Differential Lengths (Cylinder & Sphere)

[Meadman23]

This isn't a homework problem, but I was wondering if anyone could explain two things to me.

1. When you have the differential lengths of a cylinder:dl_r= dr dl_\theta = r d\theta dl_z = dz
Why is dl_\theta equal to r d\theta and not just d\theta?2. When you have the differential lengths of a sphere:

dl_R = dR dl_\theta = R d\theta dl_\varphi = R sin\theta d\varphi

Why is dl_\theta equal to R d\theta and why is \varphi equal to R sin\theta d\varphi?I really want to be able to see rather than memorize what each of these differential lengths are equal to.

 

[stephenkeiths]

As I recall, the rdθ has to do with radians. When you're on a circle, and you move through an angle θ, you have traveled a distance rθ (provided θ is measured in radians!). If you move through an infinitesimal angle dθ you have gone a distance rdθ.

As for spherical, notice that θ is the angle from the z-axis (I think?) then the radius in the X-Y plane is Rsinθ, so by the same radian argument, an infinitesmal movement in the phi direction (angle from the x-axis) should be Rsinθdψ (I can't believe they don't have phi!).

And same goes for Rdθ in the θ direction (angle from the z-axis).

hope this helps!