Engineering Mathematics Calculus Vector Calculus

Which of the following vector identities is / are always true?

1
∇ ⋅ (∇ × \(\vec{f}\)) = 0
2
∇ × (∇ × A̅) = 0
3
∇ × (∇f) = 0
4
\(\nabla \times (\log r\;\vec r) = 0\)

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