- #1

faeriewhisper

- 5

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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