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## Main Question or Discussion Point

I calculated the christoffel symbols and know that I have them right. I want to take the covariant derivative of the basis vector field [tex]e_{r}[/tex] on the curve s(t) = (a, t/a). I differentiate it and get s' = (0, 1/a) and according to the metric, this is a unit vector because a will always be equal to r. Every way that I take the covariant derivative, [tex]\nabla_{s'(t)}e_{r}[/tex], yields [tex]\frac{1}{r^{2}}[/tex] for the [tex]e_{\theta}[/tex] component and 0 for the [tex]e_{r}[/tex] component.

If this is correct, then I don't know how I'm supposed to interpret it. the [tex]e_{r}[/tex] component being 0 makes perfect sense to me but based on the formula for the circumfrence of a circle, I was expecting [tex]\frac{1}{r}[/tex] for the [tex]e_{\theta}[/tex] component.

Am I making a mistake in my calculations somewhere? If i'm calculating the covariant derivative correctly, whats the geometric interpretation of the square in the denominator?

Thanks for any replies.

If this is correct, then I don't know how I'm supposed to interpret it. the [tex]e_{r}[/tex] component being 0 makes perfect sense to me but based on the formula for the circumfrence of a circle, I was expecting [tex]\frac{1}{r}[/tex] for the [tex]e_{\theta}[/tex] component.

Am I making a mistake in my calculations somewhere? If i'm calculating the covariant derivative correctly, whats the geometric interpretation of the square in the denominator?

Thanks for any replies.

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