- #1
PhilDSP
- 643
- 15
I have a number of books which give a vector identity equation for the curl of a cross product thus:
[tex]\nabla \times \left(a \times b \right) = a \left( \nabla \cdot b \right) + \left( b \cdot \nabla \right) a - b \left( \nabla \cdot a \right) - \left( a \cdot \nabla \right) b[/tex]
But doesn't
[tex]b \left( \nabla \cdot a \right) = \left( a \cdot \nabla \right) b[/tex]
If that is true then
[tex]\nabla \times \left(a \times b \right) = 2a \left( b \cdot \nabla \right) - 2b \left( a \cdot \nabla \right)[/tex]
Or is there something I'm missing? (Since nabla is an operator the last equation as it's written might only make sense if it was multiplied by a vector)
[tex]\nabla \times \left(a \times b \right) = a \left( \nabla \cdot b \right) + \left( b \cdot \nabla \right) a - b \left( \nabla \cdot a \right) - \left( a \cdot \nabla \right) b[/tex]
But doesn't
[tex]b \left( \nabla \cdot a \right) = \left( a \cdot \nabla \right) b[/tex]
If that is true then
[tex]\nabla \times \left(a \times b \right) = 2a \left( b \cdot \nabla \right) - 2b \left( a \cdot \nabla \right)[/tex]
Or is there something I'm missing? (Since nabla is an operator the last equation as it's written might only make sense if it was multiplied by a vector)
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