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What you have to do is selecting a parametrization with respect to the arc length (s).

Thanks for your answer. The problem is that I have two points in spherical coordinates: $$P_{1}=(\frac{\pi}{4},0)$$ $$P_{2}=(\frac{\pi}{3},\frac{\pi}{2})$$ The great circle which passes through these two points is: $$\cot(\theta)=\frac{2}{\sqrt{3}}\cdot \sin(\phi+\frac{\pi}{3}-n \cdot...

I'm trying to evaluate the arc length between two points on a 2-sphere. The geodesic equation of a 2-sphere is: $$\cot(\theta)=\sqrt{\frac{1-K^2}{K^2}}\cdot \sin(\phi-\phi_{0})$$ According to this article: http://vixra.org/pdf/1404.0016v1.pdf the arc length parameterization of the...

Hi everyone! I'm trying to obtain the natural and dual basis of a circular paraboloid parametrized by: $$x = \sqrt U cos(V)$$ $$y = \sqrt U sen(V)$$ $$z = U$$ with the inverse relationship: $$V = \arctan \frac{y}{x}$$ $$U = z$$ The natural basis is: $$e_U = \frac{\partial \overrightarrow{r}}...

Thank you for your replies! My question now is related to the previous one, but has changed. Suppossing that we have the following situation: According to the picture, we have a vector in an orthogonal frame (with coordinates 2, 2). I want to obtain the contravariant (green) and covariant...

First of all, thanks for your answer, I really appreciate your quick response. So, if I have understood well, the first thing I have to do is to represent a vector and relate its components in both coordinate systems using trigonometry: But, what should I do now? I'm a little confused...

Hi everybody, This is my first post, so I apologise for all the possible mistakes that I can make now and in the future. I promise that I'll learn from them! My question is the following: It's well-known the relationship between two pair of cartesian axes when a circular rotation is made...