Sort each identity as Always True or Not Always True.

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$\nabla(fg) = f \nabla g$ + g $\nabla f$ (product rule)
∇ × ( $\nabla f$ ) = 0 (curl of gradient is zero)
$\nabla \times F$ = 0 for any F
$\nabla^{2}f$ = ∇ · ( $\nabla f$ )
∇ · ( $\nabla \times F$ ) = 0 (divergence of curl is zero)
$\nabla \cdot F$ = 0 for any F

Hints

Two of these identities follow from mixed partials commuting.
Check whether the composition even type-checks before judging it.

Show the answer

Always true: ∇ × ( $\nabla f$ ) = 0 (curl of gradient is zero), ∇ · ( $\nabla \times F$ ) = 0 (divergence of curl is zero), $\nabla(fg) = f \nabla g$ + g $\nabla f$ (product rule), $\nabla^{2}f$ = ∇ · ( $\nabla f$ )

Not always true: $\nabla \times F$ = 0 for any F, $\nabla \cdot F$ = 0 for any F

Why

The curl of a gradient and divergence of a curl are always zero by Clairaut's theorem.

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Sort each identity as Always True or Not Always True.

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