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## Main Question or Discussion Point

[Throughout we're considering the intrinsic version of the covariant derivative. The extrinsic version isn't of any concern.]

I'm having trouble reconciling different versions of the properties to be satisfied by the covariant derivative. Essentially ##\nabla## sends ##(p,q)##-tensors to ##(p,q+1)##-tensors. I'll write down the required properties for ##\nabla## from the two sources.

Source 1: This lecture (relevant timestamp linked):

If ##X## is a vector field,

As for the last two properties from the two sources, I have no idea on how to relate them. Are these requirements incomplete for either of the sources?

I'm having trouble reconciling different versions of the properties to be satisfied by the covariant derivative. Essentially ##\nabla## sends ##(p,q)##-tensors to ##(p,q+1)##-tensors. I'll write down the required properties for ##\nabla## from the two sources.

Source 1: This lecture (relevant timestamp linked):

If ##X## is a vector field,

Source 2: Core principles of special and general relativity (Luscombe):1. ##\nabla_Xf=Xf##, for a scalar field ##f##

2. ##\nabla_X(T+S)=\nabla_XT+\nabla_XS##

3. ##\nabla_X(T(\omega,Y))=(\nabla_XT)(\omega,Y)+T(\nabla_X\omega,Y)+T(\omega,\nabla_XY)##

4. ##\nabla_{fX+Z}\ T=f\nabla_XT+\nabla_ZT##

At least the second property is consistent. The first property from the book is a more restrictive version of the first property from the lecture. In fact, ##\nabla_i## means ##\nabla_{\partial_i}## and ##\partial_i## isn't even a vector field!1. ##\nabla_if=\partial_if##

2. ##\nabla(aT+bS)=a\nabla T+b\nabla S## for real ##a,b##

3. ##\nabla(S\otimes T)=(\nabla S)\otimes T+S\otimes (\nabla T)##

4. ##\nabla## commutes with contractions, ##\nabla_i(T^j_{\ \ jk})=(\nabla T)^j_{\ \ ijk}##

As for the last two properties from the two sources, I have no idea on how to relate them. Are these requirements incomplete for either of the sources?

*If not, how can these two sets of requirements be shown to be equivalent?*