In much of the material of related to physics, one finds it expedient to adapt the summation convention first introduced by Einstein. Let us consider first the set of linear equations
![]() |
(1) |
We shall find it to our advantage to set
,
,
. The superscripts do not denote powers
but are simply a means for distinguishing between the three quantities
,
, and
. One immediate advantage is obvious. If we were dealing with 29 variables, it would be foolish to use 29 different letters, one letter for each variable. The single letter
with a set of superscripts ranging from 1 to 29 would suffice to yield the 29 variables, written
,
,
,
,
. Our reason for using superscripts rather than subscripts will soon become evident. Equations (1) can now be written
![]() |
(2) |
Equations (2) still leave something to be desired, for if there were 29 such equations, our patience would be exhausted in trying to deal with the coefficients of
,
,
,
,
. Let us note that in (2) the coefficients of
,
,
may be expressed by the matrix
![]() |
(3) |
By defining
,
,
,
,
,
,
,
,
, the matrix (3) becomes
![]() |
(4) |
One advantage is immediately evident. The single element
lies in the ith row and jth column of the matrix (4). Equations (1) can now be written
![]() |
(5) |
Using the familiar summation notation of mathematics, we rewrite (5) as
![]() |
(6) |
or in even shorter form
![]() |
(7) |
The system of equations
![]() |
(8) |
represents
linear equations.
Einstein noticed that it was excessive to carry along the
sign in (8). we may rewrite (8) as
![]() |
(9) |
provided it is understood that whenever an index occurs exactly once both as a subscript and superscript a summation is indicated for this index over its full range of definition. In (9) the index
occurs both as a subscript (in
) and as a superscript (in
), so that we sum on
from
to
. In a four-dimensional spacetime
(
) summation indices range from 1 to 4. The index of summation is a dummy index since the final result is independent of the letter used. We can write
![]() |
(10) |
We may also write (9) as
![]() |
(11) |
where the element
belongs to the ith row and jth column of the matrix
![]() |
(12) |
[1] Lass, Harry. "Elements of pure and applied mathematics" New York: McGraw-Hill Companies, 1957.
This entry is a derivative of the Public domain work [1].
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