Are Differential Distances Along Curved Surfaces Larger Than Fixed Distances?

[aaaa202]

I recently did a problem with some electron constraint to move on a hoop. It kind of surprised me that you just could take the old Schrödinger-equation with and let your
dx ->dβ, where β is the distance along the hoop.
Saying it in a less mathematical way, isn't a differential distance along something curved larger than a differential distance in a fixed direction? I do realize that a rigorous mathematician would shoot me for saying something like this, so how would he say it?

 

[kevinferreira]

In a problem like this the best suited thing to do is to pass to polar coordinates, so you can describe a loop more simply. Then in these coordinates with origin at the center of the loop which is at a fixed radius r, you have $dx^2+dy^2=r^2d\theta^2=d\beta^2$. So it's simply the old good polar coordinates.

 

[voko]

Remember this? $\displaystyle \lim_{x \rightarrow 0 } \frac {\sin x} {x} = 1$.

It ensures that the length of a chord and the corresponding arc are about the same when they are small, let alone differential. It might be useful for you to follow the proof of the statement.