- #1
LoadedAnvils
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Hello everyone!
I'm trying to learn the derivation the covariant derivative for a covector, but I can't seem to find it.
I am trying to derive this:
[itex]\nabla_{α} V_{μ} = \partial_{α} V_{μ} - \Gamma^{β}_{αμ} V_{β}[/itex]
If this is a definition, I want to know why it works with the definition of the covariant derivative of a vector:
[itex]\nabla_{α} V^{μ} = \partial_{α} V^{μ} + \Gamma^{μ}_{αβ} V^{β}[/itex]
Why is the Christoffel symbol negative for a covector? Can anyone explain this to me?
Thank you all so much.
I'm trying to learn the derivation the covariant derivative for a covector, but I can't seem to find it.
I am trying to derive this:
[itex]\nabla_{α} V_{μ} = \partial_{α} V_{μ} - \Gamma^{β}_{αμ} V_{β}[/itex]
If this is a definition, I want to know why it works with the definition of the covariant derivative of a vector:
[itex]\nabla_{α} V^{μ} = \partial_{α} V^{μ} + \Gamma^{μ}_{αβ} V^{β}[/itex]
Why is the Christoffel symbol negative for a covector? Can anyone explain this to me?
Thank you all so much.