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Parallel Transport of a Vector

Solve the parallel transport equations for a vector v = (v^r, v^theta) being transported around a circle of radius a about r = 0 in the 2D space with the metric
ds^2 = dr^2 + (1 - beta)r^2 dtheta^2
where beta is a constant.
Use the parametrization of the circle with theta, 0 <= theta <= 2pi.

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