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 Emeritus Sci Advisor PF Gold P: 9,355 One of the conditions on $\nabla$ mentioned in the definition of a connection is that $$\nabla_X(fY)=(Xf)Y+f\nabla_XY$$ Note that Xf and fX are not the same. Xf is the function $p\mapsto X_pf$ and fX is the vector field $p\mapsto(p,f(p)X_p)$. The effect of $\nabla_X$ on a function is defined as $$\nabla_Xf=Xf$$ Use this definition with $X=\partial_\mu$, and you get $$\nabla_{\partial_\mu}f=\partial_\mu f$$ so $$t(f)=t^\mu\partial_\mu f=t^\mu\nabla_{\partial_\mu}f$$ Note also that the definition allows us to rewrite the first equation as $$\nabla_X(fY)=(\nabla_Xf)Y+f\nabla_XY$$ The other two requirements in the definition of a connection are $$\nabla_X(Y+Z)=\nabla_XY+\nabla_XZ$$ $$\nabla_{fX+gY}Z=f\nabla_XZ+g\nabla_YZ$$