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Index Notation

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 \[c_{i j}=a_{i k} b_{k j}\]