big theorem (see e.g. this question and links therein) tells us that the only such operator is the exterior derivative!
You can find references to restore the details in my answer here, and also please notice Andy Čap's important remark.
Yuri Vyatkin
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