Kronecker Delta

The “Kronecker Delta”, $\delta_{ij}$ , is a tool that we’ll be using throughout this text. It is simply defined as follows:

(1) $\begin{equation*} \delta_{ij} = \left\{ \begin{array}{rl} 1 & \text{if } i = j,\\ 0 & \text{if } i \ne j,\\ \end{array} \right. \end{equation*}$

The usefulness of the Kronecker Delta lies in its ability to transform indices:

$\begin{equation*} \delta_{ik}a_k=\delta_{i1}a_1+\delta_{i2}a_2+\delta_{i3}a_3 \end{equation*}$

If $\: i=1,\delta_{ik}a_k=a_1$
If $\: i=2,\delta_{ik}a_k=a_2$
If $\: i=3,\delta_{ik}a_k=a_3$

(2) $\begin{equation*} \underbrace{\therefore \: \delta_{ik}a_k=a_i \: and \: \delta_{ik}A_{kj}=A_{ij}}_{\text{These, and similar ideas involving the Kronecker Delta, will be used extensively in later derivations}} \end{equation*}$

note: $\mathbf{e_i \cdot e_j}=\delta_{ij}$
In other words, the dot product of a unit vector along the “ $i$ ” axis and a unit vector along the “ $j$ ” axis is equal to zero, unless $i$ and $j$ are equal, in which case the dot product would be $1$ .

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