Einstein summation convention confusion

[dyn]

Hi
If i have a vector r = ( x₁ , x₂ , x₃) then i can write r² as x_ix_i where the i is summed over because it occurs twice. Now is x_ix_i the same as x_i² ? I have come across an example where they are used as equivalent but i am confused because x_i² seems to be the square of just one component of r but x_i² also seems to be logically the same as x_ix_i

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 m_kv_kv_k but obviously k appears 3 times here. This applies to small oscillations where the r_k is differentiated with respect to different variables . Are some quantities impossible to write in summation convention ?

Thanks

 

[mathman]

My understanding of the Einstein convention is that it would be xⁱx_i.

 

[dyn]

Thanks. My questions are just in reference to classical mechanics so in both questions i have asked all indices are lower indices

 

[mfb]

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.

 

[Office_Shredder]

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

 

[PeroK]

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 m_kv_kv_k but obviously k appears 3 times here. This applies to small oscillations where the r_k 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: $$\exists i: x_i = y_i$$