PlanetPhysics/Einstein Summation Notation

In much of the material of related to physics, one finds it expedient to adapt the summation convention first introduced by [[../AlbertEinstein/|Einstein]]. Let us consider first the set of linear equations

$\begin{array}{ccc}{a}_{1}x+b_{1}y+c_{1}z& =& d_1\\ a_{2}x+b_{2}y+c_{2}z& =& d_2\\ a_{3}x+b_{3}y+c_{3}z& =& d_3\end{array}$

We shall find it to our advantage to set $x=x^1$, $y=x^2$, $z=x^3$. The superscripts do not denote [[../Power/|powers]] but are simply a means for distinguishing between the three quantities $x$, $y$, and $z$. 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 $x$ with a set of superscripts ranging from 1 to 29 would suffice to yield the 29 variables, written $x^1$, $x^2$, $x^3$, $\dots$, $x^{29}$. Our reason for using superscripts rather than subscripts will soon become evident. Equations (1) can now be written

$\begin{array}{ccc}{a}_1{x}^1+b_1{x}^2+c_1{x}^3& =& d_1\\ a_2{x}^1+b_2{x}^2+c_2{x}^3& =& d_2\\ a_3{x}^1+b_3{x}^2+c_3{x}^3& =& d_3\end{array}$

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 $x^1$, $x^2$, $x^3$, $\dots$, $x^{29}$. Let us note that in (2) the coefficients of $x^1$, $x^2$, $x^3$ may be expressed by the [[../Matrix/|matrix]]

$\left(\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right)$

By defining $a_1=a_{11}$, $b_1=a_{12}$, $c_1=a_{13}$, $a_2=a_{21}$, $b_2=a_{22}$, $c_2=a_{2}3$, $a_3=a_{31}$, $b_3=a_{32}$, $c_3=a_{33}$, the matrix (3) becomes

$\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)$

One advantage is immediately evident. The single element $a_{ij}$ lies in the i th row and j th column of the matrix (4). Equations (1) can now be written

$\begin{array}{ccc}{a}_{11}{x}^1+a_{12}{x}^2+a_{13}{x}^3& =& d_1\\ a_{21}{x}^1+a_{22}{x}^2+a_{33}{x}^3& =& d_2\\ a_{31}{x}^1+a_{32}{x}^2+a_{33}{x}^3& =& d_3\end{array}$

Using the familiar summation notation of mathematics, we rewrite (5) as

$\sum _{r=1}^3{a}_{1r}{x}^r=d_1\sum _{r=1}^3{a}_{2r}{x}^r=d_2\sum _{r=1}^3{a}_{3r}{x}^r=d_3$

or in even shorter form

$\sum _{r=1}^3{a}_{ir}{x}^r=d_{i}i=1,2,3$

The [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] of equations

$\sum _{r=1}^3{a}_{ir}{x}^r=d_{i}i=1,2,3,\dots ,n$

represents $n$ linear equations.

Einstein noticed that it was excessive to carry along the $\sum$ sign in (8). we may rewrite (8) as

$a_{ir}{x}^r=d_{i}i=1,2,3,\dots ,n$

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 $r$ occurs both as a subscript (in $a_{ir}$) and as a superscript (in $x^r$), so that we sum on $r$ from $r=1$ to $r=n$. In a four-dimensional [[../SR/|spacetime]] ($x^1=x,x^2=y,x^3=z,x^4=ct$) 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

$a_{ir}{x}^r\equiv a_{ij}{x}^j\equiv a_{i\alpha }{x}^{\alpha }$

We may also write (9) as

$a_r^i{x}^r=d^{i}i=1,2,3,\dots ,n$

where the element $a_r^i$ belongs to the i th row and j th column of the matrix

$\left(\begin{array}{cccc}{a}_1^1& a_2^1& \dots & a_n^1\\ a_1^2& a_2^2& \dots & a_n^2\\ \dots & \dots & \dots & \dots \\ a_1^n& a_2^n& \dots & a_n^n\end{array}\right)$

References

[1] Lass, Harry. "Elements of pure and applied mathematics" New York: McGraw-Hill Companies, 1957.

This entry is a derivative of the Public [[../Bijective/|domain]] [[../Work/|work]] [1].