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I saw this exercise that we have to calculate the covariant derivative of a vector field (in polar coordinates). Most of them equals zero, but two of them are non-zero, sugesting that this vector field is not constant. What i want to understand is the physical meaning of this values.

The exercise itself is:

Consider in polar coordinates, the vector field with the following component:

A = (A^{ρ},A^{∅}) = (0,1)

Is this vector field constant? Compute the covariant derivative.

The Christoffel symbols gave me (assuming ζ as the connection symbol):

ζ^{σ}_{με}= 0

Except for:

ζ^{ρ}_{∅∅}=-ρ

ζ^{∅}_{ρ∅}=ζ^{∅}_{∅ρ}=1/ρ

The covariant derivate:

A^{σ};_{ε}= A^{σ},_{ε}in most cases

Except for:

A^{∅};_{∅}= A^{∅},_{∅}+ ζ^{∅}_{ρ∅}A_{∅}= 1/p

A^{∅};_{ρ}= A^{∅},_{p}+ ζ^{∅}_{∅ρ}A_{∅}= 1/p

The question is, what is the physical meaning of this values? What do they really represent?

And be soft please guys, i'm only starting :)

Thanks for the help

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# Whats the physical meaning of a covariant derivative?

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