Finding volumes from infinitesimal displacements

[Niles]

Homework Statement

In spherical polar coordinates, the infinitesimal displacement ds is given by:

$$ds^2 = dr^2 + r^2 d\theta ^2 + r^2 \sin \left( \theta \right)^2 d\phi ^2$$

Can I find the volume of a sphere using ds?

The Attempt at a Solution

I know the spherical volume-element is given by $$dV = r^2 \sin \left( \theta \right)drd\theta d\phi$$.

I can always make a figure showing the geometry of spherical coordinates, but is there a general way of finding the volume from ds? Perhaps finding the area, and then the volume?

 

[Niles]

No hints at all?

I thought of ds^3, but ds is a vector, so that won't work.