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.