Interpreting Ambiguous HW Question on Curve Length and Surface Patch Domains

[quasar987]

I got this HW question made up by the professor that I find ambiguous. It says

Consider the curve \theta (t)=\pi/2-t, \phi(t)=\log \cot(\pi/4-t/2) on the sphere r(\theta,\phi)=(\sin\theta \cos\phi,\sin\theta\sin\phi,cos\theta)

Find the length of the curve btw the points t=pi/6 and pi/4

He did not specify domains for either the curve nor the "surface" r. On one hand, if we take r to be a surface patch, this requires that the (maximum) domain be 0 < \theta < \pi, 0 < \phi < 2\pi. Anything bigger and the domain is not open or r is not injective. But this surface patch does not cover the whole sphere.

I could also consider two other surface patches of the form r_{2,3}(\theta,\phi)=(\sin\theta \cos\phi,\sin\theta\sin\phi,cos\theta) with appropriate domains, that together with r above form an atlas for the unit sphere.

Any thoughts? How would you interpret this question?

 

[StatusX]

I don't know what you mean. The domain is pi/6<t<pi/4. From the above relations you can get r as a function of t, and this is just some curve.

 

[quasar987]

There is another question after that:

Find the angles of intersection btw this curve and the parallels \theta = const.

would you still say that the curve's domain is (pi/6,pi/4)? Or would you study it more carefully to find what is the maximum domain where the curve is defined and thus find all the possible intersectino points?

 

[StatusX]

I would stick to (pi/6,pi/4), at least for this question. If you're curious, keep going, but then you're doing more than what's asked.