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- I am having trouble following Neuenschwander's logic in deriving the components of angular momentum ... as he proceed to derive the components of the inertia tensor

I am reading Tensor Calculus for Physics by Dwight E. Neuenschwander and am having difficulties in following his logic regarding proceeding to derive the components of Angular Momentum and from there the components of the Inertia Tensor ...

On page 36 we read the following:

In the above text by Neuenschwander we read the following:

" ... ... ## L = \int [ ( r \bullet r ) \omega - r ( r \bullet \omega ) ] dm ## ... ... ... (2.9)

Relative to an xyz coordinate system and noting that

## \omega^i = \sum_j \omega^j \delta^{ij} ## ... ... ... (2.10)

the ith component of L may be written

## L^i =\sum_j \omega^j \ \int [ \delta^{ij} ( r \bullet r ) - x^i x^j] dm ## ... ... ... (2.11) ... ... ... "

Can some please (... preferably in some detail) show me how Neuenschwander derives the expression for ## L^i ### (that is equation 2.11 from the equations above 2.11 ...

Help will be much appreciated ...

Now I think it will be helpful for readers of the above post to have access to the start of Section 2.2 so i am providing this ... as follows:

'

Hope that helps

Peter

On page 36 we read the following:

In the above text by Neuenschwander we read the following:

" ... ... ## L = \int [ ( r \bullet r ) \omega - r ( r \bullet \omega ) ] dm ## ... ... ... (2.9)

Relative to an xyz coordinate system and noting that

## \omega^i = \sum_j \omega^j \delta^{ij} ## ... ... ... (2.10)

the ith component of L may be written

## L^i =\sum_j \omega^j \ \int [ \delta^{ij} ( r \bullet r ) - x^i x^j] dm ## ... ... ... (2.11) ... ... ... "

Can some please (... preferably in some detail) show me how Neuenschwander derives the expression for ## L^i ### (that is equation 2.11 from the equations above 2.11 ...

Help will be much appreciated ...

Now I think it will be helpful for readers of the above post to have access to the start of Section 2.2 so i am providing this ... as follows:

'

Hope that helps

Peter

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