Apply the ordinary product rule to the components of the curl. For example, the first component is

[\nabla \times (ϕ F)]_{x} = \partial_{y} (ϕ F_{z}) - \partial_{z} (ϕ F_{y}) = ϕ (\partial_{y} F_{z} - \partial_{z} F_{y}) + ϕ_{y} F_{z} - ϕ_{z} F_{y} .

The last two terms are the first component of

\nabla ϕ \times F

. The other components follow cyclically, so

\nabla \times (ϕ F) = ϕ \nabla \times F + \nabla ϕ \times F = ϕ \nabla \times F - F \times \nabla ϕ .

The sign in the last form follows from antisymmetry of the cross product, not from a different curl convention. Continuous first derivatives suffice for this identity.

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