Vector Calculus proof

1. The problem statement, all variables and given/known data

Use your knowledge of vector algebra to verify the following identity:



[tex]

\vec{\Omega} \cdot \nabla n = \nabla \cdot \vec{\Omega} n



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2. Relevant equations



Divergence product rule

[tex]

\nabla \cdot (\vec{F} \phi) = \nabla (\phi) \cdot \vec{F} + \phi (\nabla \cdot \vec{F})

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3. The attempt at a solution



By the product rule,



[tex]

\nabla \cdot (\vec{\Omega} n) = \nabla n \cdot \vec{\Omega} + n (\nabla \cdot \vec{\Omega})

[/tex]



Therefore,



[tex]

\vec{\Omega} \cdot \nabla n = \nabla \cdot \vec{\Omega} n = \nabla \cdot (\vec{\Omega} n) = \nabla n \cdot \vec{\Omega} + n (\nabla \cdot \vec{\Omega})

[/tex]



and



[tex]

0 = n (\nabla \cdot \vec{\Omega})

[/tex]



I'm not quite sure what I'm doing wrong. Maybe it's a grouping thing. Any help would be appreciated.