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.(adsbygoogle = window.adsbygoogle || []).push({});

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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# Covariant derivative in polar coordinates

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