- #1
nsiderbam
- 2
- 0
Homework Statement
Use your knowledge of vector algebra to verify the following identity:
[tex]
\vec{\Omega} \cdot \nabla n = \nabla \cdot \vec{\Omega} n
[/tex]
Homework Equations
Divergence product rule
[tex]
\nabla \cdot (\vec{F} \phi) = \nabla (\phi) \cdot \vec{F} + \phi (\nabla \cdot \vec{F})
[/tex]
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.