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Secondly, say we have some other vector field ##U \in \mathfrak{X}(M): M \rightarrow TM##, then how do we define ##\nabla_{U} V##, which is a function ##\mathfrak{X}(M) \times \mathfrak{X}(M) \rightarrow \mathfrak{X}(M)##? Maybe something like$$\nabla(U,V) = \nabla(U^{\mu} e_{\mu}, V)\equiv \nabla_{U} V = \nabla_{U^{\mu} e_{\mu}} V = U^{\mu} \nabla_{e_{\mu}}V = U^{\mu} \partial_{\mu} V$$and I was wondering if this is correct? Also, if ##f## is some function and ##W## a vector field, then why does ##\nabla_{fW} = f \nabla_W##?

Thanks!