TITLE: is it possible to motivate higher differential forms without integration?
QUESTION [26 upvotes]: You have been teaching Dennis and Inez math using the Moore method for their entire lives, and you're currently deep into topology class.
You've convinced them that topological manifolds aren't quite the right object of study because you "can't do calculus on them," and you've managed to motivate the definition of a smooth manifold.
Dennis and Inez take your motivation about "doing calculus" seriously. They pose to you that they asked themselves how to take the derivative of a function $f: M \to \mathbb{R}$. They figured out independently that such a thing couldn't meaningfully be another smooth function. Instead, it has to say how the function is changing in the direction of a given vector field.
You are thrilled at their work: they've independently invented $d$ and the cotangent bundle, and they arrive at the correct formal definition of both with little prodding.
You want to point them in the direction of generalization, so you ask them to consider how to extend $d$ to higher exterior powers of the cotangent bundle. You get blank stares. Why should they think about higher tensor powers of the cotangent bundle at all? Why the alternating tensors, in particular? Yes, they know what that means (you led them to it in their linear algebra course), but the cotangent bundle just naturally showed up when they went thinking about derivatives, and the alternating tensors don't.
Oops! Now's the time to confess that your students never had a traditional 3-dimensional vector calculus course culminating in Stokes's theorem. Actually, it's worse: they have never heard of integrals. You see, you were torn about whether to start with the Lebesgue theory or spend time building the more intuitive Riemann integral, so you just skipped from the first half of Calc One to a topics course on higher topos theory. Lesson learned.
So, how do you teach them? (Alternatively, answer this question in the negative by proving that Dennis and Inez will always invent integration if you make them think about higher exterior derivatives enough.)

REPLY [2 votes]: Regarding the sentence:
"Alternatively, answer this question in the negative by proving that Dennis and Inez will always invent integration if you make them think about higher exterior derivatives enough."
The answer is "yes" (but "long enough" might take longer than their life expectancy). There are two ways to approach this:
a. Suppose they study smooth maps of $k$-manifolds into an $m$-manifold $M$ and come up with an idea of making a vector space out of this by working with chains (with real coefficients). Then they could ask "what is the dual vector space"? Under the extra analytical assumptions on the element of the dual space which I will not spell out here, the elements of the dual space are differential $k$-forms on $M$ (and vice versa). See
Whitney, Hassler, Geometric integration theory, Princeton Mathematical Series. Princeton, N. J.: Princeton University Press; London: Oxford University Press. XV, 387 p. (1957). ZBL0083.28204.
Theorem 10A, page 167, for details. Then they might discover the exterior differential as the dual of the boundary operator on chains (Stokes' Theorem).
b. Conversely, if they discover (somehow) differential forms, they might ask themselves "what is the dual space" of $\Omega^k(M)$?" The elements of the dual are know as currents. They probably will then try to find "concrete examples" of currents which will be, most likely, Lipschitz currents, given by integration over Lipschitz submanifolds.