Editing Einstein notation (section)

===Statement of convention===

According to this convention, when an index variable appears twice in a single [[Addend|term]] and is not otherwise defined (see [[Free and bound variables]]), it implies summation of that term over all the values of the index. So where the indices can range over the [[Set (mathematics)|set]] {{math|{1, 2, 3}<nowiki/>}},
<math display="block">y = \sum_{i = 1}^3 x^i e_i = x^1 e_1 + x^2 e_2 + x^3 e_3 </math>
is simplified by the convention to:
<math display="block">y = x^i e_i </math>

The upper indices are not [[Exponentiation|exponents]] but are indices of coordinates, [[coefficient]]s or [[basis vector]]s.  That is, in this context {{math|''x''<sup>2</sup>}} should be understood as the second component of {{math|''x''}} rather than the square of {{math|''x''}} (this can occasionally lead to ambiguity).  The upper index position in {{math|''x''<sup>''i''</sup>}} is because, typically, an index occurs once in an upper (superscript) and once in a lower (subscript) position in a term (see ''{{section link|#Application}}'' below).  Typically, {{math|(''x''<sup>1</sup> ''x''<sup>2</sup> ''x''<sup>3</sup>)}} would be equivalent to the traditional {{math|(''x'' ''y'' ''z'')}}.

In [[general relativity]], a common convention is that
* the [[Greek alphabet]] is used for space and time components, where indices take on values 0, 1, 2, or 3 (frequently used letters are {{math|''μ'', ''ν'', ...}}),
* the [[Latin alphabet]] is used for spatial components only, where indices take on values 1, 2, or 3 (frequently used letters are {{math|''i'', ''j'', ...}}),

In general, indices can range over any [[Indexed family|indexing set]], including an [[infinite set]]. This should not be confused with a typographically similar convention used to distinguish between [[tensor index notation]] and the closely related but distinct basis-independent [[abstract index notation]].

An index that is summed over is a ''summation index'', in this case "{{math|''i''&hairsp;}}". It is also called a [[bound variable|dummy index]] since any symbol can replace "{{math|''i''&hairsp;}}" without changing the meaning of the expression (provided that it does not collide with other index symbols in the same term).

An index that is not summed over is a [[free variable|''free index'']] and should appear only once per term.  If such an index does appear, it usually also appears in every other term in an equation. An example of a free index is the "{{math|''i''&hairsp;}}" in the equation <math>v_i = a_i b_j x^j</math>, which is equivalent to the equation <math display="inline">v_i = \sum_j(a_{i} b_{j} x^{j})</math>.