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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?*