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latentcorpse
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The metric of Euclidean [itex]\mathbb{R}^3[/itex] in spherical coordinates is [itex]ds^2=dr^2+r^2(d \theta^2 + \sin^2{\theta} d \phi^2)[/itex].
I am asked to calculate the Christoffel components [itex]\Gamma^{\sigma}{}_{\mu \nu}[/itex] in this coordinate system.
i'm not too sure how to go about this.
it talks about [itex]ds^2[/itex] being the metric but normally the metric is of the form [itex]g_{ab}[/itex] i.e. a 2-form but ds^2 isn't a 2-form. are these metrics different or do i make [itex]g_{\mu \nu}=ds^2 \omega_{\mu} \omega_{\nu}[/itex] where [itex]\omega_i[/itex] is a 1 form?
i think I'm missing some key point here...
I am asked to calculate the Christoffel components [itex]\Gamma^{\sigma}{}_{\mu \nu}[/itex] in this coordinate system.
i'm not too sure how to go about this.
it talks about [itex]ds^2[/itex] being the metric but normally the metric is of the form [itex]g_{ab}[/itex] i.e. a 2-form but ds^2 isn't a 2-form. are these metrics different or do i make [itex]g_{\mu \nu}=ds^2 \omega_{\mu} \omega_{\nu}[/itex] where [itex]\omega_i[/itex] is a 1 form?
i think I'm missing some key point here...
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