# Differential Lengths (Cylinder & Sphere)

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:

dlr= dr dl$\theta$ = r d$\theta$ dlz = 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:

dlR = 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.

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!

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