Talk:PlanetPhysics/Einstein Summation Notation

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In much of the material of related to physics, one finds it expedient to adapt the summation convention first introduced by \htmladdnormallink{Einstein}{http://planetphysics.us/encyclopedia/AlbertEinstein.html}. Let us consider first the set of linear equations

\begin{equation} \begin{array}{ccc} a_1x + b_1y +c_1z & = & d_1 \\ a_2x + b_2y + c_2z & = & d_2 \\ a_3x + b_3y + c_3z & =& d_3 \end{array} \end{equation}

We shall find it to our advantage to set $x=x^1$, $y = x^2$, $z=x^3$. The superscripts do \textbf{not} denote \htmladdnormallink{powers}{http://planetphysics.us/encyclopedia/Power.html} 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{equation} \begin{array}{ccc} a_1x^1 + b_1x^2 +c_1x^3 & = & d_1 \\ a_2x^1 + b_2x^2 + c_2x^3 & = & d_2 \\ a_3x^1 + b_3x^2 + c_3x^3 & = & d_3 \end{array} \end{equation}

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 \htmladdnormallink{matrix}{http://planetphysics.us/encyclopedia/Matrix.html} \begin{equation} \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 ) \end{equation}

By defining $a_1 = a_{11}$, $b_1 = a_{12}$, $c_1 = a_{13}$, $a_2 = a_{21}$, $b_2 = a_{22}$, $c_2 = a_23$, $a_3 = a_{31}$, $b_3 = a_{32}$, $c_3 = a_{33}$, the matrix (3) becomes

\begin{equation} \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 ) \end{equation}

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

\begin{equation} \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} \end{equation}

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

\begin{equation} \sum_{r=1}^3 a_{1r} x^r = d_1 \qquad \sum_{r=1}^3 a_{2r} x^r = d_2 \qquad \sum_{r=1}^3 a_{3r} x^r = d_3 \end{equation}

or in even shorter form

\begin{equation} \sum_{r=1}^3 a_{ir} x^r = d_i \qquad i = 1,2,3 \end{equation}

The \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} of equations

\begin{equation} \sum_{r=1}^3 a_{ir} x^r = d_i \qquad i = 1,2,3,\dots,n \end{equation}

represents $n$ linear equations.

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

\begin{equation} a_{ir} x^r = d_i \qquad i = 1,2,3,\dots,n \end{equation}

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 \htmladdnormallink{spacetime}{http://planetphysics.us/encyclopedia/SR.html} ($x^1 = x, x^2 = y, x^3 = z, x^4 = ct$) summation indices range from 1 to 4. The index of summation is a \textbf{dummy index} since the final result is independent of the letter used. We can write

\begin{equation} a_{ir}x^r \equiv a_{ij}x^j \equiv a_{i\alpha}x^{\alpha} \end{equation}

We may also write (9) as

\begin{equation} a_r^i x^r = d^i \qquad i = 1,2,3,\dots,n \end{equation}

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

\begin{equation} \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 ) \end{equation}

\subsection{References}

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

This entry is a derivative of the Public \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} [1].

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