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I know that this comment will be somewhat controversial, but I strongly believe that the standard (algebraic) textbook definition of d of a differential form is unpedagogical.

I much prefer the route taken in Arnold's GTM book on classical mechanics: Just define d of a form as the thing that makes stokes Stokes theorem true!

Then one derives the algebraic formula for d of a form. Everything is motivated at every step, and the student isn't confronted with a confusing algebraic definition of unknown origin.
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I know that this comment will be somewhat controversial, but I strongly believe that the standard (algebraic) textbook definition of d of a differential form is unpedagogical.

I much prefer the route taken in Arnold's GTM book on classical mechanics: Just define d of a form as the thing that makes stokes theorem true!

Then one derives the algebraic formula for d of a form. Everything is motivated at every step, and the student isn't confronted with a confusing algebraic definition of unknown origin.