Talk:Einstein notation

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[edit] ??

1. The text explaining the meaning of notations like "aμνbα" is rather confusing.

2. The examples given seem to illustrate Einstein notation, whereas the formulae in the discussion above them do not (I see no superscripts in them at all).

3. Also, please write for English clear and proper grammar and complete senten

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The text I copied from tensor said: both raised and lowered on the same side. When I did Lagrangian mechanics, it was twice on the same side, which allow dot and cross products to be written in this way -- Tarquin 20:13 Mar 13, 2003 (UTC)

In general, either are used but in several fields (i.e. spacetime geometry) the up/down is required and correspond to a 'tensor contraction'. -- looxix 20:36 Mar 13, 2003 (UTC)

Can we add this funny Einstein's comment stolen from Wolfram:

The convention was introduced by Einstein (1916, sec. 5), who later jested to a friend, "I have made a great discovery in mathematics; I have suppressed the summation sign every time that the summation must be made over an index which occurs twice..." (Kollros 1956; Pais 1982, p. 216).
-- looxix 00:14 Mar 14, 2003 (UTC)


Don't see why not -- Tarquin 11:40 Mar 14, 2003 (UTC)

http://mathworld.wolfram.com/EinsteinSummation.html

I will write the next section, but not until tomorrow at least. I may not be up to the last section, however. Both of these are briefly described in Appendix 3 of Wald's 1994 QFT in CS & BHT book. So you can look there if you want to write them! ^_^ -- Toby 07:16 Mar 18, 2003 (UTC)

IE6 does not show ⊗ (& otimes;) - Patrick 11:00 Mar 18, 2003 (UTC)

I try to explain in the surrounding context what the missing symbol would be when I do something like this. But there are alternatives short of going to full-blown texvc. Can you give some advice for your browser:

  1. V ⊗ W (relying on context to know what the box represents);
  2. V (x) W (long used by mathematicians in ASCII contexts like Usenet);
  3. V x W (relying on context to know what kind of multiplication is involved);
  4. V Image:ProduitTensoriel.png W (a less distorted picture than that produced by texvc).

-- Toby 00:05 Mar 20, 2003 (UTC)

The box is not clear, the same one appears for every symbol that can not be represented. #3 and #4 both are fine. - Patrick 01:06 Mar 20, 2003 (UTC)

[edit] some details

The basic rule is:

   v = vi ei.

In this expression, it is assumed that the term on the right side is to be summed as i goes from 1 to n, because the index i appears on both sides. In that case, the equation is indeed true.

I guess what is meant is "because the index i appears twice", or maybe "because the index i doesn't appear on both sides".


Here, the Levi-Civita symbol e (or ε) satisfies eijk is 1 if (i,j,k) is a positive permutation of (1,2,3), -1 if it's a negative permutation, and 0 if it's not a permutation of (1,2,3) at all.

Shouldn't it be odd and even permutations, rahter than positive and negative? (I don't know if the terms 'positive' and 'negative' are common use, but if it is the case then they should appear in the "permutations" page...)

I think this question from a year or two ago has been answered -- the main page uses a different phrasing when presenting the Levi-Civita symbol. However, just in cast: the 1 and -1 are not describing the permutation, they're describing the value of ε based on the indices of ε. Permutations are a relevant concept, but you don't need to think about permutations to use the definition. In other words, ε has values from the set {1, 0, -1} -- the system would not make sense if the 1 and -1 were replaced with odd and even. For example, how do you multiply by "odd"? RaulMiller 17:04, 3 October 2005 (UTC)

[edit] Make more simple?

I think this article could be made simpler. For example, it would be greatly enhanced with the Riemman summation symbol at least once in the article (preferably in the definition), and a strategic use of the word "implicit". (preferably in the definition) Kevin Baas | talk 22:11, 2004 Jul 31 (UTC)

I'm removing the entire "examples" section following the Formal Definitions section. Overall, the presentation was haphazard, and confused. For example, discussion of the Levi-Civita symbol \ \epsilon does not belong in a section titled Elementary vector algebra and matrix algebra, as it is neither a vector nor a matrix. (Column vectors and row vectors are a mechanism to cast vector operations as matrix operations and simplify the scope of the discussion, but if you're going to talk about a rank 3 tensor this is a complication not a simplification, if it could be said to be valid at all.) If we're going to have examples, they should be real, concrete examples, not formally broken attempts at formal presentation. As it is, the only people capable of following these "examples" would have more than enough information from the preceeding sections. However, numerical examples and/or functional examples might be appropriate, if I have time, perhaps I'll add some of those. RaulMiller 14:58, 3 October 2005 (UTC)

I've removed the following two paragraphs from the main page:

We have also used a superscript for the dual basis, which fits in with a convention requiring summed indices to appear once as a subscript and once as a superscript. In this case, if L is an element in V*, then:
L = Li ei.
If instead every index is required to be a subscript, then a different letter must be used for the dual basis, say di = ei.
The real purpose of the Einstein notation is for formulas and equations that make no mention of the chosen basis. For example, if L and v are as above, then
L(v) = Li vi,
and this is true for every basis.

This seems unclear. In particular, L is defined as both a vector and a scalar -- e is a rank 2 tensor, and v is a rank 1 tensor. RaulMiller 17:54, 3 October 2005 (UTC)

[edit] References

The article needs some solid references. I've added one already, namely Einstein's 1916 Annalen der Physik paper. I found this post http://www.lepp.cornell.edu/spr/2003-05/msg0051086.html which contains some useful info. about quotes from Abrahams Pais' biography of Einstein. Perhaps some of this material can be used. MP (talk) 21:20, 12 September 2006 (UTC)

I don't agree. There are no controversial claims in this article. A reference at the bottom to one or two handbooks should suffice. Classical geographer 14:27, 31 January 2007 (UTC)
Agree with Classical geographer --Berland 09:30, 2 April 2007 (UTC)

[edit] Vector identity proofs

Might it be a good idea to have some brief proofs of vector identities (calculus ones too) using this notation? After all, the Einstein summation convention is an excellent way to prove CAB-BAC rule and so on.. (Aihadley 19:06, 2 April 2007 (UTC))

I think this is a very good idea. On top of that, we should stress the point that the summation convention was motivated by the fact that you can only sum together a covariant with a contravariant element, that means that aibi does not make any sense in this context. Hence, this distinction gave rise to the summation convention in the first place, so it should be noted. When calculating in Euclidean space, one often disregards the fact, because gij = δij or wherever you like to put the indices. The article could be more specific and give an example for that.Tprosser 19:21, 14 May 2007 (UTC)

[edit] Scope

Although this article is associated to physics, but in fact it is talking about a notation which applicable to any linear space. So, for any multidimension linear space, this notation is useful. Any objection? Jackzhp 13:54, 2 June 2007 (UTC)

[edit] three items

if V=ai*bi*ci sum for all i, then how to write it in einstein notation? Jackzhp 13:54, 2 June 2007 (UTC)

[edit] Template

There are a number of pages that have a boilerplate note saying they use this convention. There should be a Template:Einstein notation. 155.212.242.34 21:40, 2 October 2007 (UTC)

Created: Template:Einstein_summation_convention

[edit] Examples

Suggest:

[edit] Verification of claim/weasel word problem

The article states in the vector representations section: "This point is frequently confused." Are there any citations about this and who confuses this point so often? --Hydraton31 (talk) 20:58, 7 May 2008 (UTC)