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I'm a math guy, just wanting to clarify some notation. enter image description here

This is the Euler Lagrange equation associated with a charge in an electric field I found in a book. Where $\phi$ and $A$ are the scalar and vector potential related to Maxwell Equations. I am just curious, this last term, the one with the subscript $j$, is this last term suppose to be a sum over $j=1,2,3$ of that last quantity, i.e it will be three terms? I just wanted clarification for this, since I'm not sure why this sum of omitted if there should be a summation here. If there is not a summation here. Can someone tell me what $j$ is referencing.

I know i is referencing different components of the position vector.

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    $\begingroup$ Yes, it sums over $j = 1, 2, 3$. $\endgroup$
    – knzhou
    May 24, 2017 at 2:50
  • $\begingroup$ Thank you for the response. Just curious in physics is this omittance of the sum typical? Does the change in subscript over another variable just imply sum? $\endgroup$
    – Vogtster
    May 24, 2017 at 2:55
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    $\begingroup$ Yes, the convention is that any index repeated twice is summed. Also, if it's a regular letter (as opposed to Greek) it's over $1, 2, 3$ instead of $0, 1, 2, 3$. $\endgroup$
    – knzhou
    May 24, 2017 at 2:56
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    $\begingroup$ The sum is omitted because a lot of equations would become bulky and almost unreadable with all the sums included. Einstein notation is very neat and compact :) $\endgroup$
    – koldrakan
    May 24, 2017 at 7:46
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    $\begingroup$ Actually, beware, because in Special and General relativity, the implicit sum is only for "up-down" indices in tensors. For example, in $F_{\mu\nu}F^{\mu\nu}$, there is an implicit sum $\sum_{\mu=1}^4\sum_{\nu=1}^4$. But if you write $F_{\mu\nu}F_{\mu\nu}$, there is no implicit sum. $\endgroup$
    – user154997
    May 24, 2017 at 8:05

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I'd add a small comment, which is to give a name to the notation to possibly help in the future, but I don't have enough reputation yet.

Therefore I'm writing this as an answer: Yes, it's extremely powerful, and it's known as the Einstein Summation Convention. There are nice articles on Wikipedia and Wolfram's MathWorld on the topic.

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