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I'm currently trying to practice finding the inertia tensor for simple rigid bodies, with the inertia tensor elements given by: $$I_{ij}=\int_{V}^{}\rho(\delta_{ij}\sum_{k}^{}x_{k}^2-x_{i}x_{j})dv$$

I understand the basic ideas behind summations, Einstein notation, the Kronecker delta function, etc., but for some reason the combination of the delta symbol with the summation over an arbitrary variable $k$ is confusing me.

How would this integral play out for, say, $I_{11}$? I would greatly appreciate any explanation you all could offer.

OldWorldBlues
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    $I_{11} = \int_V \rho \left(\delta_{11}\sum_k x_k^2 - x_1^2\right)dv = \int_V \rho \sum_{k\neq 1}x_k^2 dv$ – John Barber Apr 16 '24 at 15:46
  • @JohnBarber Thanks! I think what's really confusing me is the variable 'k'. does that correspond to the physical component, or is it a placeholder variable? – OldWorldBlues Apr 16 '24 at 17:08
  • Since the $k$ is inside of a sum (over $k$), it is a placeholder variable. That is, you could replace $k$ with any other variable ($m$, say), as long as you don't replace it with any variable that exists outside the sum (i.e. $i$ or $j$). – John Barber Apr 16 '24 at 17:14

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