In my struggles to understand index notation, I am trying to figure out how my book came up with the following transformation.(adsbygoogle = window.adsbygoogle || []).push({});

[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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Index transformation

**Physics Forums | Science Articles, Homework Help, Discussion**