Index notation is used for Tensors
\[\begin{array}{ccc}
& \begin{array}{c}
\text { Vector } \\
\text { Notation }
\end{array} & \begin{array}{c}
\text { Index } \\
\text { Notation }
\end{array} \\
\hline \text { scalar } & a & a \\
\text { vector } & \vec{a} & a_{i} \\
\text { tensor } & \underline{A} & A_{i j}
\end{array}\]
Definitions:
Scalar: magnitude with no direction
Vector: magnitude and direction
Tensor: A vector with direction (a moving vector)
What The Index Means
Each Index represents the respective xyz coordinate \[\mathbf{u}=u_{i}=\left\{\begin{array}{l}
u_{1} \\
u_{2} \\
u_{3}
\end{array}\right\}=\left\{\begin{array}{l}
u_{x} \\
u_{y} \\
u_{z}
\end{array}\right\}\]
Index Notation in Use
Index notation can be used to represents tensors of 3 dimensions such as the stress, strain, moment of inertia and curvature tensor or this stress tensor:
\[[\sigma]=\left[\begin{array}{lll}
\sigma_{11} & \sigma_{12} & \sigma_{13} \\
\sigma_{21} & \sigma_{22} & \sigma_{23} \\
\sigma_{31} & \sigma_{32} & \sigma_{33}
\end{array}\right]\]
\(\sigma_{23}\) would represent the stress on the 2 face and acting in the z direction. And because the tensor is symmetric \[\sigma_{i j}=\sigma_{j i}\] only 6 of the 9 components are independent.
Index Notation Convention
Instead of writing this: \[c_{i j}=\sum_{k=1}^{N} a_{i k} b_{k j}=a_{i 1} b_{1 j}+a_{i 2} b_{2 j}+\cdots+a_{i N} b_{N k}\] We would write it like this \[c_{i j}=a_{i k} b_{k j}\]