Differential operators: Exterior derivative vs Covariant derivative

Question

The classical differential operators in vector calculus can be expressed via differential forms: $$\mathrm{grad}f=\mathrm{d}f$$ $$\mathrm{div}X=\star\mathrm{d}\star (X^{\flat})$$ $$\mathrm{curl}X=(\star\mathrm{d}(X^{\flat}))^{\sharp}$$ But they can also be expressed via the covariant derivative: $$(\mathrm{grad}f)_i=\nabla_if$$ $$\mathrm{div}X= \nabla_iX^i $$ $$(\mathrm{curl}X)^i=\epsilon^{ijk}\nabla_jX_k$$ My question is now if the two expressions are equal when the connection is torsion-free or do we need stricter conditions?