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Peregrine

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In my struggles to understand index notation, I am trying to figure out how my book came up with the following transformation.

[tex] \frac {D \omega}{Dt} \cdot \omega = \omega_j \partial_j v_i \cdot \omega + \nu \partial_j \partial_j \omega_i \cdot \omega [/tex]

=

[tex] \frac {D \frac{\omega^2}{2}}{Dt} = \omega_i \omega_j S_{ji} + \nu \partial_j \partial_j \frac {\omega^2}{2} - \nu \partial_j \omega_i \partial_j \omega_i[/tex]

This is how far I got.

changing [tex] \frac {D \omega}{Dt} \cdot \omega[/tex] to index notation yields

[tex] \omega_i \partial_o \omega_i + \omega_i v_j \partial_j \omega_i = \omega_i \omega_j \partial_j v_i + \omega_i \nu \partial_j \partial_j \omega_i [/tex]

Looking back to the book's solution, this appears to say [tex]\omega_i \omega_i = 1/2 \omega^2[/tex]? I thought it would be simple [tex]\omega^2[/tex]?

Then, it is pretty clear to me that [tex]\omega_i \omega_j \partial_j v_i = S_{ji}[/tex], so the term

[tex] \omega_i \omega_j \partial_j v_i = \omega_i \omega_j S_{ji} [/tex]

Finally, looking at [tex] \omega_i \nu \partial_j \partial_j \omega_i [/tex], I get, from the chain rule,

[tex] \nu \omega_i \omega_i \partial_j \partial_j + \nu \omega_i \partial_j \partial_j \omega_i [/tex]

which can be rewritten as

[tex] \nu \partial_j \partial_j \omega_i \omega_i + \nu \partial_j \omega_i \partial_j \omega_i [/tex]

So again, I run into [tex]\omega_i \omega_i = 1/2 \omega^2[/tex], which I don't understand.

First, have I don this correctly, and second, any ideas? Thanks much.

[tex] \frac {D \omega}{Dt} \cdot \omega = \omega_j \partial_j v_i \cdot \omega + \nu \partial_j \partial_j \omega_i \cdot \omega [/tex]

=

[tex] \frac {D \frac{\omega^2}{2}}{Dt} = \omega_i \omega_j S_{ji} + \nu \partial_j \partial_j \frac {\omega^2}{2} - \nu \partial_j \omega_i \partial_j \omega_i[/tex]

This is how far I got.

changing [tex] \frac {D \omega}{Dt} \cdot \omega[/tex] to index notation yields

[tex] \omega_i \partial_o \omega_i + \omega_i v_j \partial_j \omega_i = \omega_i \omega_j \partial_j v_i + \omega_i \nu \partial_j \partial_j \omega_i [/tex]

Looking back to the book's solution, this appears to say [tex]\omega_i \omega_i = 1/2 \omega^2[/tex]? I thought it would be simple [tex]\omega^2[/tex]?

Then, it is pretty clear to me that [tex]\omega_i \omega_j \partial_j v_i = S_{ji}[/tex], so the term

[tex] \omega_i \omega_j \partial_j v_i = \omega_i \omega_j S_{ji} [/tex]

Finally, looking at [tex] \omega_i \nu \partial_j \partial_j \omega_i [/tex], I get, from the chain rule,

[tex] \nu \omega_i \omega_i \partial_j \partial_j + \nu \omega_i \partial_j \partial_j \omega_i [/tex]

which can be rewritten as

[tex] \nu \partial_j \partial_j \omega_i \omega_i + \nu \partial_j \omega_i \partial_j \omega_i [/tex]

So again, I run into [tex]\omega_i \omega_i = 1/2 \omega^2[/tex], which I don't understand.

First, have I don this correctly, and second, any ideas? Thanks much.

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