Why isn't line element differential form, while area element and volume element are?

Question

Area element and volume element are differential form, but why isn't line element?

Answer 1

The area element is a differential only in $\mathbb{R}^2$. The line element is the differential $dx$ in $\mathbb{R}$, but it needs to be modified to curves in $\mathbb{R}^n$ as you mention. Similarly, the area element is also not a differential for curved surfaces in $\mathbb{R}^3$, say.

The problem you're encountering is precisely what kept the Greeks tied up with lengths of curves. Euclid has theorems about the area of a circle and volumes of cones and pyramids, but not about the perimeter of a circle! It's not that they didn't know what it was. This problem persisted until the Renaissance when Descartes famously suggested that curves cannot be rectified.