Index Transformation: Understanding Notation & Index Rules

[Peregrine]

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.

 

[coalquay404]

Can you perhaps explain your notation? For example, in your first equation you are using capital Ds for differentiation (I assume that there is some particular reason for this). So, if you can answer the following questions then somebody here should be able to help you:

1) What's the significance of the upper case D in the differentiation?
2) What types of objects are [itex]\omega[/itex], [itex]v[/itex], and [itex]S[/itex]?
3) What's your understanding of the summation rule? For instance, in the first equation you have a term in which three quantities all have the same index. Given that it seems that both [itex]\omega[/itex] and [itex]v[/itex] are either one-forms or vectors, this term doesn't make sense.

 

[Peregrine]

1. Capital D refers to the substantial derivative, in the notation of Stokes. It boils down to:

[tex] \frac {D()}{Dt} = \frac {\partial ()}{\partial t} + v_i \partial_i () [/tex]

Also, in the index notation of my book, [tex] \frac {\partial ()}{\partial t} = \partial_o [/tex]

2. w is a vector. v is a vector. And Sij is a 2nd order tensor.

3. You caught a typo in my writing. The term you speak of should have been [tex] \omega_j \partial_j v_i \cdot \omega [/tex]. I corrected this in the initial post.

Also, I do understand that the first equation I gave mixes index and symbolic notation, but that's how the book presented it, even though it seems to me to be bad form.

Thanks for taking a look at this. I understand this is probably a pretty simplistic problem for most here, but I am having a tough time understanding.

 

[Peregrine]

Okay, so looking at this backwards makes sense.

Since w is general, and thus can be f(x,y,z,t)

[tex] \frac {D \frac{\omega^2}{2}}{Dt}[/tex] differentiated by the chain rule gives:

[tex] \frac{2 \omega}{2} \frac{D \omega}{Dt}[/tex]

[tex] = \omega \cdot \frac{D \omega}{Dt}[/tex]

So, looking at it backwards makes it apparent that

[tex]\omega_i \frac{D}{Dt} \omega_i = \omega \cdot \frac {D \omega}{Dt}[/tex]

But I still don't see the transformation in index notation.