Proper usage of Einstein sum notation

  • #1

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


[/B]
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!
 

Answers and Replies

  • #2
vela
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One error is that the same index shouldn't show up more than twice in a term, so ##P_i = P_{ij}\delta_{jj}## doesn't make sense because ##j## appears three times. It's not clear to me what you're trying to do there. What is ##P_i## supposed to be equal to in normal summation notation?
 
  • #3
One error is that the same index shouldn't show up more than twice in a term, so ##P_i = P_{ij}\delta_{jj}## doesn't make sense because ##j## appears three times. It's not clear to me what you're trying to do there. What is ##P_i## supposed to be equal to in normal summation notation?
In regular notation,

$$P_i = \sum_j P_{ij} $$

I suppose I could get the same result by multiplying by an array of ones with a single index, I just don't know if there's a conventional symbol for that.
 
  • #4
vela
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That's probably the most straightforward way. You can define a vector of ones, say ##e = (1, 1, \dots, 1)##, then ##P_i = P_{ij}e_j##.

You also need to clean up the notation for the derivative. The chain rule gives you (with no implied summation here)
$$\frac{d}{dx_n} P_{ij} = \frac{\partial P_{ij}}{\partial x_i}\frac{dx_i}{dx_n} + \frac{\partial P_{ij}}{\partial x_j}\frac{dx_j}{dx_n}.$$
 
  • #5
That's probably the most straightforward way. You can define a vector of ones, say ##e = (1, 1, \dots, 1)##, then ##P_i = P_{ij}e_j##.

You also need to clean up the notation for the derivative. The chain rule gives you (with no implied summation here)
$$\frac{d}{dx_n} P_{ij} = \frac{\partial P_{ij}}{\partial x_i}\frac{dx_i}{dx_n} + \frac{\partial P_{ij}}{\partial x_j}\frac{dx_j}{dx_n}.$$
Yes, right, I'll fix that. The minus sign came from me knowing it appears in the end but it's not correct there.

EDIT: apparently I can't edit the first post in the thread? Sorry for that.
 

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