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In mathematics, we define $dx^i$ as linear functionals, when speaking of integration. However, in physics, we interpret $dx^i$ as very small quantities.

There is nothing inherently small about a basis functional (covector). So why can we treat them as such? Is there any bridge between the two forms of intuition?

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Here's how I think about it: $dx^i$ measures vectors, and the tangent vectors it's going to be measuring are the thing that's very small. More precisely: the integral of a one-form $\theta$ along a curve $\gamma$ is

$$ \int_\gamma \theta = \int \theta ( \gamma'(t) ) dt. $$

In the Riemann-integral style approximation, the curve becomes a sequence of small displacements, you measure each small displacement using $\theta$ and then add them all up. In the conceptual limit the displacements become the velocity vector of the curve.

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