Einstein summation convention confusion

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  • #1
dyn
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Hi
If i have a vector r = ( x1 , x2 , x3) then i can write r2 as xixi where the i is summed over because it occurs twice. Now is xixi the same as xi2 ? I have come across an example where they are used as equivalent but i am confused because xi2 seems to be the square of just one component of r but xi2 also seems to be logically the same as xixi

My other question is ; are there some quantities that cannot be written in summation convention ? Such the kinetic energy of many particles . I have seen it written using sigma notation as the sum over k from 1 to N as mkvkvk but obviously k appears 3 times here. This applies to small oscillations where the rk is differentiated with respect to different variables . Are some quantities impossible to write in summation convention ?

Thanks
 

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  • #2
mathman
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My understanding of the Einstein convention is that it would be xixi.
 
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  • #3
dyn
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Thanks. My questions are just in reference to classical mechanics so in both questions i have asked all indices are lower indices
 
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Use whatever can be read unambiguously without confusing the reader too much. I wouldn't expect the Einstein sum convention in classical mechanics at all, so a footnote or other comment would be useful anyway. Specify how you want to use it there.
 
  • #5
Office_Shredder
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I think a lot of this depends on context too. If you wrote ##y_i=x_i^2## it's pretty clear you're not summing, and if you write ##y=x_i^2## then you are. Assuming the book doesn't have a typo 😬
 
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PeroK
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My other question is ; are there some quantities that cannot be written in summation convention ? Such the kinetic energy of many particles . I have seen it written using sigma notation as the sum over k from 1 to N as mkvkvk but obviously k appears 3 times here. This applies to small oscillations where the rk is differentiated with respect to different variables . Are some quantities impossible to write in summation convention ?
Because the convention assumes the universal quantifier, it can't express the existential quantifier. You can't say: $$\exists i: x_i = y_i$$
 

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