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## Main Question or Discussion Point

In covariant derivative, we have a quantity called affinity.

The book says: When defining an affine connection, a coordinate frame must first be selected and the choice of the components of the affinity is then arbitrary within this frame. However, when these are all determined, the component of the affinity with respect to any other frame are completely fixed by the transformation law.

My question is why can the components be arbitrarily chosen?

Thank you very much!!

The book says: When defining an affine connection, a coordinate frame must first be selected and the choice of the components of the affinity is then arbitrary within this frame. However, when these are all determined, the component of the affinity with respect to any other frame are completely fixed by the transformation law.

My question is why can the components be arbitrarily chosen?

Thank you very much!!