- #1

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## Homework Statement

I'm dealing with some pretty complex derivatives of a kernel function; long story short, there's a lot of summations going on, so I'm trying to write it down using the Einstein notation, for shortness and hopefully reduction of errors (also for the sake of a paper in which I have to write all this stuff down and possibly do it without blowing past the page's margins). Right now I was testing something that's relatively simple, but I'm not sure I'm using this correctly.

## Homework Equations

My test example was a relatively simple derivative. For reference, these are the symbols I am using:

$$ P_{ij} = exp[-(x_i-x_j)^2]

\qquad

P_{ij}' = \frac{dP_{ij}}{dx_i} = -\frac{dP_{ij}}{dx_j}

\qquad

P_i = P_{ij}\delta_{jj}

\qquad

P_i' = P_{ij}'\delta_{jj}

$$

I'm already unsure about the use of ##\delta_{jj}## there, but then comes the problem. As a first exercise I'm trying an example of a derivative, with an additional index ##n##:

$$\frac{d(P_iP_i)}{dx_n}$$

## The Attempt at a Solution

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Here's my solution:

$$\frac{d(P_iP_i)}{dx_n} = 2P_i\frac{dP_i}{dx_n} = 2P_i\left[\frac{dP_i}{dx_i}\delta_{in}-\frac{dP_i}{dx_j}\delta_{jn}\right] = 2P_nP_n' - 2P_iP_{in}'

$$

Which actually works (tested numerically), but seems ugly and wrong to me due to those repeated ##n## indices which seem to imply a summation that isn't really there. Did I do something wrong? Is there some other symbol I'm disregarding or some rule I don't know? Thanks!