Length of sinusoid on a sphere

[Vrbic]

Homework Statement

Originally the statement:
Find a length of two points on sphere. It was easy.
$\int \sqrt{g_{\phi\phi}}d\phi$
I hope you agree :-) But I have idea, how to find a length of path which is NOT a part of arc (circle). For example sinusoid. Is possible to find length of sinusoid on the sphere and how?

Homework Equations

$ds^2=g_{rr}dr^2+g_{\theta\theta}d\theta^2+g_{\phi\phi}d\phi^2$

The Attempt at a Solution

My attempt hit the snag very early :-)
A took $\theta=\pi/2+\sin{\phi}$
$d\theta=\cos{\phi}d\phi$
$ds^2=0+r^2d\theta^2+r^2\sin^2{\theta}d\phi^2$
$ds=r\sqrt{\cos^2{\phi}+\sin^2{(\pi/2+\sin{\phi})}}d\phi$
And now I don't know. I'm not sure if my procedure is so naive, and it exists better, or such problem doesn't have an analytical solution.
Please advice.

 

[Buffu]

I don't agree until you tell me what does those symbols mean.

What does length mean here ? arc length or something else ?

 

[Vrbic]

$r,\theta,\phi$ are spherical coordinates and $\theta=\pi/2$ is equator. $g_{ij}$ is metric tensor in these coordinates. By length I mean arc length (I hope it is same number when you take a ruler and measure sinusoid on a ball).

 

[Mark44]

Thread moved.
@Vrbic, please post questions involving integrals and tensors in the Calculus & Beyond section. These concepts are well beyond the Precalculus level.

 

[Buffu]

@Vrbic Do you really require calculus here ? I think this question is perfectly feasible without calculus.

 

[Vrbic]

@Vrbic Do you really require calculus here ? I think this question is perfectly feasible without calculus.

Yes, I would like calculations (maybe both:-) ). I believe it is a training for work in general relativity. No?

 

[Ray Vickson]

@Vrbic Do you really require calculus here ? I think this question is perfectly feasible without calculus.

Calculus is definitely needed. Even the problem of the length of a sinusoid in a plane involves the (non-elementary) elliptic function.

 

[Buffu]

Calculus is definitely needed. Even the problem of the length of a sinusoid in a plane involves the (non-elementary) elliptic function.

Not that problem. I was talking about,

Find a length of two points on sphere. It was easy.

 

[Vrbic]

Ok, elliptic function are needed for final solution, it seems not trivial. But how to get to them?
1) Is all right my procedure for finding length between two points on a sphere?
2) How to find length of path between two points connected by sinusoid (or sinusoid along all equator?)