Einstein summation convention confusion

In summary, the conversation discusses the use of the Einstein sum convention in classical mechanics and whether certain quantities, such as kinetic energy, can be expressed in summation notation. It is noted that the convention assumes the universal quantifier and cannot express the existential quantifier. There is also a mention of using footnotes or comments to clarify usage of the convention.
  • #1
dyn
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62
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
My understanding of the Einstein convention is that it would be xixi.
 
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  • #3
Thanks. My questions are just in reference to classical mechanics so in both questions i have asked all indices are lower indices
 
  • #4
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
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 😬
 
  • #6
dyn said:
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: $$\exists i: x_i = y_i$$
 

FAQ: Einstein summation convention confusion

1. What is the Einstein summation convention?

The Einstein summation convention is a mathematical notation used in tensor calculus to simplify the representation of equations involving summation over multiple indices. It was introduced by Albert Einstein in his theory of general relativity.

2. How does the Einstein summation convention work?

The convention states that whenever an index appears twice in a single term of an equation, it implies summation over all possible values of that index. This eliminates the need for explicit summation signs and allows for more concise and elegant notation.

3. What is the purpose of the Einstein summation convention?

The purpose of the convention is to simplify the representation of equations in tensor calculus, making them easier to read and understand. It also helps to reduce the number of terms in an equation, making it more efficient to work with.

4. Why is the Einstein summation convention sometimes confusing?

The convention can be confusing for those who are not familiar with it, as it requires a different way of thinking about and writing equations. It can also be confusing when dealing with more complex equations involving multiple indices and tensors.

5. Are there any limitations to the Einstein summation convention?

While the convention is a useful tool, it does have its limitations. It can only be applied to equations involving summation over indices, and it may not be suitable for all types of mathematical problems. Additionally, it may not be compatible with certain computer programs or software used for calculations.

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