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I was reading Tensors by Feynman, where he said:

[...]$$u_P=\tfrac{1}{2}\sum_i\sum_j\alpha_{ij}E_iE_j\;.\tag{31.7}$$

The energy density $u_P$ is a number independent of the choice of axes, so it is a scalar. A tensor has then the property that when it is summed over one index (with a vector), it gives a new vector; and when it is summed over both indexes (with two vectors), it gives a scalar.

Got the point what Feynman wanted to say.

Then he discussed about the tensor of inertia and wrote the kinetic energy as:

$$\text{KE}=\tfrac{1}{2}\sum_{ij}I_{ij}\omega_i\omega_j.\tag{31.17}$$

Now, we know $\text{KE}$ is scalar; so it must be 'summed over both indexes (with two vectors)' as said by Feynman.

However, the notation of summation is not the same he used in $(31.7)$ as in $(31.17);$ in the former, the notation is $\sum_i\sum_j$ while in the later the notation is $\sum_{ij}\;.$ Since both are scalar and the summation is computed over two vectors in both the equations, can I infer from Feynman's statement 'when it is summed over both indexes (with two vectors), it gives a scalar' that $\sum_{ij}$ and $\sum_i\sum_j$ are necessarily the same?

Community
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    In the instance that Feynman uses yes they are the same, have you seen Einstein summation notation? – Triatticus May 18 '16 at 05:01
  • Whatsoever, the terms to be summed are all scalars, aren't they? – Vim May 18 '16 at 05:01
  • @Dan: Thanks for the comment; no for the second question. –  May 18 '16 at 05:02
  • I want to say that in general they are the same thing but it may or may not be abuse of notation, upvoting the question for a more mathematical answer – Triatticus May 18 '16 at 05:05
  • @Dan: Can you tell when they are different? What was the need of Feynman to use different notations? –  May 18 '16 at 05:06
  • Feynman as is typical of most physicists (myself included) love to abuse mathematical notation, I can't tell you his aims or implication beyond that he assumed they are the same. – Triatticus May 18 '16 at 05:08
  • Thanks, @Dan. However, I would appreciate if someone tells me when these two notations act differently. –  May 18 '16 at 05:09
  • This is why I reserved my information for a comment as I had no real mathematically based answer, only intuition – Triatticus May 18 '16 at 05:11

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The way I interpret your question:

If $a_{ij}$ are scalars indexed by some subset of $\Bbb N^2$, then are the two summation signs the same thing?

I'll assume you know some basics of sequence and series theory. (If not, then this answer is not helpful to you.)

In the absence of absolute convergence, $\sum_{ij}$ itself makes little sense because it's ambiguous as to the order in which to carry out the summation. And check out this famous theorem which says that if you have only conditional convergence, the order does matter. However, when we indeed have a.c., this problem is automatically resolved since the order doesn't matter anyway. So, if the author has explicitly written $\sum_{ij}$ in the first place, and if he's mathematically literate (which is of course true in your case), then he's already (maybe implicitly) assumed absolute convergence, so the summation order doesn't matter and the two signs are essentially saying the same thing.

Vim
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$\sum_{ij}$ is just a shorthand for $\sum_i\sum_j$. They both mean "sum over index $i$ and index $j$".

In some cases the order counts, in that case the notation with two sums is preferable.

Andrea
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    When the order counts, the "double sum" isn't even well-defined, I think – Vim May 18 '16 at 05:26
  • @Vim, this is true; however, this will not stop the physicists from writing the sum this way :) – zhoraster May 18 '16 at 05:28
  • @zhoraster I think it's because they assume God has created the world absolutely convergent. – Vim May 18 '16 at 05:30
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    Let's not forget that the indexing set is generally finite. In this case it literally is ${1,2,3}$. – Andrea May 18 '16 at 06:16
  • @Vim, you are right, but often physicists use the single sum notation informally as a shorthand anyways. – Andrea May 18 '16 at 06:17